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?