FEBRERO 1940 ABRIL 1940 JUNIO

The result known as Tennenbaum’s Theorem was given by Stanley Tennenbaum in 1959 and appeared as a one-page abstract under Tennenbaum(1959). The result sharply contrasts the computational properties of arithmetical operations defined in standard and non-standard models of _pa:

Theorem: (Tennenbaum, 1959) If M is a countable model of pa such that M 6∼=N, thenMis not recursive.

The standard modelN where order, addition and multiplication are interpreted stan- dardly is recursive. Tennenbaum shows that it is the only recursive model, up to isomorphism. The proof of the above statement will mostly follow Kaye(2011). The strategy is normally to assume that there is a recursive non-standard model of _pa and show this leads to contradiction; we do this by building a non-recursive set and prove, under the assumption that such recursive non-standard model exists, that the set is recursive. First, we define:

Definition: We call anL_pa-structureM recursive iff there are recursive functions S :_N → _N, + : _N2 _→

N and ×: N2 → N, a binary recursive relation<⊆_N2_{and 0∈}

Nsuch thatM ∼=hM,0, S,+,×i.

The proof of the Theorem may be separated into two subproblems: first, define when a setA⊆Niscoded in a model, and, second, the implications of non-recursive coded sets. The coding of a non-recursive set in the domain of a model can be done by a prime number technique where a set of numbers n is defined by an element c ∈ M and a two-place formulaϕsuch thatM |=ϕ(n, c)↔ ∃k(c=k×pn) wherepn is the

nth_{prime number. Then we can define thestandard system of sets coded in a model}

as the set of all sets coded by an elementcin the model.

Definition: (SSy(M)) Given a non-standard modelMof _pawe call the standard systems of sets inM, SSy(M), the set defined as:

SSy(M) ={A⊆N| ∃c∈ M:A={n∈N| M |=ϕ(n, c)}}

The Theorem is also related with the following classical results from recursion theory: Definition: (Recursively Inseparable) We say that two disjoint setsA, B⊆ Narerecursively inseparable iff there is no recursive setC⊆Nsuch that A⊆CandC∩B=∅.

Lemma: There exist recursively enumerable recursively inseparable sets.

The traditional approach to Tennenbaum’s Theorem now proves the existence of a non-recursive set coded in a non-standard model. For this, we recall the principle:

(Overspill Principle) Let ϕ(x) be a LP A formula. Let M be a non-

standard model. If M |=ϕ(n) for alln∈N, then there is a non-standard number asuch thatM |=ϕ(a).

Now, we have:

Theorem: LetMbe a non-standard model of_pa. Then SSy(M) contains a non-recursive set.

Proof. Consider A, B ⊆ N recursively enumerable recursively insepara- ble sets, as given by the above Lemma. Consider a, b the µ-recursive functions that enumerate them. Given that every µ-recursive function is Σ1-representable inpa, there are Σ1-formulas∃yα(x, y) and∃zβ(x, z)

that define A andB, respectively, where αand β are ∆0. We regard N

as an initial segment of any non-standard model. Since N ≺ M, there is an embedding π : _N → M that preserves the Σ1 formulas. That is,

since the interpretations ofa andb are preserved upwards fromN to its extension (i.e. a(π(x)) = π(a(x)) and b(π(x)) = π(b(x))), the same Σ1

formulas expressAandBinM. Then, from this latter fact and from the disjointness ofA, B it follows that for anyk∈N:

M |=∀x<k,∀y<k,∀z<k ¬(α(x, y)∧β(x, z))

By applying the Overspill Principle, there is a non-standard c∈M such that:

M |=∀x<c,∀y<c,∀z<c ¬(α(x, y)∧β(x, z))

Define the set C ⊆Nwith C ={n∈N| M |=∃y<c (α(n, y))}. Then, by preservation of Σ1-fromulas and sincec is non-standard, we have that

A ⊆ C and that C∩B = ∅. Hence, since A and B are recursively inseparable, Cis a non-recursive set.

From the above we can prove: Tennenbaum’s Theorem:

Proof. Let M be a non-standard model of pa. By the above Theorem there is a set C∈SSy(M) such thatC is non-recursive. Now, there is a c∈M such that

C={n∈N| M |=ϕ(n, c)}

whereϕ(n, c) =∃k(c=k×pn) uniquely codesCwithc. We want to show

that if +M _{is recursive, thenC is recursive, against assumption. Assume} then, for contradiction, that +M:_N2_→

Nis recursive. Define: ψ(n, c) = (c=k+...+k | {z } pn times )∨(c=k+...+k | {z } pn times +1)∨(c=k+...+k | {z } pn times + 1 +...+ 1 | {z } pn−1 times )

We now note thatpa`Euclidean Division. Then, sinceM |=pa, we have that M also proves Euclidean Division. Hence, ∃!pMn ∃!r such that c =

(k×pn)+rfor 0≤r < pn. Now, for any inputn, we may computepnand

search for ak∈M andr < pnsuch that (k+...+k

| {z }

pn times

) +r=ϕ(n, c). This search is bounded to terminate since Euclidean division is computable. Now, if r= 0, thenc =k+...+k

| {z }

pn times

is the case andψ(n, c) is true; in this case ϕ(n, c) is true and n∈C. If r6= 0, then one of the other disjuncts of ψ(n, c) is true; in which case ϕ(n, c) is false and n 6∈ C. This shows that it is decidable inMifn∈Corn6∈C. Hence,Cis recursive against assumption. Since we derive a contradiction, we conclude that +M_{is not} recursive.

An interesting question is how far can the argument be extended to theories weaker thanpa. The proof above only requires overspill for ∆0-relations andI∆0 is strong

enough to prove enough properties concerning Euclidean Divison and primes for the above argument to go through. In fact, McAlloon(1982) showed that if we replace

pawith the weaker subsystemI∆0, with induction restricted to bounded quantifiers,

an analogue to Tennenbaum’s Theorem also holds: addition and multiplication will be non-recursive in any non-standard model of I∆0. Also, from Wilmers(1985) we

know the same result holds for the subtheory IE1 ofI∆0, with induction restricted

to bounded existential quantifiers.