# 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 ${\displaystyle M}$ in the language of PA is recursive if there are recursive functions + and × from ${\displaystyle N\times N}$ to ${\displaystyle N}$, a recursive two-place relation < on ${\displaystyle N}$, and distinguished constants ${\displaystyle n_{0},n_{1}}$ such that

${\displaystyle (N,\oplus ,\otimes ,<_{M},n_{0},n_{1})\equiv M,}$

where ${\displaystyle \equiv }$ indicates isomorphism and ${\displaystyle N}$ 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 ${\displaystyle x\in M}$ can be viewed as a code for a set ${\displaystyle S_{x}}$ of elements of M. In particular, if we let ${\displaystyle p_{i}}$ be the ith prime in M, then ${\displaystyle z\in S_{x}\leftrightarrow M\vDash p_{z}|x}$. Each set ${\displaystyle S_{x}}$ will be bounded in M, but if x is nonstandard then the set ${\displaystyle S_{x}}$ may contain infinitely many standard natural numbers. The standard system of the model is the collection ${\displaystyle \{S_{x}\cap N:x\in M\}}$. 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 ${\displaystyle A,B\subseteq N}$ so that that there is no recursive set ${\displaystyle C\subseteq N}$ with ${\displaystyle A\subseteq C}$ and ${\displaystyle B\cap C=\emptyset }$.

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 ${\displaystyle i\in A}$ then ${\displaystyle p_{i}|y}$ and if ${\displaystyle i\in B}$ then ${\displaystyle p_{i}\not |y}$. 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 ${\displaystyle m\in M}$ with ${\displaystyle m<_{M}x}$, we have

${\displaystyle M\vDash (m\in A\to p_{m}|y)\land (m\in B\to p_{m}\not |y)}$

Let ${\displaystyle S=N\cap S_{y}}$ be the corresponding set in the standard system of M. Because A and B are r.e., one can show that ${\displaystyle A\subseteq S}$ and ${\displaystyle B\cap S=\emptyset }$. 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 ${\displaystyle S=N\cap S_{a}}$ 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 ${\displaystyle \oplus }$ of M as an oracle. In particular, if ${\displaystyle n_{0}}$ is the element of M corresponding to 0, and ${\displaystyle n_{1}}$ is the element of M corresponding to 1, then for each ${\displaystyle i\in N}$ we can compute ${\displaystyle n_{i}=n_{1}\oplus \cdots \oplus n_{1}}$ (i times). To decide if a number n is in S, first compute p, the nth prime in N. Then, search for an element y of M so that

${\displaystyle a=\underbrace {y\oplus y\oplus \cdots \oplus y} _{p{\text{ times}}}\oplus n_{i}}$

for some ${\displaystyle i. This search will halt because the Euclidean algorithm can be applied to any model of PA. Finally, we have ${\displaystyle n\in S}$ 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).

## References

• George Boolos, John P. Burgess, and Richard Jeffrey (2002) Computability and Logic, 4th ed. Cambridge University Press. ISBN 0-521-00758-5
• Richard Kaye (1991) Models of Peano arithmetic. Oxford University Press. ISBN 0-19-853213-X.
• Richard Kaye (Sep 2011). "Tennenbaum's Theorem for Models of Arithmetic". In Juliette Kennedy and Roman Kossak. Set theory, arithmetic, and foundations of mathematics - theorems, philosophies (PDF). Lecture Notes in Logic. 36. ISBN 9781107008045.
• Stanley Tennenbaum (1959) Non-Archimedean models for arithmetic, In: Notices of the American Mathematical Society 6, p. 270