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ygert comments on Standard and Nonstandard Numbers - Less Wrong
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Well, this was very interesting, formal logic is a very fun topic. I just spent ~10 minutes trying to find a way in first order logic to write that axiom, as it intuitively feels (to someone who has studied formal logic at least) that there should be a way... Of course I failed, all the axioms I attempted turned out to be no more powerful then "0 is not the successor of any number". I am deeply intrigued by this problem, and I am looking forward to your next post where you explain exactly why it's impossible.
If you like spoilers, google "Lowenheim-Skoler" -- the same technique as the proof for the "upwards" part allows you to generate non-standard models for the First-order logic version of Peano axioms in a fairly straight-forward manner.