Arithmetic and Recursion

Definition. An $n$-ary relation $R$ on the universe of a structure $ \mathfrak{} \mathfrak{A} $ is definable in the structure if there is a formula $ \phi $ in the language of the structure such that for all $n$-tuples $\overline{a}$ of members of the universe of the structure

\[\overline{a} \in R\text{ if and only if } \mathfrak{A} \models \phi [\overline{a}]\]
A function is definable in a structure if ... .

A relation or function on the natural numbers is said to be arithmetical if it is definable in $ \mathfrak{} \mathfrak{N} $.

We proved last time (theorem 6.12) that recursive relations and functions on the natural numbers are arithmetical.

Definition. A bounded quantifier is one of the form

\[\forall x<y{ \text{ which is short for }} \forall x(x<y \rightarrow \bullet )\]
or of the form
\[\exists x<y{ \text{ which is short for }} \exists x(x<y \land \bullet )\]

Definition. A $\Delta_0$ formula (of the language of arithmetic) is a formula of the language of arithmetic in which all quantifiers are bounded.

Definition. A $\Sigma$ formula (of the language of arithmetic) is a formula that has some existential quantifiers in front of a $\Delta_0$ formula.

I sketched last time a proof that a relation on the natural numbers is r.e. (and a function on the natural numbers is recursive) if and only if it is definable in $ \mathfrak{} \mathfrak{N} $ by a $\Sigma$ formula. Thus, a relation on the natural numbers is recursive if and only if both it and its complement are definable using $\Sigma$ formulas.

The proof consisted in observing that the godawful formulas I wrote out last time were in fact obviously equivalent to $\Sigma$ formulas. And $\Sigma$-definable relations are r.e. by Church's thesis.

A formula of set theory is bounded if it is of one of the forms $ \forall x \in y $ or $ \exists x \in y $. Most of what we have done is not special to recursion, but holds for any system that has a universal function and a version of the $s$-$m$-$n$ theorem, and so practically everything has an analog about $\Sigma$ formulas, whether of arithmetic or of set theory.

We are now done with what can done in terms of arithmetic truth. We turn to a related but different topic, what can be done in terms of provability in a theory of arithmetic.

Definition. Given a set of formulas $\Phi$ that includes $S$ and $0$ in its language, an $n$-ary relation on numbers $R$ is said to be $\Phi$ representable if there is a formula $\phi$ in the language of $\Phi$ such that for all $n$-tuples $\overline{a}$ of numbers,

\[ { \text{If }}\overline{a}\in R{ \text{ then }} \Phi \vdash \phi (\overline{\mathbf{a}}) \]
and
\[ { \text{If }}\overline{a}\notin R{ \text{ then }} \Phi \vdash \lnot \phi (\overline{\mathbf{a}}) \]

I just did this for $ S,0 $, but it can be extended to any system of words.

Definition. A function is representable if it is representable as a relation and you can prove it is a function, that is

\[ \Phi \vdash \forall \overline{x}\exists ^{=1} y \phi (\overline{x}y) \]

Fact. (proved below) A relation or function is recursive if and only if it is representable in a finite $\Phi$.

Claim. (controversial) Representability is the most natural and fundamental definition of recursive, all the others (Turing machines, register machines, $\Sigma$ formulas) are derivative of this one.

-- ShaughanLavine - 08 Apr 2005