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I'm looking for an axiomatization of the natural numbers similar to ZFC or the standard axiomatization of set theory in the language of the first order predicate calculus such that the axiomatization only has one unique model of countable cardinality. I want one that excludes the nonstandard models of the natural numbers. Can anyone cite a reference?

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marked as duplicate by Asaf Karagila set-theory Apr 26 at 13:00

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    $\begingroup$ That's impossible! By the Löwenheim-Skolem Theorem every theory that has an infinite model, has models in all infinite cardinalities. So in particular it has two non-isomorphic models. $\endgroup$ – Achilles Apr 26 at 12:39
  • $\begingroup$ Thanks for your response. I updated the question to ask about countable models only. Also, does Lowenheim Skolem apply to higher order languages as well? $\endgroup$ – David Warren Katz Apr 26 at 12:41
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    $\begingroup$ It doesn't. For example, second order arithmetic has only one model (up to isomorphism). $\endgroup$ – mihaild Apr 26 at 12:55

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