# [FOM] The use of replacement in model theory

John Baldwin jbaldwin at uic.edu
Thu Jan 28 17:23:49 EST 2010

Byunghan Kim proved that for a simple first order theory non-forking is
equivalent to
non-dividing. The notions of simple, non-forking, and non-dividing are all
statements about countable sets of formulas.  Nevertheless, the argument
for the result employs Morley's technique for omitting types; that is it
uses the Erdos-Rado theorem on all cardinals less than $\beth_{\omega_1}$.

Thus, a priori, this is a use of the replacement axiom for a result whose
statement does not require replacement.  (I use this more technical
example rather than the original Hanf number computation, precisely for
this reason).

Does any one know whether this use is essential?

An introductory account of this topic appears in the paper by Kim and
Pillay: From stability to simplicity, Bulletin of Symbolic Logic,
4, (1998), 17-36.

John T. Baldwin
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
60607