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Satiable subsets of
by
M. C. (Mack) Stanley
San Jose State
Assume the GCH for simplicity. Let \kappa = \aleph\lambda+1, where \lambda is an infinite cardinal. Say that a stationary subset of \kappa is satiable if it has a club subset in an outer model in which \kappa retains its status as the successor of \aleph\lambda, that is, in which \lambda, \kappa, and cardinals cofinal in \aleph\lambda (hence \aleph\lambda itself) are preserved.
The theorem is that in general there is no first-order characterization of the satiable subsets of \aleph\omega1+1, even if we restrict to subsets of ``bounded pattern width'' that are satiable in strongly covered outer models.
This contrasts with the case of \aleph\omega+1, where it is possible to give a first-order characterization of the bounded pattern width subsets that are satiable in strongly covered outer models.
A subset S of \kappa has bounded pattern width if it has fewer than the maximum possible number of ``patterns''. If \alpha in S has uncountable cofinality, then the pattern of S at \alpha is { \delta <
Date received: March 1, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadv-11.