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Mohammad Golshani
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The following example gives a connection between descriptive set theory and the theory of approximation by algebraic numbers:

There exists a classification, due to Mahler, of real (and complex) numbers into four classes $A, S, T$ and $U$ according to their properties of approximation by algebraic numbers.

In the paper The Borel Classes of Mahler's $A$, $S$, $T$, and $U$ Numbers, the author studies these classes from the point of view of Descriptive Set Theory, and determines their complexity in the Borel hierarchy.

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The following example gives a connection between descriptive set theory and the theory of approximation by algebraic numbers:

There exists a classification, due to Mahler, of real (and complex) numbers into four classes $A, S, T$ and $U$ according to their properties of approximation by algebraic numbers.

In the paper The Borel Classes of Mahler's $A$, $S$, $T$, and $U$ Numbers, the author studies these classes from the point of view of Descriptive Set Theory, and determines their complexity in the Borel hierarchy.

The following example gives a connection between descriptive set theory and the theory of approximation by algebraic numbers:

There exists a classification, due to Mahler, of real (and complex) numbers into four classes $A, S, T$ and $U$ according to their properties of approximation by algebraic numbers.

In the paper The Borel Classes of Mahler's $A$, $S$, $T$, and $U$ Numbers, the author studies these classes from the point of view of Descriptive Set Theory, and determines their complexity in the Borel hierarchy.

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Mohammad Golshani
Mohammad Golshani
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OneThe following example might be Addison's theoremgives a connection between descriptive set theory and the theory of approximation by algebraic numbers:

Theorem There exists a classification, due to Mahler, of real (Addisonand complex). The set of arithmetical sets is not itself arithmetical.

The above theorem is a numbers into four classes $ZFC$ result, but its proof uses forcing$A, S, T$ and $U$ according to their properties of approximation by algebraic numbers.

Here:

(a) A setIn the paper $A \subseteq \mathbb{N}$ is arithmeticalThe Borel Classes of Mahler's $A$, $S$, $T$, and $U$ Numbers, if for some formula $\phi$ of the languageauthor studies these classes from the point of arithmetic, $A=\{n: \mathbb{N}\models \phi(n) \}.$

(b) A familyview of sets $\mathcal{B} \subseteq p(\mathbb{N})$ is arithmetically definableDescriptive Set Theory, if for some fomula $\phi(X)$and determines their complexity in the language of arithmetic expanded with one unary predicate, we have $\mathcal{B}=\{G \subseteq \mathbb{N}: (\mathbb{N}, G)\models \phi(G) \}.$

A reference is the book "Computability and Logic" by Boolos, Burgess and JeffreyBorel hierarchy.

One example might be Addison's theorem:

Theorem (Addison). The set of arithmetical sets is not itself arithmetical.

The above theorem is a $ZFC$ result, but its proof uses forcing.

Here:

(a) A set $A \subseteq \mathbb{N}$ is arithmetical, if for some formula $\phi$ of the language of arithmetic, $A=\{n: \mathbb{N}\models \phi(n) \}.$

(b) A family of sets $\mathcal{B} \subseteq p(\mathbb{N})$ is arithmetically definable, if for some fomula $\phi(X)$ in the language of arithmetic expanded with one unary predicate, we have $\mathcal{B}=\{G \subseteq \mathbb{N}: (\mathbb{N}, G)\models \phi(G) \}.$

A reference is the book "Computability and Logic" by Boolos, Burgess and Jeffrey.

The following example gives a connection between descriptive set theory and the theory of approximation by algebraic numbers:

There exists a classification, due to Mahler, of real (and complex) numbers into four classes $A, S, T$ and $U$ according to their properties of approximation by algebraic numbers.

In the paper The Borel Classes of Mahler's $A$, $S$, $T$, and $U$ Numbers, the author studies these classes from the point of view of Descriptive Set Theory, and determines their complexity in the Borel hierarchy.

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Mohammad Golshani
Mohammad Golshani
  • 32.1k
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One example might be Addison's theorem:

Theorem (Addison). The set of arithmetical sets is not itself arithmetical.

The above theorem is a $ZFC$ result, but its proof uses forcing.

Here:

(a) A set $A \subseteq \mathbb{N}$ is arithmetical, if for some formula $\phi$ of the language of arithmetic, $A=\{n: \mathbb{N}\models \phi(n) \}.$

(b) A family of sets $\mathcal{B} \subseteq p(\mathbb{N})$ is arithmetically definable, if for some fomula $\phi(X)$ in the language of arithmetic expanded with one unary predicate, we have $\mathcal{B}=\{G \subseteq \mathbb{N}: (\mathbb{N}, G)\models \phi(G) \}.$

A reference is the book "Computability and Logic" by Boolos, Burgess and Jeffrey.

One example might be Addison's theorem:

Theorem (Addison). The set of arithmetical sets is not itself arithmetical.

The above theorem is a $ZFC$ result, but its proof uses forcing.

One example might be Addison's theorem:

Theorem (Addison). The set of arithmetical sets is not itself arithmetical.

The above theorem is a $ZFC$ result, but its proof uses forcing.

Here:

(a) A set $A \subseteq \mathbb{N}$ is arithmetical, if for some formula $\phi$ of the language of arithmetic, $A=\{n: \mathbb{N}\models \phi(n) \}.$

(b) A family of sets $\mathcal{B} \subseteq p(\mathbb{N})$ is arithmetically definable, if for some fomula $\phi(X)$ in the language of arithmetic expanded with one unary predicate, we have $\mathcal{B}=\{G \subseteq \mathbb{N}: (\mathbb{N}, G)\models \phi(G) \}.$

A reference is the book "Computability and Logic" by Boolos, Burgess and Jeffrey.

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Mohammad Golshani
Mohammad Golshani
  • 32.1k
  • 2
  • 99
  • 198