The Peano Axioms (partially) formalize our intuitive notion of arithmetic. Partially because they also describe the behaviour of nonstandard models and there are some theorems that they can not prove that might seem, at first sight, to be within their domain e.g. Goodstein's Theorem and some of Harvey Friedman's combinatorial theorems <http://arxiv.org/abs/math/9811187>. So there is room for some clever soul to find additional axioms which would be independent of PA and also convey a natural intuitive property of the integers. These additional axioms would prove new theorems and also restrict the class of nonstandard models.


Could such a process ever be complete in the sense that all nonstandard models would be excluded?


PS I read the FAQ and am not sure if this is a suitable question. My apologies if it is not.


Larry Wickert, Truth or Consequences, New Mexico



Extension of the Peano Axioms?
There is no hope for a *first-order* theory to eliminate non-standard models. If a first-order theory over a countable language has an infinite model then it has models of all infinite cardinalities ([Löwenheim-Skolem Theorem](http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem)). In the case of a theory $T$ of arithmetic. If the standard model satisfies $T$, then $T$ must have an uncountable model, which has to be non-standard (though there will be countable non-standard models of $T$ as well).


If you formulate PA in [second-order logic](http://en.wikipedia.org/wiki/Second-order_logic), then it is already complete with respect to the standard semantics.


