Solvability cardinals - IMJ-PRG jean.saint-raymond/DST/08/elekes... · PDF...
47
Solvability cardinals Márton Elekes [email protected] www.renyi.hu/˜emarci Rényi Institute and Eötvös University Descriptive Set Theory in Paris 2008 Joint work with M. Laczkovich. Márton Elekes [email protected] www.renyi.hu/˜ emarci Solvability cardinals
Transcript of Solvability cardinals - IMJ-PRG jean.saint-raymond/DST/08/elekes... · PDF...
Solvability cardinals
Márton [email protected]
www.renyi.hu/˜emarci
Rényi Institute and Eötvös University
Descriptive Set Theory in Paris 2008
Joint work with M. Laczkovich.
Márton Elekes [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] www.renyi.hu/˜emarci Solvability cardinals
AND THE BOREL AND BAIRE-α FUNCTIONS!
Márton Elekes [email protected] www.renyi.hu/˜emarci Solvability cardinals
