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A note on end extensions
by
C. Dimitracopoulos
University of Athens
Coauthors: Ch. Cornaros
After the work in [2], the following problem remained open:
Does every countable model M of \Sigma1 collection have a proper
end extension satisfying \Delta0 induction?
This problem was studied extensively by A. Wilkie-J. Paris ([3]), who
defined the notion of ``\Gamma-fullness'' and proved the following:
Theorem For any M as above, if M is I\Delta0-full, then M has
a proper end extension satisfying I\Delta0.
We define the notion of ``\Gamma-completeness'', which appears to be weaker than \Gamma-fullness, but still sufficient to imply the proper end extendibility of M to a model of the theory \Gamma (extending I\Delta0). In an attempt to solve a problem in [1], we investigate ways of exploiting the methods of [3] to obtain sufficient conditions for the existence of proper (\Sigman-elementary) end extensions of M satisfying certain theories extending I\Delta0.
References
[1] P. Clote: A note on the MacDowell-Specker theorem, Fund. Math. 127
(1986), 163-170.
[2] J. Paris and L. Kirby: \Sigman-collection schemas in arithmetic,
Logic Colloquium' 77 (North-Holland, Amsterdam, 1978), 199-209.
[3] A. Wilkie and J. Paris: On the existence of end extensions of models
of bounded induction, J. E. Fenstad et al., eds., Logic, Methodology
and Philosophy of Science VIII (Moscow, 1987), 143-161, Stud. Logic
Found. Math. 126, North-Holland, Amsterdam, 1989.
Date received: April 18, 2002
Copyright © 2002 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 # cail-21.