Www.vocalcom.com HYPER CONNECTED Contact Center. HYPER CONNECTED Contact Center.
On hyper-arithmetic reflection principles
Transcript of On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
On hyper-arithmetic reflection principles
Joost J. Joosten
Universitat de Barcelona
Wednesday 30-09-2014Second International Wormshop, Mexico City
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
In Memoriam: Grisha Mints
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let T be some r.e. sound theory
I By Godel 2 we know that Con(T ) is independent of T
I So, we can add it and obtain an new sound theory
I We define the Turing(-Feferman) progression along a recursiveΓ of T as follows:
I
T0 := T ;Tα+1 := Tα + Con(Tα);Tλ :=
⋃α<λ Tα for limit λ < Γ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let T be some r.e. sound theory
I By Godel 2 we know that Con(T ) is independent of T
I So, we can add it and obtain an new sound theory
I We define the Turing(-Feferman) progression along a recursiveΓ of T as follows:
I
T0 := T ;Tα+1 := Tα + Con(Tα);Tλ :=
⋃α<λ Tα for limit λ < Γ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let T be some r.e. sound theory
I By Godel 2 we know that Con(T ) is independent of T
I So, we can add it and obtain an new sound theory
I We define the Turing(-Feferman) progression along a recursiveΓ of T as follows:
I
T0 := T ;Tα+1 := Tα + Con(Tα);Tλ :=
⋃α<λ Tα for limit λ < Γ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let T be some r.e. sound theory
I By Godel 2 we know that Con(T ) is independent of T
I So, we can add it and obtain an new sound theory
I We define the Turing(-Feferman) progression along a recursiveΓ of T as follows:
I
T0 := T ;Tα+1 := Tα + Con(Tα);Tλ :=
⋃α<λ Tα for limit λ < Γ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let T be some r.e. sound theory
I By Godel 2 we know that Con(T ) is independent of T
I So, we can add it and obtain an new sound theory
I We define the Turing(-Feferman) progression along a recursiveΓ of T as follows:
I
T0 := T ;Tα+1 := Tα + Con(Tα);Tλ :=
⋃α<λ Tα for limit λ < Γ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let T be some r.e. sound theory
I By Godel 2 we know that Con(T ) is independent of T
I So, we can add it and obtain an new sound theory
I We define the Turing(-Feferman) progression along a recursiveΓ of T as follows:
I
T0 := T ;
Tα+1 := Tα + Con(Tα);Tλ :=
⋃α<λ Tα for limit λ < Γ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let T be some r.e. sound theory
I By Godel 2 we know that Con(T ) is independent of T
I So, we can add it and obtain an new sound theory
I We define the Turing(-Feferman) progression along a recursiveΓ of T as follows:
I
T0 := T ;Tα+1 := Tα + Con(Tα);
Tλ :=⋃α<λ Tα for limit λ < Γ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let T be some r.e. sound theory
I By Godel 2 we know that Con(T ) is independent of T
I So, we can add it and obtain an new sound theory
I We define the Turing(-Feferman) progression along a recursiveΓ of T as follows:
I
T0 := T ;Tα+1 := Tα + Con(Tα);Tλ :=
⋃α<λ Tα for limit λ < Γ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I The obvious way of proving things about Turing progressionsis by transfinite induction.
I How can weak theories still prove interesting statementsabout Turing progressions?
I Schmerl (1978): reflexive transfinite induction
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I The obvious way of proving things about Turing progressionsis by transfinite induction.
I How can weak theories still prove interesting statementsabout Turing progressions?
I Schmerl (1978): reflexive transfinite induction
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I The obvious way of proving things about Turing progressionsis by transfinite induction.
I How can weak theories still prove interesting statementsabout Turing progressions?
I Schmerl (1978): reflexive transfinite induction
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Transfinite induction: ∀α (∀β<α φ(β)→ φ(α)) → ∀α φ(α);
I Theorem EA proves reflexive transfinite induction (Schmerl)
If EA ` ∀α(2EA ∀β<α φ(β) → φ(α)
), then
EA ` ∀α φ(α).
I Proof By Lob’s rule
I Clearly, if
T ` ∀α(2T ∀β<α φ(β) → φ(α)
),
then alsoT ` 2T ∀α φ(α) → ∀α φ(α),
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Transfinite induction: ∀α (∀β<α φ(β)→ φ(α)) → ∀α φ(α);
I Theorem EA proves reflexive transfinite induction (Schmerl)
If EA ` ∀α(2EA ∀β<α φ(β) → φ(α)
), then
EA ` ∀α φ(α).
I Proof By Lob’s rule
I Clearly, if
T ` ∀α(2T ∀β<α φ(β) → φ(α)
),
then alsoT ` 2T ∀α φ(α) → ∀α φ(α),
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Transfinite induction: ∀α (∀β<α φ(β)→ φ(α)) → ∀α φ(α);
I Theorem EA proves reflexive transfinite induction (Schmerl)
If EA ` ∀α(2EA ∀β<α φ(β) → φ(α)
), then
EA ` ∀α φ(α).
I Proof By Lob’s rule
I Clearly, if
T ` ∀α(2T ∀β<α φ(β) → φ(α)
),
then alsoT ` 2T ∀α φ(α) → ∀α φ(α),
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Transfinite induction: ∀α (∀β<α φ(β)→ φ(α)) → ∀α φ(α);
I Theorem EA proves reflexive transfinite induction (Schmerl)
If EA ` ∀α(2EA ∀β<α φ(β) → φ(α)
), then
EA ` ∀α φ(α).
I Proof By Lob’s rule
I Clearly, if
T ` ∀α(2T ∀β<α φ(β) → φ(α)
),
then alsoT ` 2T ∀α φ(α) → ∀α φ(α),
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Transfinite induction: ∀α (∀β<α φ(β)→ φ(α)) → ∀α φ(α);
I Theorem EA proves reflexive transfinite induction (Schmerl)
If EA ` ∀α(2EA ∀β<α φ(β) → φ(α)
), then
EA ` ∀α φ(α).
I Proof By Lob’s rule
I Clearly, if
T ` ∀α(2T ∀β<α φ(β) → φ(α)
),
then also
T ` 2T ∀α φ(α) → ∀α φ(α),
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Transfinite induction: ∀α (∀β<α φ(β)→ φ(α)) → ∀α φ(α);
I Theorem EA proves reflexive transfinite induction (Schmerl)
If EA ` ∀α(2EA ∀β<α φ(β) → φ(α)
), then
EA ` ∀α φ(α).
I Proof By Lob’s rule
I Clearly, if
T ` ∀α(2T ∀β<α φ(β) → φ(α)
),
then alsoT ` 2T ∀α φ(α) → ∀α φ(α),
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We can generalize Turing progressions to stronger notions ofconsistency.
I For n ∈ ω:
I We will denote “provable in T using all true Πn sentences” by[n]T
(of logical complexity
Σ0n+1
)
I The dual notion “consistent with T and all true Πn
sentences” is denoted 〈n〉T .
(
Π0n+1
)
I Then
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We can generalize Turing progressions to stronger notions ofconsistency.
I For n ∈ ω:
I We will denote “provable in T using all true Πn sentences” by[n]T
(of logical complexity
Σ0n+1
)
I The dual notion “consistent with T and all true Πn
sentences” is denoted 〈n〉T .
(
Π0n+1
)
I Then
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We can generalize Turing progressions to stronger notions ofconsistency.
I For n ∈ ω:
I We will denote “provable in T using all true Πn sentences” by[n]T
(of logical complexity
Σ0n+1
)
I The dual notion “consistent with T and all true Πn
sentences” is denoted 〈n〉T .
(
Π0n+1
)
I Then
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We can generalize Turing progressions to stronger notions ofconsistency.
I For n ∈ ω:
I We will denote “provable in T using all true Πn sentences” by[n]T (of logical complexity
Σ0n+1
)
I The dual notion “consistent with T and all true Πn
sentences” is denoted 〈n〉T .
(
Π0n+1
)
I Then
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We can generalize Turing progressions to stronger notions ofconsistency.
I For n ∈ ω:
I We will denote “provable in T using all true Πn sentences” by[n]T (of logical complexity Σ0
n+1)
I The dual notion “consistent with T and all true Πn
sentences” is denoted 〈n〉T .
(
Π0n+1
)
I Then
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We can generalize Turing progressions to stronger notions ofconsistency.
I For n ∈ ω:
I We will denote “provable in T using all true Πn sentences” by[n]T (of logical complexity Σ0
n+1)
I The dual notion “consistent with T and all true Πn
sentences” is denoted 〈n〉T .
(
Π0n+1
)I Then
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We can generalize Turing progressions to stronger notions ofconsistency.
I For n ∈ ω:
I We will denote “provable in T using all true Πn sentences” by[n]T (of logical complexity Σ0
n+1)
I The dual notion “consistent with T and all true Πn
sentences” is denoted 〈n〉T . (
Π0n+1
)
I Then
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We can generalize Turing progressions to stronger notions ofconsistency.
I For n ∈ ω:
I We will denote “provable in T using all true Πn sentences” by[n]T (of logical complexity Σ0
n+1)
I The dual notion “consistent with T and all true Πn
sentences” is denoted 〈n〉T . ( Π0n+1 )
I Then
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We can generalize Turing progressions to stronger notions ofconsistency.
I For n ∈ ω:
I We will denote “provable in T using all true Πn sentences” by[n]T (of logical complexity Σ0
n+1)
I The dual notion “consistent with T and all true Πn
sentences” is denoted 〈n〉T . ( Π0n+1 )
I Then
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We can generalize Turing progressions to stronger notions ofconsistency.
I For n ∈ ω:
I We will denote “provable in T using all true Πn sentences” by[n]T (of logical complexity Σ0
n+1)
I The dual notion “consistent with T and all true Πn
sentences” is denoted 〈n〉T . ( Π0n+1 )
I ThenI T i
0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We can generalize Turing progressions to stronger notions ofconsistency.
I For n ∈ ω:
I We will denote “provable in T using all true Πn sentences” by[n]T (of logical complexity Σ0
n+1)
I The dual notion “consistent with T and all true Πn
sentences” is denoted 〈n〉T . ( Π0n+1 )
I ThenI T i
0 := T ;I T i
α+1 := T iα ∪ {〈i〉T i
α>};
I Tλ :=⋃α<λ Tαfor limit λ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We can generalize Turing progressions to stronger notions ofconsistency.
I For n ∈ ω:
I We will denote “provable in T using all true Πn sentences” by[n]T (of logical complexity Σ0
n+1)
I The dual notion “consistent with T and all true Πn
sentences” is denoted 〈n〉T . ( Π0n+1 )
I ThenI T i
0 := T ;I T i
α+1 := T iα ∪ {〈i〉T i
α>};
I Tλ :=⋃α<λ Tαfor limit λ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Poly-modal provability logics turn out to be suitably wellequipped to talk about Turing progressions
I Already just the language with one modality [0] is expressive
I Godel II: 3T> → ¬2T3T>I Godel II: 2T (2T⊥ → ⊥)→ 2T⊥I For n ∈ N we see Tn ≡ T + 3n
T>I Transfinite progressions are not expressible in the modal
language with just one modal operator.
I However:
I Proposition: T + 〈1〉T> is a Π1 conservative extension ofT + {〈0〉kT> | k ∈ ω}.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Poly-modal provability logics turn out to be suitably wellequipped to talk about Turing progressions
I Already just the language with one modality [0] is expressive
I Godel II: 3T> → ¬2T3T>I Godel II: 2T (2T⊥ → ⊥)→ 2T⊥I For n ∈ N we see Tn ≡ T + 3n
T>I Transfinite progressions are not expressible in the modal
language with just one modal operator.
I However:
I Proposition: T + 〈1〉T> is a Π1 conservative extension ofT + {〈0〉kT> | k ∈ ω}.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Poly-modal provability logics turn out to be suitably wellequipped to talk about Turing progressions
I Already just the language with one modality [0] is expressive
I Godel II: 3T> → ¬2T3T>
I Godel II: 2T (2T⊥ → ⊥)→ 2T⊥I For n ∈ N we see Tn ≡ T + 3n
T>I Transfinite progressions are not expressible in the modal
language with just one modal operator.
I However:
I Proposition: T + 〈1〉T> is a Π1 conservative extension ofT + {〈0〉kT> | k ∈ ω}.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Poly-modal provability logics turn out to be suitably wellequipped to talk about Turing progressions
I Already just the language with one modality [0] is expressive
I Godel II: 3T> → ¬2T3T>I Godel II: 2T (2T⊥ → ⊥)→ 2T⊥
I For n ∈ N we see Tn ≡ T + 3nT>
I Transfinite progressions are not expressible in the modallanguage with just one modal operator.
I However:
I Proposition: T + 〈1〉T> is a Π1 conservative extension ofT + {〈0〉kT> | k ∈ ω}.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Poly-modal provability logics turn out to be suitably wellequipped to talk about Turing progressions
I Already just the language with one modality [0] is expressive
I Godel II: 3T> → ¬2T3T>I Godel II: 2T (2T⊥ → ⊥)→ 2T⊥I For n ∈ N we see Tn ≡ T + 3n
T>
I Transfinite progressions are not expressible in the modallanguage with just one modal operator.
I However:
I Proposition: T + 〈1〉T> is a Π1 conservative extension ofT + {〈0〉kT> | k ∈ ω}.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Poly-modal provability logics turn out to be suitably wellequipped to talk about Turing progressions
I Already just the language with one modality [0] is expressive
I Godel II: 3T> → ¬2T3T>I Godel II: 2T (2T⊥ → ⊥)→ 2T⊥I For n ∈ N we see Tn ≡ T + 3n
T>I Transfinite progressions are not expressible in the modal
language with just one modal operator.
I However:
I Proposition: T + 〈1〉T> is a Π1 conservative extension ofT + {〈0〉kT> | k ∈ ω}.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Poly-modal provability logics turn out to be suitably wellequipped to talk about Turing progressions
I Already just the language with one modality [0] is expressive
I Godel II: 3T> → ¬2T3T>I Godel II: 2T (2T⊥ → ⊥)→ 2T⊥I For n ∈ N we see Tn ≡ T + 3n
T>I Transfinite progressions are not expressible in the modal
language with just one modal operator.
I However:
I Proposition: T + 〈1〉T> is a Π1 conservative extension ofT + {〈0〉kT> | k ∈ ω}.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Poly-modal provability logics turn out to be suitably wellequipped to talk about Turing progressions
I Already just the language with one modality [0] is expressive
I Godel II: 3T> → ¬2T3T>I Godel II: 2T (2T⊥ → ⊥)→ 2T⊥I For n ∈ N we see Tn ≡ T + 3n
T>I Transfinite progressions are not expressible in the modal
language with just one modal operator.
I However:
I Proposition: T + 〈1〉T> is a Π1 conservative extension ofT + {〈0〉kT> | k ∈ ω}.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
DefinitionThe logic GLPΛ is the propositional normal modal logic that hasfor each ξ < Λ a modality [ξ] and is axiomatized by the followingschemata:
[ξ](A→ B)→ ([ξ]A→ [ξ]B)[ξ]([ξ]A→ A)→ [ξ]A〈ξ〉A→ [ζ]〈ξ〉A for ξ < ζ,[ξ]A→ [ζ]A for ξ < ζ.
The rules of inference are Modus Ponens and necessitation foreach modality: ψ
[ζ]ψ .
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
DefinitionThe logic GLPΛ is the propositional normal modal logic that hasfor each ξ < Λ a modality [ξ] and is axiomatized by the followingschemata:
[ξ](A→ B)→ ([ξ]A→ [ξ]B)[ξ]([ξ]A→ A)→ [ξ]A〈ξ〉A→ [ζ]〈ξ〉A for ξ < ζ,[ξ]A→ [ζ]A for ξ < ζ.
The rules of inference are Modus Ponens and necessitation foreach modality: ψ
[ζ]ψ .
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
DefinitionThe logic GLPΛ is the propositional normal modal logic that hasfor each ξ < Λ a modality [ξ] and is axiomatized by the followingschemata:
[ξ](A→ B)→ ([ξ]A→ [ξ]B)[ξ]([ξ]A→ A)→ [ξ]A〈ξ〉A→ [ζ]〈ξ〉A for ξ < ζ,[ξ]A→ [ζ]A for ξ < ζ.
The rules of inference are Modus Ponens and necessitation foreach modality: ψ
[ζ]ψ .
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I GLP0Λ denotes the closed fragment (no propositional variables)
I Iterated consistency statements in GLP0Λ are called worms
I 〈ξ0〉 . . . 〈ξn〉>I We write W for the class of all worms
I We write Wξ for the class of all worms all of whose modalitiesare at least ξ
I We can define natural orderings <ξ on W by
A <ξ B :⇔ GLP ` B → 〈ξ〉A
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I GLP0Λ denotes the closed fragment (no propositional variables)
I Iterated consistency statements in GLP0Λ are called worms
I 〈ξ0〉 . . . 〈ξn〉>I We write W for the class of all worms
I We write Wξ for the class of all worms all of whose modalitiesare at least ξ
I We can define natural orderings <ξ on W by
A <ξ B :⇔ GLP ` B → 〈ξ〉A
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I GLP0Λ denotes the closed fragment (no propositional variables)
I Iterated consistency statements in GLP0Λ are called worms
I 〈ξ0〉 . . . 〈ξn〉>
I We write W for the class of all worms
I We write Wξ for the class of all worms all of whose modalitiesare at least ξ
I We can define natural orderings <ξ on W by
A <ξ B :⇔ GLP ` B → 〈ξ〉A
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I GLP0Λ denotes the closed fragment (no propositional variables)
I Iterated consistency statements in GLP0Λ are called worms
I 〈ξ0〉 . . . 〈ξn〉>I We write W for the class of all worms
I We write Wξ for the class of all worms all of whose modalitiesare at least ξ
I We can define natural orderings <ξ on W by
A <ξ B :⇔ GLP ` B → 〈ξ〉A
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I GLP0Λ denotes the closed fragment (no propositional variables)
I Iterated consistency statements in GLP0Λ are called worms
I 〈ξ0〉 . . . 〈ξn〉>I We write W for the class of all worms
I We write Wξ for the class of all worms all of whose modalitiesare at least ξ
I We can define natural orderings <ξ on W by
A <ξ B :⇔ GLP ` B → 〈ξ〉A
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I GLP0Λ denotes the closed fragment (no propositional variables)
I Iterated consistency statements in GLP0Λ are called worms
I 〈ξ0〉 . . . 〈ξn〉>I We write W for the class of all worms
I We write Wξ for the class of all worms all of whose modalitiesare at least ξ
I We can define natural orderings <ξ on W by
A <ξ B :⇔ GLP ` B → 〈ξ〉A
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
A <ξ B :⇔ GLP ` B → 〈ξ〉A
I For <0 defines a well-order on the class of worms moduloprovable GLP equivalence.
(Beklemishev, Fernandez Duque, JjJ)
I For <ξ with ξ > 0 the relation is no longer linear (mod prov.equivalence) but is still well-founded
(infinite anti-chains)
I Definition By oα(A) we denote the order type of A under <α
and we write o(A) instead of o0(A).
I Worms of GLPω are known to be useful for Turingprogressions:
I Proposition (Beklemishev) For each ordinal α < ε0 there issome GLPω-worm A such that o(A) = α, and T + A is Π1
equivalent to Tα.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
A <ξ B :⇔ GLP ` B → 〈ξ〉A
I For <0 defines a well-order on the class of worms moduloprovable GLP equivalence.(Beklemishev, Fernandez Duque, JjJ)
I For <ξ with ξ > 0 the relation is no longer linear (mod prov.equivalence) but is still well-founded
(infinite anti-chains)
I Definition By oα(A) we denote the order type of A under <α
and we write o(A) instead of o0(A).
I Worms of GLPω are known to be useful for Turingprogressions:
I Proposition (Beklemishev) For each ordinal α < ε0 there issome GLPω-worm A such that o(A) = α, and T + A is Π1
equivalent to Tα.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
A <ξ B :⇔ GLP ` B → 〈ξ〉A
I For <0 defines a well-order on the class of worms moduloprovable GLP equivalence.(Beklemishev, Fernandez Duque, JjJ)
I For <ξ with ξ > 0 the relation is no longer linear (mod prov.equivalence) but is still well-founded
(infinite anti-chains)
I Definition By oα(A) we denote the order type of A under <α
and we write o(A) instead of o0(A).
I Worms of GLPω are known to be useful for Turingprogressions:
I Proposition (Beklemishev) For each ordinal α < ε0 there issome GLPω-worm A such that o(A) = α, and T + A is Π1
equivalent to Tα.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
A <ξ B :⇔ GLP ` B → 〈ξ〉A
I For <0 defines a well-order on the class of worms moduloprovable GLP equivalence.(Beklemishev, Fernandez Duque, JjJ)
I For <ξ with ξ > 0 the relation is no longer linear (mod prov.equivalence) but is still well-founded (infinite anti-chains)
I Definition By oα(A) we denote the order type of A under <α
and we write o(A) instead of o0(A).
I Worms of GLPω are known to be useful for Turingprogressions:
I Proposition (Beklemishev) For each ordinal α < ε0 there issome GLPω-worm A such that o(A) = α, and T + A is Π1
equivalent to Tα.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
A <ξ B :⇔ GLP ` B → 〈ξ〉A
I For <0 defines a well-order on the class of worms moduloprovable GLP equivalence.(Beklemishev, Fernandez Duque, JjJ)
I For <ξ with ξ > 0 the relation is no longer linear (mod prov.equivalence) but is still well-founded (infinite anti-chains)
I Definition By oα(A) we denote the order type of A under <α
and we write o(A) instead of o0(A).
I Worms of GLPω are known to be useful for Turingprogressions:
I Proposition (Beklemishev) For each ordinal α < ε0 there issome GLPω-worm A such that o(A) = α, and T + A is Π1
equivalent to Tα.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
A <ξ B :⇔ GLP ` B → 〈ξ〉A
I For <0 defines a well-order on the class of worms moduloprovable GLP equivalence.(Beklemishev, Fernandez Duque, JjJ)
I For <ξ with ξ > 0 the relation is no longer linear (mod prov.equivalence) but is still well-founded (infinite anti-chains)
I Definition By oα(A) we denote the order type of A under <αand we write o(A) instead of o0(A).
I Worms of GLPω are known to be useful for Turingprogressions:
I Proposition (Beklemishev) For each ordinal α < ε0 there issome GLPω-worm A such that o(A) = α, and T + A is Π1
equivalent to Tα.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
A <ξ B :⇔ GLP ` B → 〈ξ〉A
I For <0 defines a well-order on the class of worms moduloprovable GLP equivalence.(Beklemishev, Fernandez Duque, JjJ)
I For <ξ with ξ > 0 the relation is no longer linear (mod prov.equivalence) but is still well-founded (infinite anti-chains)
I Definition By oα(A) we denote the order type of A under <αand we write o(A) instead of o0(A).
I Worms of GLPω are known to be useful for Turingprogressions:
I Proposition (Beklemishev) For each ordinal α < ε0 there issome GLPω-worm A such that o(A) = α, and T + A is Π1
equivalent to Tα.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
A <ξ B :⇔ GLP ` B → 〈ξ〉A
I For <0 defines a well-order on the class of worms moduloprovable GLP equivalence.(Beklemishev, Fernandez Duque, JjJ)
I For <ξ with ξ > 0 the relation is no longer linear (mod prov.equivalence) but is still well-founded (infinite anti-chains)
I Definition By oα(A) we denote the order type of A under <αand we write o(A) instead of o0(A).
I Worms of GLPω are known to be useful for Turingprogressions:
I Proposition (Beklemishev) For each ordinal α < ε0 there issome GLPω-worm A such that o(A) = α, and T + A is Π1
equivalent to Tα.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I For the first part of this talk we shall focus on GLPω
I For a worm A we define hn(A) as the n head as the largestpart on the left of A where all modalities are at least n
I Example: h2(34245) = 34245 and h3(34) = 34
I We define an Ignatiev sequence to be a sequence ~A = {Ai} ofworms so that
I Each An ∈ Bn
I An+1 ≤n+1 hn+1(An)
I Example: 〈22022, 22〉I but also 〈22022, 2〉
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I For the first part of this talk we shall focus on GLPωI For a worm A we define hn(A) as the n head as the largest
part on the left of A where all modalities are at least n
I Example: h2(34245) = 34245 and h3(34) = 34
I We define an Ignatiev sequence to be a sequence ~A = {Ai} ofworms so that
I Each An ∈ Bn
I An+1 ≤n+1 hn+1(An)
I Example: 〈22022, 22〉I but also 〈22022, 2〉
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I For the first part of this talk we shall focus on GLPωI For a worm A we define hn(A) as the n head as the largest
part on the left of A where all modalities are at least n
I Example: h2(34245) = 34245 and h3(34) = 34
I We define an Ignatiev sequence to be a sequence ~A = {Ai} ofworms so that
I Each An ∈ Bn
I An+1 ≤n+1 hn+1(An)
I Example: 〈22022, 22〉I but also 〈22022, 2〉
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I For the first part of this talk we shall focus on GLPωI For a worm A we define hn(A) as the n head as the largest
part on the left of A where all modalities are at least n
I Example: h2(34245) = 34245 and h3(34) = 34
I We define an Ignatiev sequence to be a sequence ~A = {Ai} ofworms so that
I Each An ∈ Bn
I An+1 ≤n+1 hn+1(An)
I Example: 〈22022, 22〉I but also 〈22022, 2〉
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I For the first part of this talk we shall focus on GLPωI For a worm A we define hn(A) as the n head as the largest
part on the left of A where all modalities are at least n
I Example: h2(34245) = 34245 and h3(34) = 34
I We define an Ignatiev sequence to be a sequence ~A = {Ai} ofworms so that
I Each An ∈ Bn
I An+1 ≤n+1 hn+1(An)
I Example: 〈22022, 22〉I but also 〈22022, 2〉
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I For the first part of this talk we shall focus on GLPωI For a worm A we define hn(A) as the n head as the largest
part on the left of A where all modalities are at least n
I Example: h2(34245) = 34245 and h3(34) = 34
I We define an Ignatiev sequence to be a sequence ~A = {Ai} ofworms so that
I Each An ∈ Bn
I An+1 ≤n+1 hn+1(An)
I Example: 〈22022, 22〉I but also 〈22022, 2〉
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I For the first part of this talk we shall focus on GLPωI For a worm A we define hn(A) as the n head as the largest
part on the left of A where all modalities are at least n
I Example: h2(34245) = 34245 and h3(34) = 34
I We define an Ignatiev sequence to be a sequence ~A = {Ai} ofworms so that
I Each An ∈ Bn
I An+1 ≤n+1 hn+1(An)
I Example: 〈22022, 22〉
I but also 〈22022, 2〉
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I For the first part of this talk we shall focus on GLPωI For a worm A we define hn(A) as the n head as the largest
part on the left of A where all modalities are at least n
I Example: h2(34245) = 34245 and h3(34) = 34
I We define an Ignatiev sequence to be a sequence ~A = {Ai} ofworms so that
I Each An ∈ Bn
I An+1 ≤n+1 hn+1(An)
I Example: 〈22022, 22〉I but also 〈22022, 2〉
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I For two Ignatiev sequences ~A and ~B we define an accessibilityrelation <n:
I ~A <n~B if and only if
I Am = Bm for all m < nI An <n Bn
I Example: 〈202, 22〉 >1 〈202, 2〉I but also 〈202, 22〉 >1 〈202, 2, 2〉
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I For two Ignatiev sequences ~A and ~B we define an accessibilityrelation <n:
I ~A <n~B if and only if
I Am = Bm for all m < nI An <n Bn
I Example: 〈202, 22〉 >1 〈202, 2〉I but also 〈202, 22〉 >1 〈202, 2, 2〉
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I For two Ignatiev sequences ~A and ~B we define an accessibilityrelation <n:
I ~A <n~B if and only if
I Am = Bm for all m < n
I An <n Bn
I Example: 〈202, 22〉 >1 〈202, 2〉I but also 〈202, 22〉 >1 〈202, 2, 2〉
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I For two Ignatiev sequences ~A and ~B we define an accessibilityrelation <n:
I ~A <n~B if and only if
I Am = Bm for all m < nI An <n Bn
I Example: 〈202, 22〉 >1 〈202, 2〉I but also 〈202, 22〉 >1 〈202, 2, 2〉
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I For two Ignatiev sequences ~A and ~B we define an accessibilityrelation <n:
I ~A <n~B if and only if
I Am = Bm for all m < nI An <n Bn
I Example: 〈202, 22〉 >1 〈202, 2〉
I but also 〈202, 22〉 >1 〈202, 2, 2〉
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I For two Ignatiev sequences ~A and ~B we define an accessibilityrelation <n:
I ~A <n~B if and only if
I Am = Bm for all m < nI An <n Bn
I Example: 〈202, 22〉 >1 〈202, 2〉I but also 〈202, 22〉 >1 〈202, 2, 2〉
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let I denote the set of all Ignatiev sequences
I We define a Kripke frame:
〈I, {>n}n∈ω〉
I We shall denote this frame also by II We define by ~A >, for no ~A, ~A ⊥.
I commutes with Boolean connectives: ~A φ ∧ ψ if and onlyif ~A φ and ~A ψ, etc
I ~A 〈n〉φ if and only if there is some ~B with ~A >n~B so that
~B φ
I Theorem GLP0ω ` φ ⇔ I ` φ
I Proof by a p-morphic embedding of this structure into thegeneralization of Ignatiev’s model.
I Let’s see a picture
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let I denote the set of all Ignatiev sequences
I We define a Kripke frame:
〈I, {>n}n∈ω〉
I We shall denote this frame also by II We define by ~A >, for no ~A, ~A ⊥.
I commutes with Boolean connectives: ~A φ ∧ ψ if and onlyif ~A φ and ~A ψ, etc
I ~A 〈n〉φ if and only if there is some ~B with ~A >n~B so that
~B φ
I Theorem GLP0ω ` φ ⇔ I ` φ
I Proof by a p-morphic embedding of this structure into thegeneralization of Ignatiev’s model.
I Let’s see a picture
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let I denote the set of all Ignatiev sequences
I We define a Kripke frame:
〈I, {>n}n∈ω〉
I We shall denote this frame also by I
I We define by ~A >, for no ~A, ~A ⊥.
I commutes with Boolean connectives: ~A φ ∧ ψ if and onlyif ~A φ and ~A ψ, etc
I ~A 〈n〉φ if and only if there is some ~B with ~A >n~B so that
~B φ
I Theorem GLP0ω ` φ ⇔ I ` φ
I Proof by a p-morphic embedding of this structure into thegeneralization of Ignatiev’s model.
I Let’s see a picture
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let I denote the set of all Ignatiev sequences
I We define a Kripke frame:
〈I, {>n}n∈ω〉
I We shall denote this frame also by II We define by ~A >, for no ~A, ~A ⊥.
I commutes with Boolean connectives: ~A φ ∧ ψ if and onlyif ~A φ and ~A ψ, etc
I ~A 〈n〉φ if and only if there is some ~B with ~A >n~B so that
~B φ
I Theorem GLP0ω ` φ ⇔ I ` φ
I Proof by a p-morphic embedding of this structure into thegeneralization of Ignatiev’s model.
I Let’s see a picture
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let I denote the set of all Ignatiev sequences
I We define a Kripke frame:
〈I, {>n}n∈ω〉
I We shall denote this frame also by II We define by ~A >, for no ~A, ~A ⊥.
I commutes with Boolean connectives: ~A φ ∧ ψ if and onlyif ~A φ and ~A ψ, etc
I ~A 〈n〉φ if and only if there is some ~B with ~A >n~B so that
~B φ
I Theorem GLP0ω ` φ ⇔ I ` φ
I Proof by a p-morphic embedding of this structure into thegeneralization of Ignatiev’s model.
I Let’s see a picture
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let I denote the set of all Ignatiev sequences
I We define a Kripke frame:
〈I, {>n}n∈ω〉
I We shall denote this frame also by II We define by ~A >, for no ~A, ~A ⊥.
I commutes with Boolean connectives: ~A φ ∧ ψ if and onlyif ~A φ and ~A ψ, etc
I ~A 〈n〉φ if and only if there is some ~B with ~A >n~B so that
~B φ
I Theorem GLP0ω ` φ ⇔ I ` φ
I Proof by a p-morphic embedding of this structure into thegeneralization of Ignatiev’s model.
I Let’s see a picture
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let I denote the set of all Ignatiev sequences
I We define a Kripke frame:
〈I, {>n}n∈ω〉
I We shall denote this frame also by II We define by ~A >, for no ~A, ~A ⊥.
I commutes with Boolean connectives: ~A φ ∧ ψ if and onlyif ~A φ and ~A ψ, etc
I ~A 〈n〉φ if and only if there is some ~B with ~A >n~B so that
~B φ
I Theorem GLP0ω ` φ ⇔ I ` φ
I Proof by a p-morphic embedding of this structure into thegeneralization of Ignatiev’s model.
I Let’s see a picture
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let I denote the set of all Ignatiev sequences
I We define a Kripke frame:
〈I, {>n}n∈ω〉
I We shall denote this frame also by II We define by ~A >, for no ~A, ~A ⊥.
I commutes with Boolean connectives: ~A φ ∧ ψ if and onlyif ~A φ and ~A ψ, etc
I ~A 〈n〉φ if and only if there is some ~B with ~A >n~B so that
~B φ
I Theorem GLP0ω ` φ ⇔ I ` φ
I Proof by a p-morphic embedding of this structure into thegeneralization of Ignatiev’s model.
I Let’s see a picture
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Let I denote the set of all Ignatiev sequences
I We define a Kripke frame:
〈I, {>n}n∈ω〉
I We shall denote this frame also by II We define by ~A >, for no ~A, ~A ⊥.
I commutes with Boolean connectives: ~A φ ∧ ψ if and onlyif ~A φ and ~A ψ, etc
I ~A 〈n〉φ if and only if there is some ~B with ~A >n~B so that
~B φ
I Theorem GLP0ω ` φ ⇔ I ` φ
I Proof by a p-morphic embedding of this structure into thegeneralization of Ignatiev’s model.
I Let’s see a picture
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We define the Πn+1 proof-theoretic ordinal of a theory U asfollows:
I |U|Πn+1 = sup{ξ | T nξ ⊆ U}.
I For U a arithmetical theory we define its Turing-Taylorexpansion by
I tt(U) :=⋃∞
n=0 T n|U|Πn+1
I In case U ≡ tt(U) we say that U has a convergentTuring-Taylor expansion.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We define the Πn+1 proof-theoretic ordinal of a theory U asfollows:
I |U|Πn+1 = sup{ξ | T nξ ⊆ U}.
I For U a arithmetical theory we define its Turing-Taylorexpansion by
I tt(U) :=⋃∞
n=0 T n|U|Πn+1
I In case U ≡ tt(U) we say that U has a convergentTuring-Taylor expansion.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We define the Πn+1 proof-theoretic ordinal of a theory U asfollows:
I |U|Πn+1 = sup{ξ | T nξ ⊆ U}.
I For U a arithmetical theory we define its Turing-Taylorexpansion by
I tt(U) :=⋃∞
n=0 T n|U|Πn+1
I In case U ≡ tt(U) we say that U has a convergentTuring-Taylor expansion.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We define the Πn+1 proof-theoretic ordinal of a theory U asfollows:
I |U|Πn+1 = sup{ξ | T nξ ⊆ U}.
I For U a arithmetical theory we define its Turing-Taylorexpansion by
I tt(U) :=⋃∞
n=0 T n|U|Πn+1
I In case U ≡ tt(U) we say that U has a convergentTuring-Taylor expansion.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We define the Πn+1 proof-theoretic ordinal of a theory U asfollows:
I |U|Πn+1 = sup{ξ | T nξ ⊆ U}.
I For U a arithmetical theory we define its Turing-Taylorexpansion by
I tt(U) :=⋃∞
n=0 T n|U|Πn+1
I In case U ≡ tt(U) we say that U has a convergentTuring-Taylor expansion.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We will now link Ignatiev’s model to Turing-Taylor expansions
I Let us recall:
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
I We shall use the ordinal notation system 〈Bn, <n〉 to label theTuring progression based on n-consistency
I Thus, T 13 denotes T 1
ωω ,
I and T 23 denotes T 2
ω
I and T 33 denotes T 2
1
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We will now link Ignatiev’s model to Turing-Taylor expansions
I Let us recall:
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
I We shall use the ordinal notation system 〈Bn, <n〉 to label theTuring progression based on n-consistency
I Thus, T 13 denotes T 1
ωω ,
I and T 23 denotes T 2
ω
I and T 33 denotes T 2
1
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We will now link Ignatiev’s model to Turing-Taylor expansions
I Let us recall:
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
I We shall use the ordinal notation system 〈Bn, <n〉 to label theTuring progression based on n-consistency
I Thus, T 13 denotes T 1
ωω ,
I and T 23 denotes T 2
ω
I and T 33 denotes T 2
1
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We will now link Ignatiev’s model to Turing-Taylor expansions
I Let us recall:
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
I We shall use the ordinal notation system 〈Bn, <n〉 to label theTuring progression based on n-consistency
I Thus, T 13 denotes T 1
ωω ,
I and T 23 denotes T 2
ω
I and T 33 denotes T 2
1
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We will now link Ignatiev’s model to Turing-Taylor expansions
I Let us recall:
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
I We shall use the ordinal notation system 〈Bn, <n〉 to label theTuring progression based on n-consistency
I Thus, T 13 denotes T 1
ωω ,
I and T 23 denotes T 2
ω
I and T 33 denotes T 2
1
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We will now link Ignatiev’s model to Turing-Taylor expansions
I Let us recall:
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
I We shall use the ordinal notation system 〈Bn, <n〉 to label theTuring progression based on n-consistency
I Thus, T 13 denotes T 1
ωω ,
I and T 23 denotes T 2
ω
I and T 33 denotes T 2
1
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We will now link Ignatiev’s model to Turing-Taylor expansions
I Let us recall:
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
I We shall use the ordinal notation system 〈Bn, <n〉 to label theTuring progression based on n-consistency
I Thus, T 13 denotes T 1
ωω ,
I and T 23 denotes T 2
ω
I and T 33 denotes T 2
1
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We will now link Ignatiev’s model to Turing-Taylor expansions
I Let us recall:
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
I We shall use the ordinal notation system 〈Bn, <n〉 to label theTuring progression based on n-consistency
I Thus, T 13 denotes T 1
ωω ,
I and T 23 denotes T 2
ω
I and T 33 denotes T 2
1
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We will now link Ignatiev’s model to Turing-Taylor expansions
I Let us recall:
I T i0 := T ;
I T iα+1 := T i
α ∪ {〈i〉T iα>};
I Tλ :=⋃α<λ Tαfor limit λ.
I We shall use the ordinal notation system 〈Bn, <n〉 to label theTuring progression based on n-consistency
I Thus, T 13 denotes T 1
ωω ,
I and T 23 denotes T 2
ω
I and T 33 denotes T 2
1
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I TheoremFor each worm A : T + A ≡
⋃∞n=0 T n
hn(A)
I TheoremFor each worm A : T + A ≡
⋃∞n=0 T n
A
I Compare this to
f (x) :=∞∑
n=0
f (n)(0)
n!xn
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I TheoremFor each worm A : T + A ≡
⋃∞n=0 T n
hn(A)
I TheoremFor each worm A : T + A ≡
⋃∞n=0 T n
A
I Compare this to
f (x) :=∞∑
n=0
f (n)(0)
n!xn
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I TheoremFor each worm A : T + A ≡
⋃∞n=0 T n
hn(A)
I TheoremFor each worm A : T + A ≡
⋃∞n=0 T n
A
I Compare this to
f (x) :=∞∑
n=0
f (n)(0)
n!xn
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Theorem The Ignatiev sequences exactly correspond to thosesub-theories of PA that have a convergent Turing-Taylorexpansion
I That is, for each such theory U, we have that tt(U) ∈ II and for each ~A ∈ I, there is a theory U so that tt(U) = ~A
I This yields a roadmap to conservation results!
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Theorem The Ignatiev sequences exactly correspond to thosesub-theories of PA that have a convergent Turing-Taylorexpansion
I That is, for each such theory U, we have that tt(U) ∈ I
I and for each ~A ∈ I, there is a theory U so that tt(U) = ~A
I This yields a roadmap to conservation results!
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Theorem The Ignatiev sequences exactly correspond to thosesub-theories of PA that have a convergent Turing-Taylorexpansion
I That is, for each such theory U, we have that tt(U) ∈ II and for each ~A ∈ I, there is a theory U so that tt(U) = ~A
I This yields a roadmap to conservation results!
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Theorem The Ignatiev sequences exactly correspond to thosesub-theories of PA that have a convergent Turing-Taylorexpansion
I That is, for each such theory U, we have that tt(U) ∈ II and for each ~A ∈ I, there is a theory U so that tt(U) = ~A
I This yields a roadmap to conservation results!
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We proof of the theorem uses three main results
I We shall see why worms are better than the more familiarordinal notations in this context
I For each number n and each GLPω worm A,GLPω ` A↔ hn(A) ∧ rn(A)
(here, rn(A) denotes the nremainder of A so that A = hn(A)rn(A))
I For each worm A ∈Wn we have T + A ≡n T nA (Beklemishev)
I For each worm A : T + A ≡⋃∞
n=0 T nhn(A) (JjJ)
I Corollaries:
I For each worm A ∈Wn we have T + nA ≡ T nnA
I For each worm A ∈Wn T nA ` T m
A for m < n
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We proof of the theorem uses three main results
I We shall see why worms are better than the more familiarordinal notations in this context
I For each number n and each GLPω worm A,GLPω ` A↔ hn(A) ∧ rn(A)
(here, rn(A) denotes the nremainder of A so that A = hn(A)rn(A))
I For each worm A ∈Wn we have T + A ≡n T nA (Beklemishev)
I For each worm A : T + A ≡⋃∞
n=0 T nhn(A) (JjJ)
I Corollaries:
I For each worm A ∈Wn we have T + nA ≡ T nnA
I For each worm A ∈Wn T nA ` T m
A for m < n
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We proof of the theorem uses three main results
I We shall see why worms are better than the more familiarordinal notations in this context
I For each number n and each GLPω worm A,GLPω ` A↔ hn(A) ∧ rn(A)
(here, rn(A) denotes the nremainder of A so that A = hn(A)rn(A))
I For each worm A ∈Wn we have T + A ≡n T nA (Beklemishev)
I For each worm A : T + A ≡⋃∞
n=0 T nhn(A) (JjJ)
I Corollaries:
I For each worm A ∈Wn we have T + nA ≡ T nnA
I For each worm A ∈Wn T nA ` T m
A for m < n
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We proof of the theorem uses three main results
I We shall see why worms are better than the more familiarordinal notations in this context
I For each number n and each GLPω worm A,GLPω ` A↔ hn(A) ∧ rn(A) (here, rn(A) denotes the nremainder of A so that A = hn(A)rn(A))
I For each worm A ∈Wn we have T + A ≡n T nA (Beklemishev)
I For each worm A : T + A ≡⋃∞
n=0 T nhn(A) (JjJ)
I Corollaries:
I For each worm A ∈Wn we have T + nA ≡ T nnA
I For each worm A ∈Wn T nA ` T m
A for m < n
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We proof of the theorem uses three main results
I We shall see why worms are better than the more familiarordinal notations in this context
I For each number n and each GLPω worm A,GLPω ` A↔ hn(A) ∧ rn(A) (here, rn(A) denotes the nremainder of A so that A = hn(A)rn(A))
I For each worm A ∈Wn we have T + A ≡n T nA (Beklemishev)
I For each worm A : T + A ≡⋃∞
n=0 T nhn(A) (JjJ)
I Corollaries:
I For each worm A ∈Wn we have T + nA ≡ T nnA
I For each worm A ∈Wn T nA ` T m
A for m < n
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We proof of the theorem uses three main results
I We shall see why worms are better than the more familiarordinal notations in this context
I For each number n and each GLPω worm A,GLPω ` A↔ hn(A) ∧ rn(A) (here, rn(A) denotes the nremainder of A so that A = hn(A)rn(A))
I For each worm A ∈Wn we have T + A ≡n T nA (Beklemishev)
I For each worm A : T + A ≡⋃∞
n=0 T nhn(A) (JjJ)
I Corollaries:
I For each worm A ∈Wn we have T + nA ≡ T nnA
I For each worm A ∈Wn T nA ` T m
A for m < n
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We proof of the theorem uses three main results
I We shall see why worms are better than the more familiarordinal notations in this context
I For each number n and each GLPω worm A,GLPω ` A↔ hn(A) ∧ rn(A) (here, rn(A) denotes the nremainder of A so that A = hn(A)rn(A))
I For each worm A ∈Wn we have T + A ≡n T nA (Beklemishev)
I For each worm A : T + A ≡⋃∞
n=0 T nhn(A) (JjJ)
I Corollaries:
I For each worm A ∈Wn we have T + nA ≡ T nnA
I For each worm A ∈Wn T nA ` T m
A for m < n
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We proof of the theorem uses three main results
I We shall see why worms are better than the more familiarordinal notations in this context
I For each number n and each GLPω worm A,GLPω ` A↔ hn(A) ∧ rn(A) (here, rn(A) denotes the nremainder of A so that A = hn(A)rn(A))
I For each worm A ∈Wn we have T + A ≡n T nA (Beklemishev)
I For each worm A : T + A ≡⋃∞
n=0 T nhn(A) (JjJ)
I Corollaries:I For each worm A ∈Wn we have T + nA ≡ T n
nA
I For each worm A ∈Wn T nA ` T m
A for m < n
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I We proof of the theorem uses three main results
I We shall see why worms are better than the more familiarordinal notations in this context
I For each number n and each GLPω worm A,GLPω ` A↔ hn(A) ∧ rn(A) (here, rn(A) denotes the nremainder of A so that A = hn(A)rn(A))
I For each worm A ∈Wn we have T + A ≡n T nA (Beklemishev)
I For each worm A : T + A ≡⋃∞
n=0 T nhn(A) (JjJ)
I Corollaries:I For each worm A ∈Wn we have T + nA ≡ T n
nAI For each worm A ∈Wn T n
A ` T mA for m < n
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I
tt(U) denotes 〈|U|Π01, |U|Π0
2, |U|Π0
3, . . .〉.
I Likewise, with every sequence ~α = 〈α0, α1, . . .〉 of ordinalsbelow ε0 we can naturally associate a sub theory (~α)tt of PAas follows
(~α)tt :=∞⋃
n=0
EAnαn.
I Likewise, with every sequence ~A = 〈A0,A1, . . .〉 of GLPωworms we can naturally associate a sub theory (~A)tt of PA asfollows
(~α)tt :=∞⋃
n=0
EAnAn.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I
tt(U) denotes 〈|U|Π01, |U|Π0
2, |U|Π0
3, . . .〉.
I Likewise, with every sequence ~α = 〈α0, α1, . . .〉 of ordinalsbelow ε0 we can naturally associate a sub theory (~α)tt of PAas follows
(~α)tt :=∞⋃
n=0
EAnαn.
I Likewise, with every sequence ~A = 〈A0,A1, . . .〉 of GLPωworms we can naturally associate a sub theory (~A)tt of PA asfollows
(~α)tt :=∞⋃
n=0
EAnAn.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I
tt(U) denotes 〈|U|Π01, |U|Π0
2, |U|Π0
3, . . .〉.
I Likewise, with every sequence ~α = 〈α0, α1, . . .〉 of ordinalsbelow ε0 we can naturally associate a sub theory (~α)tt of PAas follows
(~α)tt :=∞⋃
n=0
EAnαn.
I Likewise, with every sequence ~A = 〈A0,A1, . . .〉 of GLPωworms we can naturally associate a sub theory (~A)tt of PA asfollows
(~α)tt :=∞⋃
n=0
EAnAn.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I The monomials in Turing-Taylor progressions are the T nA
I They are not entirely independent!
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I T 11 ≡ T + 〈1〉>, and
I T 001 ≡ T 0
01.
I Thus, T 11 + T 0
01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.
I Equivalent to T + 〈1〉〈0〉〈1〉>I Which is in turn equivalent to T 1
1 + T 0101
I In the classical notation system this reads
T 11 + T 0
ω+1 ≡ T 11 + T 0
ω·2
I The example shows that in general tt((~A)tt) 6= ~A !
tt((01, 1)tt) = (101, 1) 6= (01, 1)
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I The monomials in Turing-Taylor progressions are the T nA
I They are not entirely independent!
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I T 11 ≡ T + 〈1〉>, and
I T 001 ≡ T 0
01.
I Thus, T 11 + T 0
01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.
I Equivalent to T + 〈1〉〈0〉〈1〉>I Which is in turn equivalent to T 1
1 + T 0101
I In the classical notation system this reads
T 11 + T 0
ω+1 ≡ T 11 + T 0
ω·2
I The example shows that in general tt((~A)tt) 6= ~A !
tt((01, 1)tt) = (101, 1) 6= (01, 1)
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I The monomials in Turing-Taylor progressions are the T nA
I They are not entirely independent!
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I T 11 ≡ T + 〈1〉>, and
I T 001 ≡ T 0
01.
I Thus, T 11 + T 0
01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.
I Equivalent to T + 〈1〉〈0〉〈1〉>I Which is in turn equivalent to T 1
1 + T 0101
I In the classical notation system this reads
T 11 + T 0
ω+1 ≡ T 11 + T 0
ω·2
I The example shows that in general tt((~A)tt) 6= ~A !
tt((01, 1)tt) = (101, 1) 6= (01, 1)
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I The monomials in Turing-Taylor progressions are the T nA
I They are not entirely independent!
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I T 11 ≡ T + 〈1〉>, and
I T 001 ≡ T 0
01.
I Thus, T 11 + T 0
01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.
I Equivalent to T + 〈1〉〈0〉〈1〉>I Which is in turn equivalent to T 1
1 + T 0101
I In the classical notation system this reads
T 11 + T 0
ω+1 ≡ T 11 + T 0
ω·2
I The example shows that in general tt((~A)tt) 6= ~A !
tt((01, 1)tt) = (101, 1) 6= (01, 1)
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I The monomials in Turing-Taylor progressions are the T nA
I They are not entirely independent!
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I T 11 ≡ T + 〈1〉>, and
I T 001 ≡ T 0
01.
I Thus, T 11 + T 0
01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.
I Equivalent to T + 〈1〉〈0〉〈1〉>I Which is in turn equivalent to T 1
1 + T 0101
I In the classical notation system this reads
T 11 + T 0
ω+1 ≡ T 11 + T 0
ω·2
I The example shows that in general tt((~A)tt) 6= ~A !
tt((01, 1)tt) = (101, 1) 6= (01, 1)
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I The monomials in Turing-Taylor progressions are the T nA
I They are not entirely independent!
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I T 11 ≡ T + 〈1〉>, and
I T 001 ≡ T 0
01.
I Thus, T 11 + T 0
01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.
I Equivalent to T + 〈1〉〈0〉〈1〉>I Which is in turn equivalent to T 1
1 + T 0101
I In the classical notation system this reads
T 11 + T 0
ω+1 ≡ T 11 + T 0
ω·2
I The example shows that in general tt((~A)tt) 6= ~A !
tt((01, 1)tt) = (101, 1) 6= (01, 1)
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I The monomials in Turing-Taylor progressions are the T nA
I They are not entirely independent!
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I T 11 ≡ T + 〈1〉>, and
I T 001 ≡ T 0
01.
I Thus, T 11 + T 0
01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.
I Equivalent to T + 〈1〉〈0〉〈1〉>
I Which is in turn equivalent to T 11 + T 0
101
I In the classical notation system this reads
T 11 + T 0
ω+1 ≡ T 11 + T 0
ω·2
I The example shows that in general tt((~A)tt) 6= ~A !
tt((01, 1)tt) = (101, 1) 6= (01, 1)
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I The monomials in Turing-Taylor progressions are the T nA
I They are not entirely independent!
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I T 11 ≡ T + 〈1〉>, and
I T 001 ≡ T 0
01.
I Thus, T 11 + T 0
01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.
I Equivalent to T + 〈1〉〈0〉〈1〉>I Which is in turn equivalent to T 1
1 + T 0101
I In the classical notation system this reads
T 11 + T 0
ω+1 ≡ T 11 + T 0
ω·2
I The example shows that in general tt((~A)tt) 6= ~A !
tt((01, 1)tt) = (101, 1) 6= (01, 1)
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I The monomials in Turing-Taylor progressions are the T nA
I They are not entirely independent!
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I T 11 ≡ T + 〈1〉>, and
I T 001 ≡ T 0
01.
I Thus, T 11 + T 0
01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.
I Equivalent to T + 〈1〉〈0〉〈1〉>I Which is in turn equivalent to T 1
1 + T 0101
I In the classical notation system this reads
T 11 + T 0
ω+1 ≡ T 11 + T 0
ω·2
I The example shows that in general tt((~A)tt) 6= ~A !
tt((01, 1)tt) = (101, 1) 6= (01, 1)
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I The monomials in Turing-Taylor progressions are the T nA
I They are not entirely independent!
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I T 11 ≡ T + 〈1〉>, and
I T 001 ≡ T 0
01.
I Thus, T 11 + T 0
01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.
I Equivalent to T + 〈1〉〈0〉〈1〉>I Which is in turn equivalent to T 1
1 + T 0101
I In the classical notation system this reads
T 11 + T 0
ω+1 ≡ T 11 + T 0
ω·2
I The example shows that in general tt((~A)tt) 6= ~A !
tt((01, 1)tt) = (101, 1) 6= (01, 1)
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I The monomials in Turing-Taylor progressions are the T nA
I They are not entirely independent!
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I T 11 ≡ T + 〈1〉>, and
I T 001 ≡ T 0
01.
I Thus, T 11 + T 0
01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.
I Equivalent to T + 〈1〉〈0〉〈1〉>I Which is in turn equivalent to T 1
1 + T 0101
I In the classical notation system this reads
T 11 + T 0
ω+1 ≡ T 11 + T 0
ω·2
I The example shows that in general tt((~A)tt) 6= ~A !tt((01, 1)tt) = (101, 1) 6= (01, 1)
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I With a slightly more involved reasoning, we can prove
I Lemma 1: Let A ∈ Sn+1 and B ∈ Sn. We have that
I
T n+1A + T n
nB ≡n+1 T + AnB,
and
I
T n+1A + T n
nB ≡n T nAnB .
I This nicely illustrates that worms are often better than Cantor
I Using these ingredients one easily proves
I Theorem If U is some sub-theory of PA with a convergentTuring-Taylor expansion, so that U 6≡0 PA, then tt(U) definesa point in I.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I With a slightly more involved reasoning, we can prove
I Lemma 1: Let A ∈ Sn+1 and B ∈ Sn. We have that
I
T n+1A + T n
nB ≡n+1 T + AnB,
and
I
T n+1A + T n
nB ≡n T nAnB .
I This nicely illustrates that worms are often better than Cantor
I Using these ingredients one easily proves
I Theorem If U is some sub-theory of PA with a convergentTuring-Taylor expansion, so that U 6≡0 PA, then tt(U) definesa point in I.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I With a slightly more involved reasoning, we can prove
I Lemma 1: Let A ∈ Sn+1 and B ∈ Sn. We have that
I
T n+1A + T n
nB ≡n+1 T + AnB,
and
I
T n+1A + T n
nB ≡n T nAnB .
I This nicely illustrates that worms are often better than Cantor
I Using these ingredients one easily proves
I Theorem If U is some sub-theory of PA with a convergentTuring-Taylor expansion, so that U 6≡0 PA, then tt(U) definesa point in I.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I With a slightly more involved reasoning, we can prove
I Lemma 1: Let A ∈ Sn+1 and B ∈ Sn. We have that
I
T n+1A + T n
nB ≡n+1 T + AnB,
and
I
T n+1A + T n
nB ≡n T nAnB .
I This nicely illustrates that worms are often better than Cantor
I Using these ingredients one easily proves
I Theorem If U is some sub-theory of PA with a convergentTuring-Taylor expansion, so that U 6≡0 PA, then tt(U) definesa point in I.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I With a slightly more involved reasoning, we can prove
I Lemma 1: Let A ∈ Sn+1 and B ∈ Sn. We have that
I
T n+1A + T n
nB ≡n+1 T + AnB,
and
I
T n+1A + T n
nB ≡n T nAnB .
I This nicely illustrates that worms are often better than Cantor
I Using these ingredients one easily proves
I Theorem If U is some sub-theory of PA with a convergentTuring-Taylor expansion, so that U 6≡0 PA, then tt(U) definesa point in I.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I With a slightly more involved reasoning, we can prove
I Lemma 1: Let A ∈ Sn+1 and B ∈ Sn. We have that
I
T n+1A + T n
nB ≡n+1 T + AnB,
and
I
T n+1A + T n
nB ≡n T nAnB .
I This nicely illustrates that worms are often better than Cantor
I Using these ingredients one easily proves
I Theorem If U is some sub-theory of PA with a convergentTuring-Taylor expansion, so that U 6≡0 PA, then tt(U) definesa point in I.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I With a slightly more involved reasoning, we can prove
I Lemma 1: Let A ∈ Sn+1 and B ∈ Sn. We have that
I
T n+1A + T n
nB ≡n+1 T + AnB,
and
I
T n+1A + T n
nB ≡n T nAnB .
I This nicely illustrates that worms are often better than Cantor
I Using these ingredients one easily proves
I Theorem If U is some sub-theory of PA with a convergentTuring-Taylor expansion, so that U 6≡0 PA, then tt(U) definesa point in I.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions
I Example: T 11 + T 0
01 ≡ T 11 + T 0
101.
I With a slightly more involved reasoning, we can prove
I Lemma 1: Let A ∈ Sn+1 and B ∈ Sn. We have that
I
T n+1A + T n
nB ≡n+1 T + AnB,
and
I
T n+1A + T n
nB ≡n T nAnB .
I This nicely illustrates that worms are often better than Cantor
I Using these ingredients one easily proves
I Theorem If U is some sub-theory of PA with a convergentTuring-Taylor expansion, so that U 6≡0 PA, then tt(U) definesa point in I.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I We would like to extend the results of the first section beyondfirst order
I A central ingredient: syntactical complexity classes
I Like in the truth interpretation of GLP
I Omega-rule interpretation is slightly better
I However, does not tie up with the Turing jump hierarchy
I Friedman, Godlfarb and Harrington come to the rescue!
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I We would like to extend the results of the first section beyondfirst order
I A central ingredient: syntactical complexity classes
I Like in the truth interpretation of GLP
I Omega-rule interpretation is slightly better
I However, does not tie up with the Turing jump hierarchy
I Friedman, Godlfarb and Harrington come to the rescue!
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I We would like to extend the results of the first section beyondfirst order
I A central ingredient: syntactical complexity classes
I Like in the truth interpretation of GLP
I Omega-rule interpretation is slightly better
I However, does not tie up with the Turing jump hierarchy
I Friedman, Godlfarb and Harrington come to the rescue!
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I We would like to extend the results of the first section beyondfirst order
I A central ingredient: syntactical complexity classes
I Like in the truth interpretation of GLP
I Omega-rule interpretation is slightly better
I However, does not tie up with the Turing jump hierarchy
I Friedman, Godlfarb and Harrington come to the rescue!
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I We would like to extend the results of the first section beyondfirst order
I A central ingredient: syntactical complexity classes
I Like in the truth interpretation of GLP
I Omega-rule interpretation is slightly better
I However, does not tie up with the Turing jump hierarchy
I Friedman, Godlfarb and Harrington come to the rescue!
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I We would like to extend the results of the first section beyondfirst order
I A central ingredient: syntactical complexity classes
I Like in the truth interpretation of GLP
I Omega-rule interpretation is slightly better
I However, does not tie up with the Turing jump hierarchy
I Friedman, Godlfarb and Harrington come to the rescue!
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Definition (Witness-comparison relation)
For φ := ∃x φ0(x) and ψ := ∃x ψ0(x) we define
φ ≤ ψ := ∃x (φ0(x) ∧ ∀ y<x ¬ψ0(x)) and,φ < ψ := ∃x (φ0(x) ∧ ∀ y≤x ¬ψ0(x)).
I Theorem (Rosser’s Theorem)
Let T be a consistent c.e. theory extending EA. There is someρ ∈ Σ0
1 which is undecidable in T . That is,
T 0 ρ and,T 0 ¬ρ.
I Proof Consider ρ↔ ¬(2ρ < 2¬ρ).I I find it utterly amazing that something sensible can be
proven using the witness comparison techniques!
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Definition (Witness-comparison relation)
For φ := ∃x φ0(x) and ψ := ∃x ψ0(x) we define
φ ≤ ψ := ∃x (φ0(x) ∧ ∀ y<x ¬ψ0(x)) and,φ < ψ := ∃x (φ0(x) ∧ ∀ y≤x ¬ψ0(x)).
I Theorem (Rosser’s Theorem)
Let T be a consistent c.e. theory extending EA. There is someρ ∈ Σ0
1 which is undecidable in T . That is,
T 0 ρ and,T 0 ¬ρ.
I Proof Consider ρ↔ ¬(2ρ < 2¬ρ).I I find it utterly amazing that something sensible can be
proven using the witness comparison techniques!
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Definition (Witness-comparison relation)
For φ := ∃x φ0(x) and ψ := ∃x ψ0(x) we define
φ ≤ ψ := ∃x (φ0(x) ∧ ∀ y<x ¬ψ0(x)) and,φ < ψ := ∃x (φ0(x) ∧ ∀ y≤x ¬ψ0(x)).
I Theorem (Rosser’s Theorem)
Let T be a consistent c.e. theory extending EA. There is someρ ∈ Σ0
1 which is undecidable in T . That is,
T 0 ρ and,T 0 ¬ρ.
I Proof Consider ρ↔ ¬(2ρ < 2¬ρ).I I find it utterly amazing that something sensible can be
proven using the witness comparison techniques!
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Definition (Witness-comparison relation)
For φ := ∃x φ0(x) and ψ := ∃x ψ0(x) we define
φ ≤ ψ := ∃x (φ0(x) ∧ ∀ y<x ¬ψ0(x)) and,φ < ψ := ∃x (φ0(x) ∧ ∀ y≤x ¬ψ0(x)).
I Theorem (Rosser’s Theorem)
Let T be a consistent c.e. theory extending EA. There is someρ ∈ Σ0
1 which is undecidable in T . That is,
T 0 ρ and,T 0 ¬ρ.
I Proof Consider ρ↔ ¬(2ρ < 2¬ρ).I I find it utterly amazing that something sensible can be
proven using the witness comparison techniques!
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Definition (Witness-comparison relation)
For φ := ∃x φ0(x) and ψ := ∃x ψ0(x) we define
φ ≤ ψ := ∃x (φ0(x) ∧ ∀ y<x ¬ψ0(x)) and,φ < ψ := ∃x (φ0(x) ∧ ∀ y≤x ¬ψ0(x)).
I Theorem (Rosser’s Theorem)
Let T be a consistent c.e. theory extending EA. There is someρ ∈ Σ0
1 which is undecidable in T . That is,
T 0 ρ and,T 0 ¬ρ.
I Proof Consider ρ↔ ¬(2ρ < 2¬ρ).
I I find it utterly amazing that something sensible can beproven using the witness comparison techniques!
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Definition (Witness-comparison relation)
For φ := ∃x φ0(x) and ψ := ∃x ψ0(x) we define
φ ≤ ψ := ∃x (φ0(x) ∧ ∀ y<x ¬ψ0(x)) and,φ < ψ := ∃x (φ0(x) ∧ ∀ y≤x ¬ψ0(x)).
I Theorem (Rosser’s Theorem)
Let T be a consistent c.e. theory extending EA. There is someρ ∈ Σ0
1 which is undecidable in T . That is,
T 0 ρ and,T 0 ¬ρ.
I Proof Consider ρ↔ ¬(2ρ < 2¬ρ).I I find it utterly amazing that something sensible can be
proven using the witness comparison techniques!Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
LemmaLet A and B be some formulas of logical complexity Σ0
n+1 forn < ω.
1. Both A < B and A ≤ B are of complexity Σ0n+1;
2. EA ` A ∧ ¬B → (A < B) ;
3. EA ` (A < B)→ (A ≤ B);
4. EA ` (A ≤ B)→ A;
5. EA ` (A ≤ B)→ ¬(B < A) and consequently;
6. EA ` (A < B)→ ¬(B ≤ A);
7. EA ` [(B ≤ B) ∨ (A ≤ A)] → [(A ≤ B) ∨ (B < A)];
8. EA ` A ∧ ¬(A ≤ B)→ B.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Theorem (FGH theorem)
Let T be any computably enumerable theory extending EA. Foreach σ ∈ Σ0
1 we have that there is some ρ ∈ Σ01 so that
EA ` 3T> →(σ ↔ 2Tρ
).
I Proof.Consider the fixpoint ρ for which EA ` ρ ↔ (σ ≤ 2ρ).
I This shows us that we can express a syntactical class usingprovability logics!
I We wish to stretch this further
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Theorem (FGH theorem)
Let T be any computably enumerable theory extending EA. Foreach σ ∈ Σ0
1 we have that there is some ρ ∈ Σ01 so that
EA ` 3T> →(σ ↔ 2Tρ
).
I Proof.Consider the fixpoint ρ for which EA ` ρ ↔ (σ ≤ 2ρ).
I This shows us that we can express a syntactical class usingprovability logics!
I We wish to stretch this further
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Theorem (FGH theorem)
Let T be any computably enumerable theory extending EA. Foreach σ ∈ Σ0
1 we have that there is some ρ ∈ Σ01 so that
EA ` 3T> →(σ ↔ 2Tρ
).
I Proof.Consider the fixpoint ρ for which EA ` ρ ↔ (σ ≤ 2ρ).
I This shows us that we can express a syntactical class usingprovability logics!
I We wish to stretch this further
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Theorem (FGH theorem)
Let T be any computably enumerable theory extending EA. Foreach σ ∈ Σ0
1 we have that there is some ρ ∈ Σ01 so that
EA ` 3T> →(σ ↔ 2Tρ
).
I Proof.Consider the fixpoint ρ for which EA ` ρ ↔ (σ ≤ 2ρ).
I This shows us that we can express a syntactical class usingprovability logics!
I We wish to stretch this further
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Theorem (FGH theorem)
Let T be any computably enumerable theory extending EA. Foreach σ ∈ Σ0
1 we have that there is some ρ ∈ Σ01 so that
EA ` 3T> →(σ ↔ 2Tρ
).
I Proof.Consider the fixpoint ρ for which EA ` ρ ↔ (σ ≤ 2ρ).
I This shows us that we can express a syntactical class usingprovability logics!
I We wish to stretch this further
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Theorem (FGH theorem)
Let T be any computably enumerable theory extending EA. Foreach σ ∈ Σ0
1 we have that there is some ρ ∈ Σ01 so that
EA ` 3T> →(σ ↔ 2Tρ
).
I Proof.Consider the fixpoint ρ for which EA ` ρ ↔ (σ ≤ 2ρ).
I This shows us that we can express a syntactical class usingprovability logics!
I We wish to stretch this further
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Visser’s proof used A→ A ≤ A for A ∈ Σ1.
LemmaLet A ∈ Σ0
n+1, then the schema A→ (A ≤ A) is over EA provablyequivalent to the least-number principle for ∆0
n formulas.
I This can be avoided
I so FGH is generalizable over a weak base theory.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Visser’s proof used A→ A ≤ A for A ∈ Σ1.
LemmaLet A ∈ Σ0
n+1, then the schema A→ (A ≤ A) is over EA provablyequivalent to the least-number principle for ∆0
n formulas.
I This can be avoided
I so FGH is generalizable over a weak base theory.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Visser’s proof used A→ A ≤ A for A ∈ Σ1.
LemmaLet A ∈ Σ0
n+1, then the schema A→ (A ≤ A) is over EA provablyequivalent to the least-number principle for ∆0
n formulas.
I This can be avoided
I so FGH is generalizable over a weak base theory.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I Visser’s proof used A→ A ≤ A for A ∈ Σ1.
LemmaLet A ∈ Σ0
n+1, then the schema A→ (A ≤ A) is over EA provablyequivalent to the least-number principle for ∆0
n formulas.
I This can be avoided
I so FGH is generalizable over a weak base theory.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I By [n]TrueT we will denote the formalization of the predicate
“provable in T together with all true Π0n sentences”.
I It is well-known that for recursive theories T we can write[n]True
T by a Σ0n+1-formula.
I Also, we have provable Σ0n completeness for these predicates,
that is:
I propositionLet T be a computable theory extending EA and let φ be aΣ0
n+1 formula. We have that
EA ` φ→ [n]TrueT φ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I By [n]TrueT we will denote the formalization of the predicate
“provable in T together with all true Π0n sentences”.
I It is well-known that for recursive theories T we can write[n]True
T by a Σ0n+1-formula.
I Also, we have provable Σ0n completeness for these predicates,
that is:
I propositionLet T be a computable theory extending EA and let φ be aΣ0
n+1 formula. We have that
EA ` φ→ [n]TrueT φ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I By [n]TrueT we will denote the formalization of the predicate
“provable in T together with all true Π0n sentences”.
I It is well-known that for recursive theories T we can write[n]True
T by a Σ0n+1-formula.
I Also, we have provable Σ0n completeness for these predicates,
that is:
I propositionLet T be a computable theory extending EA and let φ be aΣ0
n+1 formula. We have that
EA ` φ→ [n]TrueT φ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I By [n]TrueT we will denote the formalization of the predicate
“provable in T together with all true Π0n sentences”.
I It is well-known that for recursive theories T we can write[n]True
T by a Σ0n+1-formula.
I Also, we have provable Σ0n completeness for these predicates,
that is:
I propositionLet T be a computable theory extending EA and let φ be aΣ0
n+1 formula. We have that
EA ` φ→ [n]TrueT φ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
TheoremLet T be any computably enumerable theory extending EA and letn < ω. For each σ ∈ Σ0
n+1 we have that there is some ρn ∈ Σ0n+1
so thatEA ` 〈n〉True
T > →(σ ↔ [n]True
T ρn
).
I proof The proof runs entirely analogue to the proof of theclassical FGH theorem.
I Thus, for each number n we consider the fixpoint ρn so thatEA ` ρn ↔ (σ ≤ [n]True
T ρn).I Just using Σ0
n+1-completeness now
I Corollary
Let T be a c.e. theory extending EA and let n ∈ N. For eachformulas ϕ,ψ there is some σ ∈ Σ0
n+1 so that
T ` ([n]TrueT ϕ ∨ [n]True
T ψ) ↔ [n]TrueT σ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
TheoremLet T be any computably enumerable theory extending EA and letn < ω. For each σ ∈ Σ0
n+1 we have that there is some ρn ∈ Σ0n+1
so thatEA ` 〈n〉True
T > →(σ ↔ [n]True
T ρn
).
I proof The proof runs entirely analogue to the proof of theclassical FGH theorem.
I Thus, for each number n we consider the fixpoint ρn so thatEA ` ρn ↔ (σ ≤ [n]True
T ρn).I Just using Σ0
n+1-completeness now
I Corollary
Let T be a c.e. theory extending EA and let n ∈ N. For eachformulas ϕ,ψ there is some σ ∈ Σ0
n+1 so that
T ` ([n]TrueT ϕ ∨ [n]True
T ψ) ↔ [n]TrueT σ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
TheoremLet T be any computably enumerable theory extending EA and letn < ω. For each σ ∈ Σ0
n+1 we have that there is some ρn ∈ Σ0n+1
so thatEA ` 〈n〉True
T > →(σ ↔ [n]True
T ρn
).
I proof The proof runs entirely analogue to the proof of theclassical FGH theorem.
I Thus, for each number n we consider the fixpoint ρn so thatEA ` ρn ↔ (σ ≤ [n]True
T ρn).
I Just using Σ0n+1-completeness now
I Corollary
Let T be a c.e. theory extending EA and let n ∈ N. For eachformulas ϕ,ψ there is some σ ∈ Σ0
n+1 so that
T ` ([n]TrueT ϕ ∨ [n]True
T ψ) ↔ [n]TrueT σ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
TheoremLet T be any computably enumerable theory extending EA and letn < ω. For each σ ∈ Σ0
n+1 we have that there is some ρn ∈ Σ0n+1
so thatEA ` 〈n〉True
T > →(σ ↔ [n]True
T ρn
).
I proof The proof runs entirely analogue to the proof of theclassical FGH theorem.
I Thus, for each number n we consider the fixpoint ρn so thatEA ` ρn ↔ (σ ≤ [n]True
T ρn).I Just using Σ0
n+1-completeness now
I Corollary
Let T be a c.e. theory extending EA and let n ∈ N. For eachformulas ϕ,ψ there is some σ ∈ Σ0
n+1 so that
T ` ([n]TrueT ϕ ∨ [n]True
T ψ) ↔ [n]TrueT σ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
TheoremLet T be any computably enumerable theory extending EA and letn < ω. For each σ ∈ Σ0
n+1 we have that there is some ρn ∈ Σ0n+1
so thatEA ` 〈n〉True
T > →(σ ↔ [n]True
T ρn
).
I proof The proof runs entirely analogue to the proof of theclassical FGH theorem.
I Thus, for each number n we consider the fixpoint ρn so thatEA ` ρn ↔ (σ ≤ [n]True
T ρn).I Just using Σ0
n+1-completeness now
I Corollary
Let T be a c.e. theory extending EA and let n ∈ N. For eachformulas ϕ,ψ there is some σ ∈ Σ0
n+1 so that
T ` ([n]TrueT ϕ ∨ [n]True
T ψ) ↔ [n]TrueT σ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
TheoremLet T be any computably enumerable theory extending EA and letn < ω. For each σ ∈ Σ0
n+1 we have that there is some ρn ∈ Σ0n+1
so thatEA ` 〈n〉True
T > →(σ ↔ [n]True
T ρn
).
I proof The proof runs entirely analogue to the proof of theclassical FGH theorem.
I Thus, for each number n we consider the fixpoint ρn so thatEA ` ρn ↔ (σ ≤ [n]True
T ρn).I Just using Σ0
n+1-completeness now
I Corollary
Let T be a c.e. theory extending EA and let n ∈ N. For eachformulas ϕ,ψ there is some σ ∈ Σ0
n+1 so that
T ` ([n]TrueT ϕ ∨ [n]True
T ψ) ↔ [n]TrueT σ.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I The [n]True predicates tie up with the arithmetical hierarchy:
I LemmaLet T be any c.e. theory and let A ⊆ N. The following areequivalent
1. A is c.e. in ∅(n);
2. A is 1-1 reducible to ∅(n+1);
3. A is definable on the standard model by a Σ0n+1 formula;
4. A is definable on the standard model by a formula of the form[n]True
T ρ(x);
5. A is definable on the standard model by a formula of the form[n]True
T ρ(x) where ρ(x) ∈ Σ0n+1;
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I The [n]True predicates tie up with the arithmetical hierarchy:
I LemmaLet T be any c.e. theory and let A ⊆ N. The following areequivalent
1. A is c.e. in ∅(n);
2. A is 1-1 reducible to ∅(n+1);
3. A is definable on the standard model by a Σ0n+1 formula;
4. A is definable on the standard model by a formula of the form[n]True
T ρ(x);
5. A is definable on the standard model by a formula of the form[n]True
T ρ(x) where ρ(x) ∈ Σ0n+1;
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I The [n]True predicates tie up with the arithmetical hierarchy:
I LemmaLet T be any c.e. theory and let A ⊆ N. The following areequivalent
1. A is c.e. in ∅(n);
2. A is 1-1 reducible to ∅(n+1);
3. A is definable on the standard model by a Σ0n+1 formula;
4. A is definable on the standard model by a formula of the form[n]True
T ρ(x);
5. A is definable on the standard model by a formula of the form[n]True
T ρ(x) where ρ(x) ∈ Σ0n+1;
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I The [n]True predicates tie up with the arithmetical hierarchy:
I LemmaLet T be any c.e. theory and let A ⊆ N. The following areequivalent
1. A is c.e. in ∅(n);
2. A is 1-1 reducible to ∅(n+1);
3. A is definable on the standard model by a Σ0n+1 formula;
4. A is definable on the standard model by a formula of the form[n]True
T ρ(x);
5. A is definable on the standard model by a formula of the form[n]True
T ρ(x) where ρ(x) ∈ Σ0n+1;
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I The [n]True predicates tie up with the arithmetical hierarchy:
I LemmaLet T be any c.e. theory and let A ⊆ N. The following areequivalent
1. A is c.e. in ∅(n);
2. A is 1-1 reducible to ∅(n+1);
3. A is definable on the standard model by a Σ0n+1 formula;
4. A is definable on the standard model by a formula of the form[n]True
T ρ(x);
5. A is definable on the standard model by a formula of the form[n]True
T ρ(x) where ρ(x) ∈ Σ0n+1;
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I The [n]True predicates tie up with the arithmetical hierarchy:
I LemmaLet T be any c.e. theory and let A ⊆ N. The following areequivalent
1. A is c.e. in ∅(n);
2. A is 1-1 reducible to ∅(n+1);
3. A is definable on the standard model by a Σ0n+1 formula;
4. A is definable on the standard model by a formula of the form[n]True
T ρ(x);
5. A is definable on the standard model by a formula of the form[n]True
T ρ(x) where ρ(x) ∈ Σ0n+1;
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I The [n]True predicates tie up with the arithmetical hierarchy:
I LemmaLet T be any c.e. theory and let A ⊆ N. The following areequivalent
1. A is c.e. in ∅(n);
2. A is 1-1 reducible to ∅(n+1);
3. A is definable on the standard model by a Σ0n+1 formula;
4. A is definable on the standard model by a formula of the form[n]True
T ρ(x);
5. A is definable on the standard model by a formula of the form[n]True
T ρ(x) where ρ(x) ∈ Σ0n+1;
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I The [n]True predicates tie up with the arithmetical hierarchy:
I LemmaLet T be any c.e. theory and let A ⊆ N. The following areequivalent
1. A is c.e. in ∅(n);
2. A is 1-1 reducible to ∅(n+1);
3. A is definable on the standard model by a Σ0n+1 formula;
4. A is definable on the standard model by a formula of the form[n]True
T ρ(x);
5. A is definable on the standard model by a formula of the form[n]True
T ρ(x) where ρ(x) ∈ Σ0n+1;
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I [n +1]OmegaT ϕ := ∃ψ
(∀x [n]Omega
T ψ(x) ∧ 2T (∀x ψ(x)→ ϕ))
I [n]OmegaT is a Σ0
2n+1-formula.
I LemmaLet T be a computable theory extending EA and let φ be a Σ0
2n+1
formula. We have that
EA ` φ→ [n]OmegaT φ.
Proof.By an external induction on n where each inductive step requiresthe application of an additional omega-rule.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
Corollary
Let T be any computably enumerable theory extending EA and letn < ω. For each σ ∈ Σ0
2n+1 we have that there is some ρn ∈ Σ02n+1
so thatEA ` 〈n〉Omega
T > →(σ ↔ [n]Omega
T ρn
).
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I LemmaLet T be any c.e. theory, let n be a natural number, and letA ⊆ N. The following are equivalent
1. A is definable on the standard model by a Σ02n+1 formula;
2. A is definable on the standard model by a formula of the form[n]Omega
T ρ(x);
I Runs out of phase!
I We wish to use the best of both worlds
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I LemmaLet T be any c.e. theory, let n be a natural number, and letA ⊆ N. The following are equivalent
1. A is definable on the standard model by a Σ02n+1 formula;
2. A is definable on the standard model by a formula of the form[n]Omega
T ρ(x);
I Runs out of phase!
I We wish to use the best of both worlds
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I LemmaLet T be any c.e. theory, let n be a natural number, and letA ⊆ N. The following are equivalent
1. A is definable on the standard model by a Σ02n+1 formula;
2. A is definable on the standard model by a formula of the form[n]Omega
T ρ(x);
I Runs out of phase!
I We wish to use the best of both worlds
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I LemmaLet T be any c.e. theory, let n be a natural number, and letA ⊆ N. The following are equivalent
1. A is definable on the standard model by a Σ02n+1 formula;
2. A is definable on the standard model by a formula of the form[n]Omega
T ρ(x);
I Runs out of phase!
I We wish to use the best of both worlds
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I LemmaLet T be any c.e. theory, let n be a natural number, and letA ⊆ N. The following are equivalent
1. A is definable on the standard model by a Σ02n+1 formula;
2. A is definable on the standard model by a formula of the form[n]Omega
T ρ(x);
I Runs out of phase!
I We wish to use the best of both worlds
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I LemmaLet T be any c.e. theory, let n be a natural number, and letA ⊆ N. The following are equivalent
1. A is definable on the standard model by a Σ02n+1 formula;
2. A is definable on the standard model by a formula of the form[n]Omega
T ρ(x);
I Runs out of phase!
I We wish to use the best of both worlds
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I
[0]2Tφ := 2Tφ, and
[n + 1]2Tφ := 2Tφ ∨ ∃ψ∨
0≤m≤n
(〈m〉2Tψ ∧ 2(〈m〉2Tψ → φ)
).
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
proposition
Let T be a sound c.e. theory extending EA. We have for all n ∈ Nthat
1. EA ` ∀ϕ([n]2Tϕ → [n]True
T ϕ);
2. EA ` 〈n〉TrueT > → ∀ϕ
([n + 1]2Tϕ↔ [n + 1]True
T ϕ);
3. EA ` [n]TrueT
(∀ϕ([n]2Tϕ↔ [n]True
T ϕ) )
;
4. N |= ∀ϕ([n]2Tϕ↔ [n]True
T ϕ).
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
TheoremLet T be a c.e. theory. We have for all A ⊆ N that the followingare equivalent
1. A is c.e. in ∅(n);
2. A is 1-1 reducible to ∅(n+1);
3. A is definable on the standard model by a formula of the form[n]2Tρ(x);
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I We now generalize to the transfinte
I fixing a well-behaved ordinal
I TheoremThe logic GLPΛ is sound for strong enough theories T under theinterpretation 2 7→ [λ]2,ΛT .
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I We now generalize to the transfinte
I fixing a well-behaved ordinal
I TheoremThe logic GLPΛ is sound for strong enough theories T under theinterpretation 2 7→ [λ]2,ΛT .
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I We now generalize to the transfinte
I fixing a well-behaved ordinal
I TheoremThe logic GLPΛ is sound for strong enough theories T under theinterpretation 2 7→ [λ]2,ΛT .
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics
I We now generalize to the transfinte
I fixing a well-behaved ordinal
I TheoremThe logic GLPΛ is sound for strong enough theories T under theinterpretation 2 7→ [λ]2,ΛT .
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
DefinitionLet T be a c.e. theory. We define
- ∆20 := Σ2
0 := Π20 := ∆0
0;
- Σ2α+1 = Σ2
α ∪ Π2α ∪ {[α]2Tϕ(x) | ϕ(x) ∈ Form} for α > 0;
- Π2α+1 = Σ2
α ∪ Π2α ∪ {〈α〉2Tϕ(x) | ϕ(x) ∈ Form} for α > 0;
- Σ2λ := Π2
λ :=⋃α<λ
Σ2α for λ ∈ Lim.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
I Theorem/conjecture Let T be any c.e. theory, let ξ < Λ fora natural ordinal notation system, and let A ⊆ N. Thefollowing are equivalent
1. A is c.e. in ∅(ξ);
2. A is 1-1 reducible to ∅(ξ+1);
3. A is definable on the standard model by a formula of the form[ξ]2Tρ(x);
I No longer runs out of phase
I Theorem/conjecture: So all the stuff about Turingprogressions can be generalized in a straight-forward fashion.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
I Theorem/conjecture Let T be any c.e. theory, let ξ < Λ fora natural ordinal notation system, and let A ⊆ N. Thefollowing are equivalent
1. A is c.e. in ∅(ξ);
2. A is 1-1 reducible to ∅(ξ+1);
3. A is definable on the standard model by a formula of the form[ξ]2Tρ(x);
I No longer runs out of phase
I Theorem/conjecture: So all the stuff about Turingprogressions can be generalized in a straight-forward fashion.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
I Theorem/conjecture Let T be any c.e. theory, let ξ < Λ fora natural ordinal notation system, and let A ⊆ N. Thefollowing are equivalent
1. A is c.e. in ∅(ξ);
2. A is 1-1 reducible to ∅(ξ+1);
3. A is definable on the standard model by a formula of the form[ξ]2Tρ(x);
I No longer runs out of phase
I Theorem/conjecture: So all the stuff about Turingprogressions can be generalized in a straight-forward fashion.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
I Theorem/conjecture Let T be any c.e. theory, let ξ < Λ fora natural ordinal notation system, and let A ⊆ N. Thefollowing are equivalent
1. A is c.e. in ∅(ξ);
2. A is 1-1 reducible to ∅(ξ+1);
3. A is definable on the standard model by a formula of the form[ξ]2Tρ(x);
I No longer runs out of phase
I Theorem/conjecture: So all the stuff about Turingprogressions can be generalized in a straight-forward fashion.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
I Theorem/conjecture Let T be any c.e. theory, let ξ < Λ fora natural ordinal notation system, and let A ⊆ N. Thefollowing are equivalent
1. A is c.e. in ∅(ξ);
2. A is 1-1 reducible to ∅(ξ+1);
3. A is definable on the standard model by a formula of the form[ξ]2Tρ(x);
I No longer runs out of phase
I Theorem/conjecture: So all the stuff about Turingprogressions can be generalized in a straight-forward fashion.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
I Theorem/conjecture Let T be any c.e. theory, let ξ < Λ fora natural ordinal notation system, and let A ⊆ N. Thefollowing are equivalent
1. A is c.e. in ∅(ξ);
2. A is 1-1 reducible to ∅(ξ+1);
3. A is definable on the standard model by a formula of the form[ξ]2Tρ(x);
I No longer runs out of phase
I Theorem/conjecture: So all the stuff about Turingprogressions can be generalized in a straight-forward fashion.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
DefinitionLet Γ be a class of formulas. For ordinals α, β < Λ and T ac.e. theory we define β−RFNΛ
T (Γ) to be the schema [β]2Tϕ→ ϕ forϕ ∈ Γ.
Instead of writing 0−RFNΛT (Γ) we shall just write RFNΛ
T (Γ).
We can now easily state and prove various equivalences betweenconsistency statements and reflection principles.
Joost J. Joosten On hyper-arithmetic reflection principles
Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic
Hyper-arithmetic reflection
Theorem
Let T be a c.e. theory containing ECA0.
1. ECA0 ` RFNΛT (Π2
α+1) ≡ 〈α〉2T>;
2. For β ≤ α, we have ECA0 ` β−RFNΛT (Π2
α+1) ≡ 〈α〉2T>;
3. For β > α we have that ECA0 ` β−RFNΛT (Π2
α+1) ≡ 〈β〉2T>;
4. For β > α we have thatECA0 ` β−RFNΛ
T (Π2α+1) ≡ 〈max{α, β}〉2T>.
Joost J. Joosten On hyper-arithmetic reflection principles