Classification theory and \(0^{\#}\). (English) Zbl 1063.03017

Friedman, Sy D.; Hyttinen, Tapani; Rautila, Mika

J. Symb. Log. 68, No. 2, 580-588 (2003).

This paper is part of a program investigating the solvability of problems concerning constructible sets under the assumption that \(0^{\#}\) exists.
The authors prove the following theorem. For \(T\) a constructible first-order theory that is countable in \(L\), (assuming \(0^{\#}\) exists) \(T\) is classifiable (i.e., superstable with NDOP and NOTOP) if and only if the potential-isomorphism problem is cardinal-preserving and real-preserving extensions of \(L\) for constructible models of \(T\) of size \(\omega_ 2^ L\) is solvable. By the solvability of this problem, the authors indicate that they mean the constructibility of the collection of all pairs \((\mathfrak A,\mathfrak B)\) from \(L\) where \(\mathfrak A\) and \(\mathfrak B\) are models of \(T\) with universe \(\omega_ 2^ L\) which are isomorphic in an extension of \(L\) that has the same cardinals and reals as \(L\).

Reviewer: J. M. Plotkin (East Lansing)

MSC:

03C55	Set-theoretic model theory
03C45	Classification theory, stability, and related concepts in model theory
03E45	Inner models, including constructibility, ordinal definability, and core models

Keywords:

constructibility; \(L\); 0 sharp; potential-isomorphism problem; classification theory

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