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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.ohiostate.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 SkolemLowenheim 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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