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In first order logic there are non-standard models of the natural numbers and also of the reals (which was exploited by Robinson to get a rigorous construction of infinitesimals used in the calculus, & of infinities which is different from Cantors).

However according to this article on higher order logic, there is a unique model of the natural numbers when using full semantics.

Notably, though where first order logic is sound, complete and effective; higher order logic have none of these nice properties.

a. What is the minimal order in which this happens? Is it order two?

b. Is it also true for any axiomatic system, not just the peano axioms,that they all have a unique model?

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a) It is 2nd-Order Logic with Full Semantics. The semantics is "full" because the second order variables range over the full powerset of the first-order domain. Categoricity obtains in many cases as a result of the failure of Lowenheim-Skolem in a second-order setting.

b) It is not true of every axiomatic system, or at least we do not have proofs of categoricity for every axiomatic system. Some tend to view categoricity as crucial for a proper axiomatization of non-algebraic theories like PA. It's hard to see whether algebraic theories should be categorical, though.

Section 5.2 of the SEP article on Phil of Math has some info you might find useful.

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  • Is there some nice characterisation of which axiomatic systems categoricity is achieved?
    – Mozibur Ullah
    Jun 2, 2013 at 18:35

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