Solvability cardinals

Márton Elekesemarci@renyi.hu

www.renyi.hu/˜emarci

Rényi Institute and Eötvös University

Descriptive Set Theory in Paris 2008

Joint work with M. Laczkovich.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Introduction

Definition

Let RR denote the set of R→ R functions. Let f ∈ RR.(∆hf )(x) = f (x + h)− f (x) (x ∈ R).

E.g. ∆hf = 0 iff f is periodic mod h.In a bit more generality:

Definition

A difference operator is a mapping D : RR → RR of the form

(Df )(x) =nX

i=1

ai f (x + bi ),

where ai and bi are real numbers. The set of difference operators is denoted by D.

E.g. ∆h ∈ D for every h ∈ R. This is by far the most important example for this talk.

Definition

A difference equation is a functional equation

Df = g,

where D is a difference operator, g is a given function and f is the unknown.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Introduction

Definition

Let RR denote the set of R→ R functions. Let f ∈ RR.(∆hf )(x) = f (x + h)− f (x) (x ∈ R).

E.g. ∆hf = 0 iff f is periodic mod h.In a bit more generality:

Definition

A difference operator is a mapping D : RR → RR of the form

(Df )(x) =nX

i=1

ai f (x + bi ),

where ai and bi are real numbers. The set of difference operators is denoted by D.

E.g. ∆h ∈ D for every h ∈ R. This is by far the most important example for this talk.

Definition

A difference equation is a functional equation

Df = g,

where D is a difference operator, g is a given function and f is the unknown.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Introduction

Definition

Let RR denote the set of R→ R functions. Let f ∈ RR.(∆hf )(x) = f (x + h)− f (x) (x ∈ R).

E.g. ∆hf = 0 iff f is periodic mod h.In a bit more generality:

Definition

A difference operator is a mapping D : RR → RR of the form

(Df )(x) =nX

i=1

ai f (x + bi ),

where ai and bi are real numbers. The set of difference operators is denoted by D.

E.g. ∆h ∈ D for every h ∈ R. This is by far the most important example for this talk.

Definition

A difference equation is a functional equation

Df = g,

where D is a difference operator, g is a given function and f is the unknown.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Introduction

Definition

Let RR denote the set of R→ R functions. Let f ∈ RR.(∆hf )(x) = f (x + h)− f (x) (x ∈ R).

E.g. ∆hf = 0 iff f is periodic mod h.In a bit more generality:

Definition

A difference operator is a mapping D : RR → RR of the form

(Df )(x) =nX

i=1

ai f (x + bi ),

where ai and bi are real numbers. The set of difference operators is denoted by D.

E.g. ∆h ∈ D for every h ∈ R. This is by far the most important example for this talk.

Definition

A difference equation is a functional equation

Df = g,

where D is a difference operator, g is a given function and f is the unknown.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Introduction

Definition

Let RR denote the set of R→ R functions. Let f ∈ RR.(∆hf )(x) = f (x + h)− f (x) (x ∈ R).

E.g. ∆hf = 0 iff f is periodic mod h.In a bit more generality:

Definition

A difference operator is a mapping D : RR → RR of the form

(Df )(x) =nX

i=1

ai f (x + bi ),

where ai and bi are real numbers. The set of difference operators is denoted by D.

E.g. ∆h ∈ D for every h ∈ R. This is by far the most important example for this talk.

Definition

A difference equation is a functional equation

Df = g,

where D is a difference operator, g is a given function and f is the unknown.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Definition

A system of difference equations is

Di f = gi (i ∈ I),

where I is an arbitrary set of indices.

It is not very hard to show that a system of difference equations is solvable iff everyfinite subsystem is solvable.However, if we are interested e.g. in continuous solutions then this result is no longertrue. This motivates the following definition.

Definition

Let F ⊂ RR be a class of real functions. The solvability cardinal of F is the minimal

cardinal sc(F) with the property that if every subsystem of size less than sc(F) of asystem of difference equations has a solution in F then the whole system has asolution in F .

E.g. sc(RR) ≤ ω is a reformulation of the above cited result.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Definition

A system of difference equations is

Di f = gi (i ∈ I),

where I is an arbitrary set of indices.

It is not very hard to show that a system of difference equations is solvable iff everyfinite subsystem is solvable.However, if we are interested e.g. in continuous solutions then this result is no longertrue. This motivates the following definition.

Definition

Let F ⊂ RR be a class of real functions. The solvability cardinal of F is the minimal

cardinal sc(F) with the property that if every subsystem of size less than sc(F) of asystem of difference equations has a solution in F then the whole system has asolution in F .

E.g. sc(RR) ≤ ω is a reformulation of the above cited result.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Definition

A system of difference equations is

Di f = gi (i ∈ I),

where I is an arbitrary set of indices.

It is not very hard to show that a system of difference equations is solvable iff everyfinite subsystem is solvable.However, if we are interested e.g. in continuous solutions then this result is no longertrue. This motivates the following definition.

Definition

Let F ⊂ RR be a class of real functions. The solvability cardinal of F is the minimal

cardinal sc(F) with the property that if every subsystem of size less than sc(F) of asystem of difference equations has a solution in F then the whole system has asolution in F .

E.g. sc(RR) ≤ ω is a reformulation of the above cited result.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Definition

A system of difference equations is

Di f = gi (i ∈ I),

where I is an arbitrary set of indices.

It is not very hard to show that a system of difference equations is solvable iff everyfinite subsystem is solvable.However, if we are interested e.g. in continuous solutions then this result is no longertrue. This motivates the following definition.

Definition

Let F ⊂ RR be a class of real functions. The solvability cardinal of F is the minimal

cardinal sc(F) with the property that if every subsystem of size less than sc(F) of asystem of difference equations has a solution in F then the whole system has asolution in F .

E.g. sc(RR) ≤ ω is a reformulation of the above cited result.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Fact

For every F ⊂ RR we have sc(F) ≤ (2ω)+.

Proof. |D| = 2ω . Assume F ⊂ RR, S is system of difference equations, ∀S′ ⊂ S with

|S′| ≤ 2ω is solvable in F . In particular, every pair of equations is solvable, hence forevery D ∈ D there is at most one g ∈ RR such that (Df = g) ∈ S. Therefore thecardinality of S is at most 2ω , and we are done. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Fact

For every F ⊂ RR we have sc(F) ≤ (2ω)+.

Proof. |D| = 2ω . Assume F ⊂ RR, S is system of difference equations, ∀S′ ⊂ S with

|S′| ≤ 2ω is solvable in F . In particular, every pair of equations is solvable, hence forevery D ∈ D there is at most one g ∈ RR such that (Df = g) ∈ S. Therefore thecardinality of S is at most 2ω , and we are done. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Fact

For every F ⊂ RR we have sc(F) ≤ (2ω)+.

Proof. |D| = 2ω . Assume F ⊂ RR, S is system of difference equations, ∀S′ ⊂ S with

|S′| ≤ 2ω is solvable in F . In particular, every pair of equations is solvable, hence forevery D ∈ D there is at most one g ∈ RR such that (Df = g) ∈ S. Therefore thecardinality of S is at most 2ω , and we are done. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Fact

For every F ⊂ RR we have sc(F) ≤ (2ω)+.

Proof. |D| = 2ω . Assume F ⊂ RR, S is system of difference equations, ∀S′ ⊂ S with

|S′| ≤ 2ω is solvable in F . In particular, every pair of equations is solvable, hence forevery D ∈ D there is at most one g ∈ RR such that (Df = g) ∈ S. Therefore thecardinality of S is at most 2ω , and we are done. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Fact

For every F ⊂ RR we have sc(F) ≤ (2ω)+.

Proof. |D| = 2ω . Assume F ⊂ RR, S is system of difference equations, ∀S′ ⊂ S with

|S′| ≤ 2ω is solvable in F . In particular, every pair of equations is solvable, hence forevery D ∈ D there is at most one g ∈ RR such that (Df = g) ∈ S. Therefore thecardinality of S is at most 2ω , and we are done. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Fact

For every F ⊂ RR we have sc(F) ≤ (2ω)+.

Proof. |D| = 2ω . Assume F ⊂ RR, S is system of difference equations, ∀S′ ⊂ S with

|S′| ≤ 2ω is solvable in F . In particular, every pair of equations is solvable, hence forevery D ∈ D there is at most one g ∈ RR such that (Df = g) ∈ S. Therefore thecardinality of S is at most 2ω , and we are done. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Theorem

sc(RR) = ωsc(Continuous functions) = ω1sc(Darboux functions) = (2ω)+

sc(Bounded functions) = ω1sc({f ∈ RR : |f | ≤ K}) = ωsc(Trigonometric polynomials) = ω1sc(Polynomials) = 3

Let T P denote the set of trigonometric polynomials, and let I be either the σ-ideal ofLebesgue nullsets or the meager sets. Also denote by BI the class of Lebesguemeasurable functions or the class of functions with the property of Baire.

Theorem

Suppose T P ⊂ F ⊂ F̃ ⊂ BI , where F̃ is a translation invariant linear subspace of BIsuch that whenever f ∈ F̃ and f = 0 I-a.e. then f = 0 everywhere. Then sc(F) = ω1.

As F̃ can be the class of continuous, or approximately continous functions or thederivatives, we obtain the following.

Corollary

If F equals any of the classes Cn(R), C∞(R), the class of real analytic functions,Lipschitz functions, derivatives, approximately continuous functions then sc(F) = ω1.The same is true for the subclasses {f ∈ F : f is bounded} where F is any of theclasses listed above.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Theorem

sc(RR) = ωsc(Continuous functions) = ω1sc(Darboux functions) = (2ω)+

sc(Bounded functions) = ω1sc({f ∈ RR : |f | ≤ K}) = ωsc(Trigonometric polynomials) = ω1sc(Polynomials) = 3

Let T P denote the set of trigonometric polynomials, and let I be either the σ-ideal ofLebesgue nullsets or the meager sets. Also denote by BI the class of Lebesguemeasurable functions or the class of functions with the property of Baire.

Theorem

Suppose T P ⊂ F ⊂ F̃ ⊂ BI , where F̃ is a translation invariant linear subspace of BIsuch that whenever f ∈ F̃ and f = 0 I-a.e. then f = 0 everywhere. Then sc(F) = ω1.

As F̃ can be the class of continuous, or approximately continous functions or thederivatives, we obtain the following.

Corollary

If F equals any of the classes Cn(R), C∞(R), the class of real analytic functions,Lipschitz functions, derivatives, approximately continuous functions then sc(F) = ω1.The same is true for the subclasses {f ∈ F : f is bounded} where F is any of theclasses listed above.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Theorem

sc(RR) = ωsc(Continuous functions) = ω1sc(Darboux functions) = (2ω)+

sc(Bounded functions) = ω1sc({f ∈ RR : |f | ≤ K}) = ωsc(Trigonometric polynomials) = ω1sc(Polynomials) = 3

Let T P denote the set of trigonometric polynomials, and let I be either the σ-ideal ofLebesgue nullsets or the meager sets. Also denote by BI the class of Lebesguemeasurable functions or the class of functions with the property of Baire.

Theorem

Suppose T P ⊂ F ⊂ F̃ ⊂ BI , where F̃ is a translation invariant linear subspace of BIsuch that whenever f ∈ F̃ and f = 0 I-a.e. then f = 0 everywhere. Then sc(F) = ω1.

As F̃ can be the class of continuous, or approximately continous functions or thederivatives, we obtain the following.

Corollary

If F equals any of the classes Cn(R), C∞(R), the class of real analytic functions,Lipschitz functions, derivatives, approximately continuous functions then sc(F) = ω1.The same is true for the subclasses {f ∈ F : f is bounded} where F is any of theclasses listed above.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Theorem

sc(RR) = ωsc(Continuous functions) = ω1sc(Darboux functions) = (2ω)+

sc(Bounded functions) = ω1sc({f ∈ RR : |f | ≤ K}) = ωsc(Trigonometric polynomials) = ω1sc(Polynomials) = 3

Let T P denote the set of trigonometric polynomials, and let I be either the σ-ideal ofLebesgue nullsets or the meager sets. Also denote by BI the class of Lebesguemeasurable functions or the class of functions with the property of Baire.

Theorem

Suppose T P ⊂ F ⊂ F̃ ⊂ BI , where F̃ is a translation invariant linear subspace of BIsuch that whenever f ∈ F̃ and f = 0 I-a.e. then f = 0 everywhere. Then sc(F) = ω1.

As F̃ can be the class of continuous, or approximately continous functions or thederivatives, we obtain the following.

Corollary

If F equals any of the classes Cn(R), C∞(R), the class of real analytic functions,Lipschitz functions, derivatives, approximately continuous functions then sc(F) = ω1.The same is true for the subclasses {f ∈ F : f is bounded} where F is any of theclasses listed above.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Theorem

sc(RR) = ωsc(Continuous functions) = ω1sc(Darboux functions) = (2ω)+

sc(Bounded functions) = ω1sc({f ∈ RR : |f | ≤ K}) = ωsc(Trigonometric polynomials) = ω1sc(Polynomials) = 3

Let T P denote the set of trigonometric polynomials, and let I be either the σ-ideal ofLebesgue nullsets or the meager sets. Also denote by BI the class of Lebesguemeasurable functions or the class of functions with the property of Baire.

Theorem

Suppose T P ⊂ F ⊂ F̃ ⊂ BI , where F̃ is a translation invariant linear subspace of BIsuch that whenever f ∈ F̃ and f = 0 I-a.e. then f = 0 everywhere. Then sc(F) = ω1.

As F̃ can be the class of continuous, or approximately continous functions or thederivatives, we obtain the following.

Corollary

If F equals any of the classes Cn(R), C∞(R), the class of real analytic functions,Lipschitz functions, derivatives, approximately continuous functions then sc(F) = ω1.The same is true for the subclasses {f ∈ F : f is bounded} where F is any of theclasses listed above.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Theorem

sc(RR) = ωsc(Continuous functions) = ω1sc(Darboux functions) = (2ω)+

sc(Bounded functions) = ω1sc({f ∈ RR : |f | ≤ K}) = ωsc(Trigonometric polynomials) = ω1sc(Polynomials) = 3

Let T P denote the set of trigonometric polynomials, and let I be either the σ-ideal ofLebesgue nullsets or the meager sets. Also denote by BI the class of Lebesguemeasurable functions or the class of functions with the property of Baire.

Theorem

Suppose T P ⊂ F ⊂ F̃ ⊂ BI , where F̃ is a translation invariant linear subspace of BIsuch that whenever f ∈ F̃ and f = 0 I-a.e. then f = 0 everywhere. Then sc(F) = ω1.

As F̃ can be the class of continuous, or approximately continous functions or thederivatives, we obtain the following.

Corollary

If F equals any of the classes Cn(R), C∞(R), the class of real analytic functions,Lipschitz functions, derivatives, approximately continuous functions then sc(F) = ω1.The same is true for the subclasses {f ∈ F : f is bounded} where F is any of theclasses listed above.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Theorem

sc(RR) = ωsc(Continuous functions) = ω1sc(Darboux functions) = (2ω)+

sc(Bounded functions) = ω1sc({f ∈ RR : |f | ≤ K}) = ωsc(Trigonometric polynomials) = ω1sc(Polynomials) = 3

Let T P denote the set of trigonometric polynomials, and let I be either the σ-ideal ofLebesgue nullsets or the meager sets. Also denote by BI the class of Lebesguemeasurable functions or the class of functions with the property of Baire.

Theorem

Suppose T P ⊂ F ⊂ F̃ ⊂ BI , where F̃ is a translation invariant linear subspace of BIsuch that whenever f ∈ F̃ and f = 0 I-a.e. then f = 0 everywhere. Then sc(F) = ω1.

As F̃ can be the class of continuous, or approximately continous functions or thederivatives, we obtain the following.

Corollary

If F equals any of the classes Cn(R), C∞(R), the class of real analytic functions,Lipschitz functions, derivatives, approximately continuous functions then sc(F) = ω1.The same is true for the subclasses {f ∈ F : f is bounded} where F is any of theclasses listed above.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Theorem

sc(RR) = ωsc(Continuous functions) = ω1sc(Darboux functions) = (2ω)+

sc(Bounded functions) = ω1sc({f ∈ RR : |f | ≤ K}) = ωsc(Trigonometric polynomials) = ω1sc(Polynomials) = 3

Let T P denote the set of trigonometric polynomials, and let I be either the σ-ideal ofLebesgue nullsets or the meager sets. Also denote by BI the class of Lebesguemeasurable functions or the class of functions with the property of Baire.

Theorem

Suppose T P ⊂ F ⊂ F̃ ⊂ BI , where F̃ is a translation invariant linear subspace of BIsuch that whenever f ∈ F̃ and f = 0 I-a.e. then f = 0 everywhere. Then sc(F) = ω1.

As F̃ can be the class of continuous, or approximately continous functions or thederivatives, we obtain the following.

Corollary

If F equals any of the classes Cn(R), C∞(R), the class of real analytic functions,Lipschitz functions, derivatives, approximately continuous functions then sc(F) = ω1.The same is true for the subclasses {f ∈ F : f is bounded} where F is any of theclasses listed above.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Theorem

sc(RR) = ωsc(Continuous functions) = ω1sc(Darboux functions) = (2ω)+

sc(Bounded functions) = ω1sc({f ∈ RR : |f | ≤ K}) = ωsc(Trigonometric polynomials) = ω1sc(Polynomials) = 3

Let T P denote the set of trigonometric polynomials, and let I be either the σ-ideal ofLebesgue nullsets or the meager sets. Also denote by BI the class of Lebesguemeasurable functions or the class of functions with the property of Baire.

Theorem

Suppose T P ⊂ F ⊂ F̃ ⊂ BI , where F̃ is a translation invariant linear subspace of BIsuch that whenever f ∈ F̃ and f = 0 I-a.e. then f = 0 everywhere. Then sc(F) = ω1.

As F̃ can be the class of continuous, or approximately continous functions or thederivatives, we obtain the following.

Corollary

If F equals any of the classes Cn(R), C∞(R), the class of real analytic functions,Lipschitz functions, derivatives, approximately continuous functions then sc(F) = ω1.The same is true for the subclasses {f ∈ F : f is bounded} where F is any of theclasses listed above.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Theorem

sc(RR) = ωsc(Continuous functions) = ω1sc(Darboux functions) = (2ω)+

sc(Bounded functions) = ω1sc({f ∈ RR : |f | ≤ K}) = ωsc(Trigonometric polynomials) = ω1sc(Polynomials) = 3

Let T P denote the set of trigonometric polynomials, and let I be either the σ-ideal ofLebesgue nullsets or the meager sets. Also denote by BI the class of Lebesguemeasurable functions or the class of functions with the property of Baire.

Theorem

Suppose T P ⊂ F ⊂ F̃ ⊂ BI , where F̃ is a translation invariant linear subspace of BIsuch that whenever f ∈ F̃ and f = 0 I-a.e. then f = 0 everywhere. Then sc(F) = ω1.

As F̃ can be the class of continuous, or approximately continous functions or thederivatives, we obtain the following.

Corollary

If F equals any of the classes Cn(R), C∞(R), the class of real analytic functions,Lipschitz functions, derivatives, approximately continuous functions then sc(F) = ω1.The same is true for the subclasses {f ∈ F : f is bounded} where F is any of theclasses listed above.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Theorem

sc(RR) = ωsc(Continuous functions) = ω1sc(Darboux functions) = (2ω)+

sc(Bounded functions) = ω1sc({f ∈ RR : |f | ≤ K}) = ωsc(Trigonometric polynomials) = ω1sc(Polynomials) = 3

Let T P denote the set of trigonometric polynomials, and let I be either the σ-ideal ofLebesgue nullsets or the meager sets. Also denote by BI the class of Lebesguemeasurable functions or the class of functions with the property of Baire.

Theorem

Suppose T P ⊂ F ⊂ F̃ ⊂ BI , where F̃ is a translation invariant linear subspace of BIsuch that whenever f ∈ F̃ and f = 0 I-a.e. then f = 0 everywhere. Then sc(F) = ω1.

As F̃ can be the class of continuous, or approximately continous functions or thederivatives, we obtain the following.

Corollary

If F equals any of the classes Cn(R), C∞(R), the class of real analytic functions,Lipschitz functions, derivatives, approximately continuous functions then sc(F) = ω1.The same is true for the subclasses {f ∈ F : f is bounded} where F is any of theclasses listed above.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Theorem

sc(RR) = ωsc(Continuous functions) = ω1sc(Darboux functions) = (2ω)+

sc(Bounded functions) = ω1sc({f ∈ RR : |f | ≤ K}) = ωsc(Trigonometric polynomials) = ω1sc(Polynomials) = 3

Let T P denote the set of trigonometric polynomials, and let I be either the σ-ideal ofLebesgue nullsets or the meager sets. Also denote by BI the class of Lebesguemeasurable functions or the class of functions with the property of Baire.

Theorem

Suppose T P ⊂ F ⊂ F̃ ⊂ BI , where F̃ is a translation invariant linear subspace of BIsuch that whenever f ∈ F̃ and f = 0 I-a.e. then f = 0 everywhere. Then sc(F) = ω1.

As F̃ can be the class of continuous, or approximately continous functions or thederivatives, we obtain the following.

Corollary

If F equals any of the classes Cn(R), C∞(R), the class of real analytic functions,Lipschitz functions, derivatives, approximately continuous functions then sc(F) = ω1.The same is true for the subclasses {f ∈ F : f is bounded} where F is any of theclasses listed above.

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Let us see an easy sample proof.

Theorem

sc(Continuous functions) ≤ ω1

Proof. Let S be a system such that every countable subsystem has a continuoussolution. We have to show that S itself has a countable solution.D ↪→ ∪∞n=1R

2n in a natural way.Let {Dm}m∈N be countable dense in {D : (Df = g) ∈ S}.{Dmf = gm}m∈N is a countable subsystem, let f0 be a continuous function such thatDmf0 = gm for every m ∈ N.We claim that f0 solves the whole S.Let (D∗f = g∗) ∈ S be arbitrary.{Dmf = gm}m∈N ∪ {D∗f = g∗} is still countable, hence there is a continuous f1 suchthat Dmf1 = gm for every m ∈ N and D∗f1 = g∗.Choose Dmj → D∗ (in the natural sense).Then Dmj f0 → D∗f0 pointwise (by continuity), and similarly Dmj f0 → D∗f0 pointwise.Hence gmj → D∗f0 pointwise and gmj → D∗f1 pointwise, so D∗f0 = D∗f1.But D∗f1 = g∗, hence D∗f0 = g∗, so f0 solves D∗f = g∗. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Let us see an easy sample proof.

Theorem

sc(Continuous functions) ≤ ω1

Proof. Let S be a system such that every countable subsystem has a continuoussolution. We have to show that S itself has a countable solution.D ↪→ ∪∞n=1R

2n in a natural way.Let {Dm}m∈N be countable dense in {D : (Df = g) ∈ S}.{Dmf = gm}m∈N is a countable subsystem, let f0 be a continuous function such thatDmf0 = gm for every m ∈ N.We claim that f0 solves the whole S.Let (D∗f = g∗) ∈ S be arbitrary.{Dmf = gm}m∈N ∪ {D∗f = g∗} is still countable, hence there is a continuous f1 suchthat Dmf1 = gm for every m ∈ N and D∗f1 = g∗.Choose Dmj → D∗ (in the natural sense).Then Dmj f0 → D∗f0 pointwise (by continuity), and similarly Dmj f0 → D∗f0 pointwise.Hence gmj → D∗f0 pointwise and gmj → D∗f1 pointwise, so D∗f0 = D∗f1.But D∗f1 = g∗, hence D∗f0 = g∗, so f0 solves D∗f = g∗. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Let us see an easy sample proof.

Theorem

sc(Continuous functions) ≤ ω1

Proof. Let S be a system such that every countable subsystem has a continuoussolution. We have to show that S itself has a countable solution.D ↪→ ∪∞n=1R

2n in a natural way.Let {Dm}m∈N be countable dense in {D : (Df = g) ∈ S}.{Dmf = gm}m∈N is a countable subsystem, let f0 be a continuous function such thatDmf0 = gm for every m ∈ N.We claim that f0 solves the whole S.Let (D∗f = g∗) ∈ S be arbitrary.{Dmf = gm}m∈N ∪ {D∗f = g∗} is still countable, hence there is a continuous f1 suchthat Dmf1 = gm for every m ∈ N and D∗f1 = g∗.Choose Dmj → D∗ (in the natural sense).Then Dmj f0 → D∗f0 pointwise (by continuity), and similarly Dmj f0 → D∗f0 pointwise.Hence gmj → D∗f0 pointwise and gmj → D∗f1 pointwise, so D∗f0 = D∗f1.But D∗f1 = g∗, hence D∗f0 = g∗, so f0 solves D∗f = g∗. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Let us see an easy sample proof.

Theorem

sc(Continuous functions) ≤ ω1

Proof. Let S be a system such that every countable subsystem has a continuoussolution. We have to show that S itself has a countable solution.D ↪→ ∪∞n=1R

2n in a natural way.Let {Dm}m∈N be countable dense in {D : (Df = g) ∈ S}.{Dmf = gm}m∈N is a countable subsystem, let f0 be a continuous function such thatDmf0 = gm for every m ∈ N.We claim that f0 solves the whole S.Let (D∗f = g∗) ∈ S be arbitrary.{Dmf = gm}m∈N ∪ {D∗f = g∗} is still countable, hence there is a continuous f1 suchthat Dmf1 = gm for every m ∈ N and D∗f1 = g∗.Choose Dmj → D∗ (in the natural sense).Then Dmj f0 → D∗f0 pointwise (by continuity), and similarly Dmj f0 → D∗f0 pointwise.Hence gmj → D∗f0 pointwise and gmj → D∗f1 pointwise, so D∗f0 = D∗f1.But D∗f1 = g∗, hence D∗f0 = g∗, so f0 solves D∗f = g∗. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Let us see an easy sample proof.

Theorem

sc(Continuous functions) ≤ ω1

Proof. Let S be a system such that every countable subsystem has a continuoussolution. We have to show that S itself has a countable solution.D ↪→ ∪∞n=1R

2n in a natural way.Let {Dm}m∈N be countable dense in {D : (Df = g) ∈ S}.{Dmf = gm}m∈N is a countable subsystem, let f0 be a continuous function such thatDmf0 = gm for every m ∈ N.We claim that f0 solves the whole S.Let (D∗f = g∗) ∈ S be arbitrary.{Dmf = gm}m∈N ∪ {D∗f = g∗} is still countable, hence there is a continuous f1 suchthat Dmf1 = gm for every m ∈ N and D∗f1 = g∗.Choose Dmj → D∗ (in the natural sense).Then Dmj f0 → D∗f0 pointwise (by continuity), and similarly Dmj f0 → D∗f0 pointwise.Hence gmj → D∗f0 pointwise and gmj → D∗f1 pointwise, so D∗f0 = D∗f1.But D∗f1 = g∗, hence D∗f0 = g∗, so f0 solves D∗f = g∗. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Let us see an easy sample proof.

Theorem

sc(Continuous functions) ≤ ω1

Proof. Let S be a system such that every countable subsystem has a continuoussolution. We have to show that S itself has a countable solution.D ↪→ ∪∞n=1R

2n in a natural way.Let {Dm}m∈N be countable dense in {D : (Df = g) ∈ S}.{Dmf = gm}m∈N is a countable subsystem, let f0 be a continuous function such thatDmf0 = gm for every m ∈ N.We claim that f0 solves the whole S.Let (D∗f = g∗) ∈ S be arbitrary.{Dmf = gm}m∈N ∪ {D∗f = g∗} is still countable, hence there is a continuous f1 suchthat Dmf1 = gm for every m ∈ N and D∗f1 = g∗.Choose Dmj → D∗ (in the natural sense).Then Dmj f0 → D∗f0 pointwise (by continuity), and similarly Dmj f0 → D∗f0 pointwise.Hence gmj → D∗f0 pointwise and gmj → D∗f1 pointwise, so D∗f0 = D∗f1.But D∗f1 = g∗, hence D∗f0 = g∗, so f0 solves D∗f = g∗. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Let us see an easy sample proof.

Theorem

sc(Continuous functions) ≤ ω1

Proof. Let S be a system such that every countable subsystem has a continuoussolution. We have to show that S itself has a countable solution.D ↪→ ∪∞n=1R

2n in a natural way.Let {Dm}m∈N be countable dense in {D : (Df = g) ∈ S}.{Dmf = gm}m∈N is a countable subsystem, let f0 be a continuous function such thatDmf0 = gm for every m ∈ N.We claim that f0 solves the whole S.Let (D∗f = g∗) ∈ S be arbitrary.{Dmf = gm}m∈N ∪ {D∗f = g∗} is still countable, hence there is a continuous f1 suchthat Dmf1 = gm for every m ∈ N and D∗f1 = g∗.Choose Dmj → D∗ (in the natural sense).Then Dmj f0 → D∗f0 pointwise (by continuity), and similarly Dmj f0 → D∗f0 pointwise.Hence gmj → D∗f0 pointwise and gmj → D∗f1 pointwise, so D∗f0 = D∗f1.But D∗f1 = g∗, hence D∗f0 = g∗, so f0 solves D∗f = g∗. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Let us see an easy sample proof.

Theorem

sc(Continuous functions) ≤ ω1

Proof. Let S be a system such that every countable subsystem has a continuoussolution. We have to show that S itself has a countable solution.D ↪→ ∪∞n=1R

2n in a natural way.Let {Dm}m∈N be countable dense in {D : (Df = g) ∈ S}.{Dmf = gm}m∈N is a countable subsystem, let f0 be a continuous function such thatDmf0 = gm for every m ∈ N.We claim that f0 solves the whole S.Let (D∗f = g∗) ∈ S be arbitrary.{Dmf = gm}m∈N ∪ {D∗f = g∗} is still countable, hence there is a continuous f1 suchthat Dmf1 = gm for every m ∈ N and D∗f1 = g∗.Choose Dmj → D∗ (in the natural sense).Then Dmj f0 → D∗f0 pointwise (by continuity), and similarly Dmj f0 → D∗f0 pointwise.Hence gmj → D∗f0 pointwise and gmj → D∗f1 pointwise, so D∗f0 = D∗f1.But D∗f1 = g∗, hence D∗f0 = g∗, so f0 solves D∗f = g∗. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Let us see an easy sample proof.

Theorem

sc(Continuous functions) ≤ ω1

Proof. Let S be a system such that every countable subsystem has a continuoussolution. We have to show that S itself has a countable solution.D ↪→ ∪∞n=1R

2n in a natural way.Let {Dm}m∈N be countable dense in {D : (Df = g) ∈ S}.{Dmf = gm}m∈N is a countable subsystem, let f0 be a continuous function such thatDmf0 = gm for every m ∈ N.We claim that f0 solves the whole S.Let (D∗f = g∗) ∈ S be arbitrary.{Dmf = gm}m∈N ∪ {D∗f = g∗} is still countable, hence there is a continuous f1 suchthat Dmf1 = gm for every m ∈ N and D∗f1 = g∗.Choose Dmj → D∗ (in the natural sense).Then Dmj f0 → D∗f0 pointwise (by continuity), and similarly Dmj f0 → D∗f0 pointwise.Hence gmj → D∗f0 pointwise and gmj → D∗f1 pointwise, so D∗f0 = D∗f1.But D∗f1 = g∗, hence D∗f0 = g∗, so f0 solves D∗f = g∗. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Let us see an easy sample proof.

Theorem

sc(Continuous functions) ≤ ω1

Proof. Let S be a system such that every countable subsystem has a continuoussolution. We have to show that S itself has a countable solution.D ↪→ ∪∞n=1R

2n in a natural way.Let {Dm}m∈N be countable dense in {D : (Df = g) ∈ S}.{Dmf = gm}m∈N is a countable subsystem, let f0 be a continuous function such thatDmf0 = gm for every m ∈ N.We claim that f0 solves the whole S.Let (D∗f = g∗) ∈ S be arbitrary.{Dmf = gm}m∈N ∪ {D∗f = g∗} is still countable, hence there is a continuous f1 suchthat Dmf1 = gm for every m ∈ N and D∗f1 = g∗.Choose Dmj → D∗ (in the natural sense).Then Dmj f0 → D∗f0 pointwise (by continuity), and similarly Dmj f0 → D∗f0 pointwise.Hence gmj → D∗f0 pointwise and gmj → D∗f1 pointwise, so D∗f0 = D∗f1.But D∗f1 = g∗, hence D∗f0 = g∗, so f0 solves D∗f = g∗. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Let us see an easy sample proof.

Theorem

sc(Continuous functions) ≤ ω1

Proof. Let S be a system such that every countable subsystem has a continuoussolution. We have to show that S itself has a countable solution.D ↪→ ∪∞n=1R

2n in a natural way.Let {Dm}m∈N be countable dense in {D : (Df = g) ∈ S}.{Dmf = gm}m∈N is a countable subsystem, let f0 be a continuous function such thatDmf0 = gm for every m ∈ N.We claim that f0 solves the whole S.Let (D∗f = g∗) ∈ S be arbitrary.{Dmf = gm}m∈N ∪ {D∗f = g∗} is still countable, hence there is a continuous f1 suchthat Dmf1 = gm for every m ∈ N and D∗f1 = g∗.Choose Dmj → D∗ (in the natural sense).Then Dmj f0 → D∗f0 pointwise (by continuity), and similarly Dmj f0 → D∗f0 pointwise.Hence gmj → D∗f0 pointwise and gmj → D∗f1 pointwise, so D∗f0 = D∗f1.But D∗f1 = g∗, hence D∗f0 = g∗, so f0 solves D∗f = g∗. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Let us see an easy sample proof.

Theorem

sc(Continuous functions) ≤ ω1

Proof. Let S be a system such that every countable subsystem has a continuoussolution. We have to show that S itself has a countable solution.D ↪→ ∪∞n=1R

2n in a natural way.Let {Dm}m∈N be countable dense in {D : (Df = g) ∈ S}.{Dmf = gm}m∈N is a countable subsystem, let f0 be a continuous function such thatDmf0 = gm for every m ∈ N.We claim that f0 solves the whole S.Let (D∗f = g∗) ∈ S be arbitrary.{Dmf = gm}m∈N ∪ {D∗f = g∗} is still countable, hence there is a continuous f1 suchthat Dmf1 = gm for every m ∈ N and D∗f1 = g∗.Choose Dmj → D∗ (in the natural sense).Then Dmj f0 → D∗f0 pointwise (by continuity), and similarly Dmj f0 → D∗f0 pointwise.Hence gmj → D∗f0 pointwise and gmj → D∗f1 pointwise, so D∗f0 = D∗f1.But D∗f1 = g∗, hence D∗f0 = g∗, so f0 solves D∗f = g∗. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Let us see an easy sample proof.

Theorem

sc(Continuous functions) ≤ ω1

Proof. Let S be a system such that every countable subsystem has a continuoussolution. We have to show that S itself has a countable solution.D ↪→ ∪∞n=1R

2n in a natural way.Let {Dm}m∈N be countable dense in {D : (Df = g) ∈ S}.{Dmf = gm}m∈N is a countable subsystem, let f0 be a continuous function such thatDmf0 = gm for every m ∈ N.We claim that f0 solves the whole S.Let (D∗f = g∗) ∈ S be arbitrary.{Dmf = gm}m∈N ∪ {D∗f = g∗} is still countable, hence there is a continuous f1 suchthat Dmf1 = gm for every m ∈ N and D∗f1 = g∗.Choose Dmj → D∗ (in the natural sense).Then Dmj f0 → D∗f0 pointwise (by continuity), and similarly Dmj f0 → D∗f0 pointwise.Hence gmj → D∗f0 pointwise and gmj → D∗f1 pointwise, so D∗f0 = D∗f1.But D∗f1 = g∗, hence D∗f0 = g∗, so f0 solves D∗f = g∗. �

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Which important function classes are left?The measurable functions:

Theorem

[cf(non(I))]+ ≤ sc(BI) ≤ [cof(I)]+.

Corollary

The Continuum Hypothesis implies sc(BI) = ω2.

Problem

What can we say in ZFC?

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Which important function classes are left?The measurable functions:

Theorem

[cf(non(I))]+ ≤ sc(BI) ≤ [cof(I)]+.

Corollary

The Continuum Hypothesis implies sc(BI) = ω2.

Problem

What can we say in ZFC?

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Which important function classes are left?The measurable functions:

Theorem

[cf(non(I))]+ ≤ sc(BI) ≤ [cof(I)]+.

Corollary

The Continuum Hypothesis implies sc(BI) = ω2.

Problem

What can we say in ZFC?

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Which important function classes are left?The measurable functions:

Theorem

[cf(non(I))]+ ≤ sc(BI) ≤ [cof(I)]+.

Corollary

The Continuum Hypothesis implies sc(BI) = ω2.

Problem

What can we say in ZFC?

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

Which important function classes are left?The measurable functions:

Theorem

[cf(non(I))]+ ≤ sc(BI) ≤ [cof(I)]+.

Corollary

The Continuum Hypothesis implies sc(BI) = ω2.

Problem

What can we say in ZFC?

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals

AND THE BOREL AND BAIRE-α FUNCTIONS!

Márton Elekes emarci@renyi.hu www.renyi.hu/˜emarci Solvability cardinals