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Does anyone know an example of a mathematically interesting first order theory T such that: (1) T can be formalized in the classical predicate calculus. (2) It is provable in ZFC that T is consistent and has no infinite models. (3) No upper bound is presently known for the cardinal numbers of the finite models of T, even though it has been proved that these models cannot be arbitrarily large.

flag	asked Oct 7 2011 at 22:19 Garabed Gulbenkian 3,003●6●13

Take any finite set of finite models of finite signature. The first order theory of that set is what you need (if and only if). – Mark Sapir Oct 7 2011 at 22:24

@Mark: What about (3)? – Emil Jeřábek Oct 7 2011 at 22:53

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Not overly interesting interesting, but one could encode some sort of open problem into the size of available models. For example, take sigma to be the sentence "there are at most BB(n) elements," where n is your favorite number and BB(n) is the nth busy beaver number. Obviously you could fit this to whatever natural numbers problem you like, but it's a bit contrived. – Richard Rast Oct 7 2011 at 23:17

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What I wanted to say is that your question is about first order theories of finite sets of finite models. Condition (3) is equivalent to asking that the size of the set should be unknown. There are lots of examples. Say, Vinogradov proved that almost all natural numbers are sums of three primes. Consider the set of all natural numbers which are not sums of three primes. This set is finite but we do not know if it is empty. A "number" can be easily viewed as a model in an appropriate signature. So this is (or easily converts to) an example of your theory. – Mark Sapir Oct 7 2011 at 23:28

Thanks for your answers. Using BB(n) may be just what is needed, provided that BB(n) was definable within the theory T. The closest resemlance to T that I could think of was to imagine a "theory" of simple "sporadic" groups before the discovery of the Monster. The only problem was that I do not know what axioms characterize "sporadic" groups and I do not know how to prove that there are no infinite "sporadic" groups. – Garabed Gulbenkian Oct 10 2011 at 14:44

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A question about first order theories having only finite models

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