Is there a nonstandard model of Peano's arithmetic where any element has only finitely many prime divisors?

Question

By the compactness theorem, there are nonstandard models where don't have finite prime decomposition, and some elements are divisible by infinitely many distinct primes. But what if we want a nice nonstandard model where any element is divisible by only finitely distinct primes, is it possible? It seems hard to construct one, if it is possible. I don't know how to construct models of Peano's arithmetic where I don't want something to happen.

Of course, we cannot ask for finite prime decomposition, because in any nonstandard model there are nonstandard elements whose only prime divisor is $2$, and hence they are divisible by $2^n$ for any $n \in \mathbb{N}$.

If such a model exists, can we have one of any infinite cardinality?

Answer 1

There is no such model. The following is a theorem of Peano arithmetic:

For every number $n$, there is some number $n'$ that is divisible by all $k < n$.

Consider a non-standard model $\langle \mathfrak{M}, +, \cdot, 0, S\rangle$ of Peano arithmetic. Since $\mathfrak{M}$ is non-standard, it has some non-standard element $K$ so that $0 < K, 1 < K, 2 < K, \dots$ and so on, for each standard natural.

By the theorem of Peano arithmetic quoted above, there is another number $K'$ that is divisible by all $k < K$. In particular, $K'$ is divisible by $2$, divisible by $3$, divisible by $5$, and so on for all the standard primes.

Answer 2

The question is interesting and I want to add some non-technical comments to the technically impeccable answer of Z.A.K.

To construct models of PA where something doesn't happen is a notorious difficult problem especially when this something is expressible in first-order logic.

One of the few known results of this sort is the celebrated Paris-Harrington Theorem that construct a model of PA where a (true) variant of Ramsey's Theorem does not hold.

Time ago, it was hoped that weaker versions of PA (i.e. fragments where induction only applies to formulas of low complexity) could make the task of constructing models more feasible.

E.g. Much effort has been spent on the question of whether $I\Delta_0$ (the fragment of PA where induction only applies to bounded formulas) proves the infinity of primes. But the problem was deemed intractable and abandoned some 40 years ago. (The problem is connected with some hopeless open problems in complexity theory.)