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Title Limit Models, Superlimit Models, and Two Cardinal Problems in Abstract Elementary Classes
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University/Publisher University of Illinois – Chicago
Abstract This dissertation examines three main topics, the topic of defining "superstability" for abstract elementary classes (AECs), uniqueness of limit models, and two cardinal models in abstract elementary classes. In particular we further generalize an analogue of Vaught's theorem which constructs an uncountable two cardinal model starting from the existence of a countable Vaughtian pair in an elementary class to the AEC context originally published by Lessmann, who in turn built upon the work of Grossberg and VanDieren, Shelah, and others. We also give various sufficient conditions on countable models, as well as a condition on models of size kappa that, assuming that a simplified morass,ñ allows us to construct a (kappa^++,kappa)-model. We discuss how this work in AECs to some degree parallels the proof of Jensen's Gap-2 transfer theorem for elementary classes. We also discuss difficulties inherent in proving a true gap-2 transfer theorem for AECs. Additionally, we discuss, progress that has been made toward proving the uniqueness of limit models assuming various "superstability-like" assumptions (much of the work described is due to Shelah, Villaveces, Grossberg, and VanDieren). One small original result is contributed to this discussion.
Subjects/Keywords Limit Models; Superlimit Models; Two Cardinal Problems; two cardinal models; two cardinal; 2 cardinal; 2 cardinal problems; 2 cardinal model; gap-2 transfer; gap-2; Abstract Elementary Classes; mathematical logic; uniqueness of limit models; morasses; lessmann
Contributors Baldwin, John T. (advisor); Marker, David (committee member); Takloo-Bighash, Ramin (committee member); VanDieren, Monica (committee member); Scow, Lynn (committee member)
Language en
Rights Copyright 2013 Fred R. Drueck
Country of Publication us
Record ID handle:10027/9996
Repository uic
Date Retrieved
Date Indexed 2019-12-17
Issued Date 2013-06-28 00:00:00

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