# Tennenbaum's theorem

Tennenbaum's theorem, named for Stanley Tennenbaum who presented the theorem in 1959, is a result in mathematical logic that states that no countable nonstandard model of first-order Peano arithmetic (PA) can be recursive (Kaye 1991:153ff).

## Recursive structures for PA

A structure $M$ in the language of PA is recursive if there are recursive functions $\oplus$ and $\otimes$ from $\mathbb {N} \times \mathbb {N}$ to $\mathbb {N}$, a recursive two-place relation <$_M$ on $\mathbb {N}$, and distinguished constants $n_{0},n_{1}$ such that
    $(\mathbb {N} ,\oplus ,\otimes ,<_{M},n_{0},n_{1})\cong M,$
where $\cong$ indicates isomorphism and $\mathbb {N}$ is the set of (standard) natural numbers. Because the isomorphism must be a bijection, every recursive model is countable. There are many nonisomorphic countable nonstandard models of PA.

## Statement of the theorem

Tennenbaum's theorem states that no countable nonstandard model of PA is recursive. Moreover, neither the addition nor the multiplication of such a model can be recursive.

## Proof sketch

This sketch follows the argument presented by Kaye (1991). The first step in the proof is to show that, if M is any countable nonstandard model of PA, then the standard system of M (defined below) contains at least one nonrecursive set S. The second step is to show that, if either the addition or multiplication operation on M were recursive, then this set S would be recursive, which is a contradiction.
Through the methods used to code ordered tuples, each element $x\in M$ can be viewed as a code for a set $S_{x}$ of elements of M. In particular, if we let $p_{i}$ be the ith prime in M, then $z\in S_{x}\leftrightarrow M\vDash p_{z}|x$. Each set $S_{x}$ will be bounded in M, but if x is nonstandard then the set $S_{x}$ may contain infinitely many standard natural numbers. The standard system of the model is the collection $\{S_{x}\cap \mathbb {N} :x\in M\}$. It can be shown that the standard system of any nonstandard model of PA contains a nonrecursive set, either by appealing to the incompleteness theorem or by directly considering a pair of recursively inseparable r.e. sets (Kaye 1991:154). These are disjoint r.e. sets $A,B\subseteq \mathbb {N}$ so that there is no recursive set $C\subseteq \mathbb {N}$ with $A\subseteq C$ and $B\cap C=\emptyset$.
For the latter construction, begin with a pair of recursively inseparable r.e. sets A and B. For natural number x there is a y such that, for all i < x, if $i\in A$ then $p_{i}|y$ and if $i\in B$ then $p_{i}\nmid y$. By the overspill property, this means that there is some nonstandard x in M for which there is a (necessarily nonstandard) y in M so that, for every $m\in M$ with $m<_{M}x$, we have
    $M\vDash (m\in A\to p_{m}|y)\land (m\in B\to p_{m}\nmid y)$
Let $S=\mathbb {N} \cap S_{y}$ be the corresponding set in the standard system of M. Because A and B are r.e., one can show that $A\subseteq S$ and $B\cap S=\emptyset$. Hence S is a separating set for A and B, and by the choice of A and B this means S is nonrecursive.
Now, to prove Tennenbaum's theorem, begin with a nonstandard countable model M and an element a in M so that $S=\mathbb {N} \cap S_{a}$ is nonrecursive. The proof method shows that, because of the way the standard system is defined, it is possible to compute the characteristic function of the set S using the addition function $\oplus$ of M as an oracle. In particular, if $n_{0}$ is the element of M corresponding to 0, and $n_{1}$ is the element of M corresponding to 1, then for each $i\in \mathbb {N}$ we can compute $n_{i}=n_{1}\oplus \cdots \oplus n_{1}$ (i times). To decide if a number n is in S, first compute p, the nth prime in $\mathbb {N}$. Then, search for an element y of M so that
    $a=\underbrace {y\oplus y\oplus \cdots \oplus y} _{p{\text{ times}}}\oplus n_{i}$
for some $i<p$. This search will halt because the Euclidean algorithm can be applied to any model of PA. Finally, we have $n\in S$ if and only if the i found in the search was 0. Because S is not recursive, this means that the addition operation on M is nonrecursive.
A similar argument shows that it is possible to compute the characteristic function of S using the multiplication of M as an oracle, so the multiplication operation on M is also nonrecursive (Kaye 1991:154).

## Turing degrees of models of PA

Jockush and Soare have shown there exists a model of PA with low degree.