One model of Robinson arithmetic which is obviously not our usual integers is $\mathbb{Z}[X]^+$, that is the set containing 0 and also all polynomials with coefficients in $\mathbb{Z}$ with positive lead coefficient. This ring also satisfies a bit more, since it satisfies distributivity, commutativity, and associativity, and that no number is its own successor, which Robinson arithmetic only can prove are true for specific natural numbers.


One basic property that the natural numbers have which $\mathbb{Z}[X]^+$ lacks if a division algorithm. That is, the property that for any $a, b \in \mathbb{N}$, and $b \neq 0$, there is a $q$ and $r$ such that $a=bq+r$, and where $r < b$. (Here we are for simplicity including 0 as a natural number.) We can see that $\mathbb{Z}[X]^+$ fails this by considering for example elements like $a=X$ and $b=2$. However, despite this, $\mathbb{Z}[X]^+$ has many of the properties we expect out of the natural numbers, including unique factorization; and a major reason we care about having a division algorithm in the natural numbers is to use it to prove unique factorization.


**Question:** Is there a nice model of Robinson arithmetic which satisfies commutative, associative, and distributive properties, no number is its own successor, and also the division algorithm and is not just the natural numbers?


Nice here is left vague, but at an absolute minimum requires being computable. I'm hoping to see something like some subset of some ring that an algebraist would recognize as a reasonable object.


The obvious thing to try to use that the nonnegative part of a discrete ordered ring is always a model for Robinson Arithmetic, and there are a lot of examples of things one can construct there, but I have not been able to find one that works for this purpose.



Systems intermediate in strengthen between Robinson arithmetic and PA
Yes. For example, you can take the nonnegative part of the ring of polynomials in $\mathbb Q[X]$ with integer constant coefficient (i.e., $\mathbb Q[X]X+\mathbb Z$).


For a more sophisticated example due to Shepherdson [2], the nonnegative part of the ring of *Puiseux polynomials* $\sum\_{i\le n}a\_iX^{i/k}$ over a real-closed field such as $\tilde{\mathbb Q}\cap\mathbb R$ with $a\_0\in\mathbb Z$ is a nonstandard computable model of the theory IOpen = Robinson’s arithmetic + induction for open (= quantifier-free) formulas in the language of ordered semirings. IOpen proves the existence of division with remainder and much more.


Computable nonstandard models exist for extensions of IOpen by some mild algebraic axioms, in particular GCD; see Mohsenipour [1]. However, these constructions get rather messy, and I’m not sure “an algebraist would recognize them as a reasonable object”. On the other hand, there is no computable nonstandard model of Peano arithmetic by Tennenbaum’s theorem [3], and this has been extended to models of $IE\_1$ (= Robinson’s arithmetic + induction for bounded existential formulas) by Wilmers [4].


### References


[1] Shahram Mohsenipour: *A recursive nonstandard model for open induction with GCD property and confinal primes*, Logic in Tehran (Ali Enayat et al., eds.), Lecture Notes in Logic no. 26, Association for Symbolic Logic, 2006, pp. 227–238.


[2] John C. Shepherdson: *A nonstandard model for a free variable fragment of number theory*, Bulletin de l’Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques et physiques 12 (1964), no. 2, pp. 79–86.


[3] Stanley Tennenbaum: *Non-archimedean models for arithmetic*, Notices of the American Mathematical Society 6 (1959), p. 270.


[4] George Wilmers: *Bounded existential induction*, Journal of Symbolic Logic 50 (1985), no. 1, pp. 72–90.


