countable - 1 COUNTABLE MODEL THEORY AND LARGE CARDINALS by...

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1 COUNTABLE MODEL THEORY AND LARGE CARDINALS by Harvey M. Friedman Department of Mathematics Ohio State University September, 1996 friedman@math.ohio-state.edu A familiar idea in core mathematics is to add a point at infinity, often in a canonical way. We can look at this model theoretically as follows. By the linearly ordered predicate calculus, we simply mean ordinary predicate calculus with equality and a special binary relation symbol <. It is required that in all interpretations, < be a linear ordering on the domain. Thus we have the usual completeness theorem provided we add the axioms that assert that < is a linear ordering. It will be convenient to consider subrational models M. These are models whose domain is a nonempty subset of the rationals, and whose ordering agrees with the usual ordering of rationals. We will be particularly interested in the subset N containedin Q of nonnegative integers; of course N may not be a subset of the domain of M, written dom(M). It is obvious that by the downward Skolem-Lowenheim theorem, any consistent theory in linearly ordered predicate calculus has a subrational model. Let T be a theory in linearly ordered predicate calculus. We say that “models have unique extensions (to models) at infinity” if and only if the following holds: Let M = (D,<,…) be a model of T, which consists of a nonempty domain D, a linear ordering on D, and the components of M consisting of constants from M, relations of
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countable - 1 COUNTABLE MODEL THEORY AND LARGE CARDINALS by...

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