[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

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