Title: Categoricity Without Equality
Authors: H. Jerome Keisler and Arnold W. Miller
Abstract:
We study categoricity in power for reduced models of first order logic
without equality.
The object of this paper is to study categoricity in power for theories in
first order logic without equality. Our results will reveal some surprising
differences between the model theory for logic without equality and for
logic with equality.
When we consider categoricity, it is natural to identify elements which are
indistinguishable from each other. The natural way to do this is to confine
our attention to reduced models, that is, models such that any pair of
elements which satisfy the same formulas with parameters from the model are
equal. We also confine our attention to complete theories T in a countable
language such that all models of T are infinite. T is said to be
kappa-categorical if T has exactly one reduced model of cardinality kappa up
to isomorphism.
Section 3 contains several examples of omega-categorical theories in logic
without equality which have infinitely many complete 1-types or 2-types.
The reason for this different behavior is clarified in Section 4, where we
see what happens to the Omitting Types Theorem in logic without equality.
In Section 5 we apply the Omitting Types Theorem to study omega-categoricity
and the existence of prime models of in logic without equality.
Section 3 also contains examples of bounded theories, i.e. theories for which
the class of cardinalities of infinite models is bounded. In Section 6 we
show that there are just three possibilities: All models of T are countable,
the maximum cardinality of a model of T is the continuum, or T has models of
all infinite cardinalities (i.e. T is unbounded). This shows that the Hanf
number of first order logic without equality is continuum^+. In Section 7
we show that no bounded theory is categorical in an uncountable cardinal.
Finally, the Los conjecture for logic without equality is proved in
Section 8.
This appears in the Los Memorial Volume, Fundamenta Mathematicae, 170(2001),
87-106.