Classification theory and 0<sup>#</sup>

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

doi:10.2178/jsl/1052669064

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“…L that are isomorphic to T in some cardinal preserving extension of L. The following was proved in [4]:…”

Section: Theorem 1 Let T Be a Countable First Order Theory And Let κmentioning

confidence: 98%

“…But one may consider weakening the requirement that the extension must be generic. Such notions are studied in [4], and it is shown there that this kind of notions are not always decidable. By a cardinal preserving extension of L we mean a transitive model of ZFC that contains all ordinals, is contained in a set-generic extension of V , and has the same cardinals as L. For a tree T ∈ L on (ω 1 )…”

Section: Theorem 1 Let T Be a Countable First Order Theory And Let κmentioning

confidence: 99%

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Potential isomorphism and semi-proper trees

2002

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Abstract.We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notion of weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees.We show that the existence of weakly semi-proper trees is consistent relative to ZFC by proving the existence of weakly semi-proper trees under certain cardinal arithmetic assumptions. We also prove the consistency of the non-existence of weakly semi-proper trees assuming the consistency of some large cardinals.Introduction. Two structures are said to be potentially isomorphic if they are isomorphic in some extension of the universe in which they reside. Different notions of potential isomorphism arise as restrictions are placed on the method to extend the universe. Nadel and Stavi [13] considered generic extensions in which there are no new subsets of cardinality less than κ, where κ is the cardinality of the models. They used some cardinal arithmetic assumptions on κ to show the existence of a pair of non-isomorphic but potentially isomorphic models. This kind of result can be interpreted as a non-structure theorem for the theory of the models in question.In [6] these studies were continued, with an emphasis on classification theory. One of the results obtained there concerning the notion introduced in [13] is:

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“…L that are isomorphic to T in some cardinal preserving extension of L. The following was proved in [4]:…”

Section: Theorem 1 Let T Be a Countable First Order Theory And Let κmentioning

confidence: 98%

Section: Theorem 1 Let T Be a Countable First Order Theory And Let κmentioning

confidence: 99%