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DEFINABILITY OF SATISFACTION IN OUTER MODELS

Published online by Cambridge University Press:  14 September 2016

SY-DAVID FRIEDMAN
Affiliation:
KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC WÄHRINGER STRASSE 25 1090 VIENNAAUSTRIAE-mail: sdf@logic.univie.ac.at
RADEK HONZIK
Affiliation:
KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC WÄHRINGER STRASSE 25 1090 VIENNAAUSTRIAE-mail: radek.honzik@ff.cuni.cz

Abstract

Let M be a transitive model of ZFC. We say that a transitive model of ZFC, N, is an outer model of M if MN and ORD ∩ M = ORD ∩ N. The outer model theory of M is the collection of all formulas with parameters from M which hold in all outer models of M (which exist in a universe in which M is countable; this is independent of the choice of such a universe). Satisfaction defined with respect to outer models can be seen as a useful strengthening of first-order logic. Starting from an inaccessible cardinal κ, we show that it is consistent to have a transitive model M of ZFC of size κ in which the outer model theory is lightface definable, and moreover M satisfies V = HOD. The proof combines the infinitary logic L∞,ω, Barwise’s results on admissible sets, and a new forcing iteration of length strictly less than κ+ which manipulates the continuum function on certain regular cardinals below κ. In the appendix, we review some unpublished results of Mack Stanley which are directly related to our topic.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2016 

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References

REFERENCES

Barwise, J., Admissible Sets and Structures, Springer, Berlin, 1975.CrossRefGoogle Scholar
Kunen, K., Set Theory: An Introduction to Independence Proofs, North Holland, Amsterdam, 1980.Google Scholar
Stanley, M. C., Outer model satisfiability, an unpublished manuscript.Google Scholar