A crucial difference between non-Euclidean geometries and "non-inductive" models of PA- is that any model of PA- contains a canonical copy of the true natural numbers, and in this copy of N, the induction schema is true. In other words, PA is part of the complete theory of a very canonical model of PA-, and as such it seems much more natural (so to speak) to study fragments of PA rather than extensions of PA- which contradict induction.
To make this a little more precise (and sketch a proof), the axioms of PA- say that any model M has a unique member 0_M which is not the successor of anything, and that the successor function S_M: M to M is injective; so by letting k_M (for any k in N) be the k-th successor of 0_M, the set {k_M : k in N} forms a submodel of M which is isomorphic to the usual natural numbers, N. Any extra elements of M not lying in this submodel lie in various "Z-chains," that is, infinite orbits of the model M's successor function S_M.
(In the language of categories: the "usual natural numbers" are an initial object in the category of all models of PA-, where morphisms are injective homomorphisms in the sense of model theory.)
So, while PA seems natural, I'm not sure why there would be any more motivation to study PA- plus "non-induction" than there is to study any of the other countless consistent theories you could cook up, unless you find that one of these non-inductive extensions of PA- has a particularly nice class of models.
No comments:
Post a Comment