Topological Semantics for Provability Logics - An...

Topological Semantics forProvability Logics

An Overview

Thomas IcardStanford University

September 11, 2015Berkeley Logic Colloquium

T. F. Icard: Topological Semantics for Provability Logics, An Overview 1

Outline

1. Review of Provability Logic GL

2. Polymodal Provability Logic GLP

3. The Closed Fragment of GLP

4. Ignatiev’s Model and the Canonical Model

5. Topological Completeness using ε0

6. Topological Completeness of GLB

7. More recent developments

Modal Logic of Provability

ϕ ::= P | ϕ ∧ ϕ | ¬ϕ | 2ϕ

2ϕ ≈ ‘It is provable that ϕ’

Let L be an arithmetical language and T an L-theory with anencoding of a (standard) provability predicate Prov(·), so that

N |= Prov(pAq) iff T ` A .

A realization is a function from modal to arithmetical formulas:

(ϕ ∧ ψ)∗ = ϕ∗ ∧ ψ∗ (¬ϕ)∗ = ¬ϕ∗ (2ϕ)∗ = Prov(pϕ∗q) .

ϕ ::= P | ϕ ∧ ϕ | ¬ϕ | 2ϕ

(ϕ ∧ ψ)∗ = ϕ∗ ∧ ψ∗ (¬ϕ)∗ = ¬ϕ∗ (2ϕ)∗ = Prov(pϕ∗q) .

ϕ ::= P | ϕ ∧ ϕ | ¬ϕ | 2ϕ

(ϕ ∧ ψ)∗ = ϕ∗ ∧ ψ∗ (¬ϕ)∗ = ¬ϕ∗ (2ϕ)∗ = Prov(pϕ∗q) .

Theorem (Solovay 1976)

ϕ is GL-valid, if and only if it is PA-provable under all realizations.

Axioms and rules of GL:

1. Modus ponens ;

2. Propositional tautologies ;

3. Necessitation:ϕ2ϕ ;

4. 2(ϕ→ ψ)→ (2ϕ→ 2ψ) ;

5. 2(2ϕ→ ϕ)→ 2ϕ .

Theorem (Segerberg 1971)

GL is sound and complete w.r.t. finite trees.

1. Modus ponens ;

4. 2(ϕ→ ψ)→ (2ϕ→ 2ψ) ;

5. 2(2ϕ→ ϕ)→ 2ϕ .

1. Modus ponens ;

4. 2(ϕ→ ψ)→ (2ϕ→ 2ψ) ;

5. 2(2ϕ→ ϕ)→ 2ϕ .

Polymodal Provability Logic

Consider an increasing sequence of stronger provability predicates:

Prov0 Prov1 Prov2 Prov3 . . .

where Provn(pAq) means A is provable from T together with alltrue Πn sentences of L (cf. ω-provable).

ϕ ::= P | ϕ ∧ ϕ | ¬ϕ | [0]ϕ | [1]ϕ | [2]ϕ | . . .

Theorem (Japaridze 1985)

The logic GLP is arithmetically complete:

1. GL for each modality [n] ;

2. [n]ϕ→ [n+ 1]ϕ ;

3. 〈n〉ϕ→ [n+ 1]〈n〉ϕ .

ϕ ::= P | ϕ ∧ ϕ | ¬ϕ | [0]ϕ | [1]ϕ | [2]ϕ | . . .

2. [n]ϕ→ [n+ 1]ϕ ;

3. 〈n〉ϕ→ [n+ 1]〈n〉ϕ .

ϕ ::= P | ϕ ∧ ϕ | ¬ϕ | [0]ϕ | [1]ϕ | [2]ϕ | . . .

2. [n]ϕ→ [n+ 1]ϕ ;

3. 〈n〉ϕ→ [n+ 1]〈n〉ϕ .

Applications in Ordinal Analysis

I The closed fragment GLP0 has been used in ordinal analysis.

I The L-T algebra for GLP0 possess a natural ordering:

[ϕ] <0 [ψ] iff `GLP ψ→ 〈0〉ϕ .

I It turns out <0 has order-type ε0, the ‘proof-theoretic ordinal’for PA. (Recall ε0 is the least fixed point of ωα = β.)

I Beklemishev [2004] showed how induction up to ε0, with thisalgebra providing a notation system, can be used to proveconsistency of PA.

[ϕ] <0 [ψ] iff `GLP ψ→ 〈0〉ϕ .

Frame Incompleteness

Consider a relational frame F = (W , Rnn<ω):

(a) F [n]ϕ→ [n+ 1]ϕ, iff Rn+1 ⊆ Rn ;

(b) F 〈n〉ϕ→ [n+ 1]〈n〉ϕ, iff R−1n (X ) is Rn+1-upward closed ;

Suppose F GLP and R1 6= ∅, e.g., xR1y . Then by (a), xR0y ,and by (b), yR0y , contradicting converse-well-foundedness of R0.

There is no non-trivial frame semantics for GLP.

This motivates the search for other classes of models.

(a) F [n]ϕ→ [n+ 1]ϕ, iff Rn+1 ⊆ Rn ;

I For GLP0, Ignatiev [1993] defined a single universal frame.

I It is in fact based on ε0.

I Recall the Cantor Normal Form of an ordinal κ(where λi ≥ λj for i > j):

ωλk + · · ·+ ωλ0 .

Define l(κ) = λ0.

I Let Ω be the set of ω-sequences (α1, α2, . . . ), with αi < ε0.

I Ignatiev’s frame U = (U, Rnn<ω) is defined:

U = α ∈ Ω : αn+1 ≤ l(αn)αRnβ iff αn > βn and αm = βm for m < n .

ωλk + · · ·+ ωλ0 .

Define l(κ) = λ0.

ωλk + · · ·+ ωλ0 .

Define l(κ) = λ0.

ωλk + · · ·+ ωλ0 .

Define l(κ) = λ0.

ωλk + · · ·+ ωλ0 .

Define l(κ) = λ0.

U = α ∈ Ω : αn+1 ≤ l(αn)

αRnβ iff αn > βn and αm = βm for m < n .

ωλk + · · ·+ ωλ0 .

Define l(κ) = λ0.

I How does U relate to the canonical model of GLP0?

I In Icard [2011] this question was answered using tools fromtopology and the theory of general descriptive frames.

(C0,C )algebra // T0

Lu.f .f .

(U ,A)

algebra

Lu.f .f .

(U ,A)

algebra

Lu.f .f .

(U ,A)

algebra

vvU+++

Topological Semantics

I Topological semantics of modal logic goes back to McKinseyand Tarski [1944], who interpreted 3 as topological closure.

I This is inappropriate for provability logic, since X ⊆ cl(X ),i.e., 2ϕ→ ϕ is valid on any topological space.

I Instead, we can interpret 3 as derivative. Analogous toMcKinsey & Tarski’s result on S4, Esakia showed that theclass of all spaces is axiomatized by wK4, replacing 4 with:

2ϕ ∧ ϕ→ 22ϕ .

I Esakia also showed K4 axiomatizes the Td -spaces.

2ϕ ∧ ϕ→ 22ϕ .

I The most important result from Esakia [1981] is completenessof GL w.r.t. the scattered spaces:

Every subspace has an isolated point.

I Lob’s axiom: 3ϕ→ 3(ϕ ∧ ¬3ϕ). Topologically speaking:

d(X ) = d(X \ d(X )

I A canonical example of a scattered space is any ordinal withthe interval topology (sub-basis of open rays).

Theorem (Abashidze 1985; Blass 1990)

GL is complete with respect to ωω with interval topology.

I Lob’s axiom: 3ϕ→ 3(ϕ ∧ ¬3ϕ).

Topologically speaking:

d(X ) = d(X \ d(X )

I GLP0 is also complete w.r.t. a polytopological space definedon ε0.

I Infinitely many finer and finer topologies on ε0:

• τ0 is the ‘left topology’ ;

• τ1 is the interval topology (a.k.a. ‘order topology’) ;

• τ2 adds to τ1 subbasic open sets α : l(α) > β

I In general, τn+1 adds to τn subbasic sets

α : ln(α) > β.

Theorem (Icard 2011)

GLP0 is complete with respect to (ε0, τnn<ω).

α : ln(α) > β.

I In very recent work, Fernandez-Duque and Joosten [2013]generalize the previous theorem to arbitrary ordinals.

I L0ω is a special case of L0

Λ, and the ‘logarithm’ function lmust be able to iterate into the transfinite.

I They generalize Ignatiev’s ‘universal frame’ for L0Λ, and by

eliminating ‘non-root’ points and generalizing the topologiesin the previous slide, they obtain topological completeness.

Theorem (Fernandez-Duque and Joosten 2013)

GLP0Λ is complete w.r.t. frame and topological semantics.

GLP-spaces

DefinitionA polytopological space (X , τnn<ω) is a GLP-space if, for all n:

I τn is scattered ;

I τn ⊆ τn+1 ;

I dn(X ) ∈ τn+1 .

Are there non-trivial GLP-spaces?

Given τn there is a coarsest τn+1 satisfying the above conditions.

GLP-spaces

I τn ⊆ τn+1 ;

I dn(X ) ∈ τn+1 .

GLP-spaces

I τn ⊆ τn+1 ;

I dn(X ) ∈ τn+1 .

GLP-spaces

I Beklemishev et al. [2010] proved topological completeness ofGLB w.r.t. GLP-spaces, using relational semanticsestablished in Beklemishev [2010].

I τ0 is the upset topology (standard GL model).

I τ1 is the interval topology.

GLP-spaces

Ordinal Models

Fact (Beklemishev et al. 2010)

If (X , τ0, τ1) is a GLB-space and τ0 is first-countable andHausdorff, then τ1 must be discrete.

Fact (Beklemishev 2011)

If Ω ≥ ℵω and V = L, then GLB is complete w.r.t. (Ω, τ0, τ1),where τ0 is interval topology and τ1 is club topology.

I If cf (α) ≤ ℵ0, then α is isolated ;

I If cf (α) > ℵ0, then any neighborhood of α contains a club.

I Blass [1990] showed incompleteness of GL w.r.t. τ1 isequiconsistent with existence of a weakly Mahlo cardinal.

GLP-spaces

Ordinal Models

GLP-spaces

Ordinal Models

GLP-spaces

Ordinal Models

GLP-spaces

I Even more recently Beklemishev and Gabelaia [2013] havesettled a number of questions.

I They note it is independent of ZFC whether τ2 can benon-trivial on an ordinal space, but show there areassumptions that guarantee all τn are non-discrete.

I The question of whether there are set-theoretic assumptionsthat would give ordinal completeness of GLP is open.

I Beklemishev and Gabelaia [2013] do show topologicalcompleteness of GLP w.r.t. general GLP-spaces, but (in theirown words) “it is not an example of a natural GLP-space.”

GLP-spaces

Conclusion

Summary

I The study of topological models of provability logic combineswork in proof theory, topology, algebra, set theory, and modallogic, exposing some mathematically interesting structures.

I It also promises a better grip on the polymodal provabilitylogic GLP, providing a simple non-arithmetical interpretation.

I Current work in Moscow and Barcelona is focused on usingthese models to extend Beklemishev’s original ordinal analysisof PA to stronger theories and larger ordinals.

Conclusion

Summary

Conclusion

Summary

Conclusion

Thanks for your attention!

References

Abashidze, M. (1985). Ordinal completeness of the Godel-Lobmodal system (Russian). In Intensional logics and the logicalstructure of theories, pages 49–73. Metsniereba, Tbilisi.

Beklemishev, L. D. (2004). Provability algebras and proof-theoreticordinals, I. Annals of Pure and Applied Logic, 128(1-3):103–123.

Beklemishev, L. D. (2010). Kripke semantics for provability logicGLP. Annals of Pure and Applied Logic, 161:756–774.

Beklemishev, L. D. (2011). Ordinal completeness of bimodalprovability logic GLB. In Bezhanishvili, N., editor, Logic,Language, and Computation, TbiLLC 2009, pages 1–15.Springer Lecture Notes in AI.

Beklemishev, L. D., Bezhanishvili, G., and Icard, T. F. (2010). Ontopological models of GLP. In Schindler, R., editor, Ways ofProof Theory: Festschrift for Wolfram Pohlers, pages 135–155.Ontos Series in Mathematical Logic.

Beklemishev, L. D. and Gabelaia, D. (2013). Topological

References

completeness of the provability GLP. Annals of Pure andApplied Logic, 164:1201–1223.

Blass, A. (1990). Infinitary combinatorics and modal logic. Journalof Symbolic Logic, 55(2):761–778.

Esakia, L. (1981). Diagonal constructions, Lob’s formula andCantor’s scattered space (Russian). In Studies in logic andsemantics, pages 128–143. Metsniereba, Tbilisi.

Fernandez-Duque, D. and Joosten, J. J. (2013). Models oftransfinite provability logic. Journal of Symbolic Logic,78(2):543–561.

Icard, T. F. (2008). Models of the Polymodal Provability Logic.Master’s thesis, ILLC, Universiteit van Amsterdam.

Icard, T. F. (2011). A topological study of the closed fragment ofGLP. Journal of Logic and Computation, 21(4):683–696.

Ignatiev, K. N. (1993). On strong provability predicates and theassociated modal logics. Journal of Symbolic Logic, 58:249–290.

References

Japaridze, G. (1985). The polymodal logic of provability (Russian).In Intensional logics and the logical structure of theories, pages16–48. Metsniereba, Tbilisi.

McKinsey, J. C. C. and Tarski, A. (1944). The algebra of topology.Annals of Mathematics, 45:141–191.

Segerberg, K. (1971). An essay in classical modal logic. FilosofiskaForeningen och Filosofiska Institutionen vid Uppsala Universitet,Uppsala.

Solovay, R. (1976). Provability interpretations of modal logic.Israel Journal of Mathematics, 25:287–304.

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