September 2011 Potential isomorphism of elementary substructures of a strictly stable homogeneous model
Sy-David Friedman, Tapani Hyttinen, Agatha Walczak-Typke
J. Symbolic Logic 76(3): 987-1004 (September 2011). DOI: 10.2178/jsl/1309952530


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 $\mathfrak{M}$ of some power π. We assume that in Gödel's constructible universe 𝕃, π is a regular cardinal at least the successor of the first cardinal in which $\mathfrak{M}$ is stable.

We show that the collection of pairs of submodels in 𝕃 as above which are potentially isomorphic with respect to certain cardinal-preserving extensions of 𝕃 is equiconstructible with 0#. As 0# is highly “transcendental” over 𝕃, 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.


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Sy-David Friedman. Tapani Hyttinen. Agatha Walczak-Typke. "Potential isomorphism of elementary substructures of a strictly stable homogeneous model." J. Symbolic Logic 76 (3) 987 - 1004, September 2011.


Published: September 2011
First available in Project Euclid: 6 July 2011

zbMATH: 1241.03038
MathSciNet: MR2849255
Digital Object Identifier: 10.2178/jsl/1309952530

Primary: 03C45 , 03C48 , 03C55
Secondary: 03E45

Rights: Copyright © 2011 Association for Symbolic Logic


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Vol.76 • No. 3 • September 2011
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