Talk:Prime model

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The axiomatization given for <N, S> and <N + Z, S> isn't actually enough to specify their complete theory (and thus demonstrate that they are elementarily equivalent). One can't prove from this axiomatization, for example, that no element is its own successor's successor. (Consider the model <N+Z_2, S>, where by Z_2 I mean a structure with two elements, each the other's successor, to see that this proof could not be accomplished).