Surreal numbers and omnific integers
in [Logic]
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From: George Greene on 11 Aug 2010 23:57 I had asked what made a non-standard model of the integers non- standard. On Aug 10, 9:05 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> replied: > Not being isomorphic to the integers. This is NOT a reasonable reply! > As with the naturals we can > characterize the integers (up to isomorphism as usual) as the smallest > model of a certain formal theory. Thank you; that's reasonable. "Smallest" is a little suspect here, though; There Are Always countable models, both standard AND non-, of the first-order axiomatic approximation. To the extent that non- standard models can be countable, they are not any bigger. In the case of PA, however, all the non-standard countable models have domains including the standard model's domain as a proper subset, so in that sense, the standard really is "smallest". In the analogous case for ZFC, the model with only constructible powersets was said to be "inner" (though that is simultaneously anti-analogous since the inner/smallest model in ZFC's case is NOT the standard; indeed, the standard is the opposite/widest). But, returning to the prior complaint: "a model is non-standard iff it is not isomorphic to the standard" is almost vacuous; everybody already knew what "non-" meant! The question was, what does that entail? How can you manage to become non- isomorphic while retaining agreement with the axioms? In PA's case, you had to add some elements that are both bigger than anything in the standard AND anonymous. First-order axioms are not going to suffice to distinguish standard models from non-standard ones, not completely, not in both directions. You have to invoke some sort of second-order something. The most reasonable thing to do in PA's case is to invoke finitude; you can say that every number is finite, or has finitely many numbers less than it, and every number has a numeral that is a finite number of applications of s() to 0. So I am guessing that you have just said that the distinguishing characteristic of the standard model of the integers is that it is the one where every element/number has the property that the number of numbers between it and 0 is finite. |