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Potential isomorphism of elementary substructures of a strictly stable homogeneous model

Published online by Cambridge University Press:  12 March 2014

Sy-David Friedman ,
Tapani Hyttinen  and
Agatha C. Walczak-Typke
Sy-David Friedman
Affiliation:
Kurt Gödel Research Center For Mathematical Logic, University of Vienna, A-1090 Vienna, Austria, E-mail: sdf@logic.univie.ac.at
Tapani Hyttinen
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, Fi-00014 Helsinki, Finland, E-mail: tapani.hyttinen@helsinki.fi
Agatha C. Walczak-Typke
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, Fi-00014 Helsinki, Finland, E-mail: tapani.hyttinen@helsinki.fi
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Abstract

The results herein form part of a larger project to characterize the classification properties of the class of submodels of a homogeneous stable diagram in terms of the solvability (in the sense of [1]) of the potential isomorphism problem for this class of submodels.

We restrict ourselves to locally saturated submodels of the monster model , of some power π. We assume that in Gödel's constructive universe , π is a regular cardinal at least the successor of the first cardinal in which , is stable.

We show that the collection of pairs of submodels in as above which are potentially isomorphic with respect to certain cardinal-preserving extensions of is equiconstructible with 0#. As 0# is highly “transcendental” over , this provides a very strong statement to the effect that potential isomorphism for this class of models not only fails to be set-theoretically absolute, but is of high (indeed of the highest possible) complexity.

The proof uses a novel method that does away with the need for a linear order on the skeleton.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 3 , September 2011 , pp. 987 - 1004
Copyright
Copyright © Association for Symbolic Logic 2011

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References

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