Titel
Destructibility and axiomatizability of Kaufmann models
Abstract
A Kaufmann model is an ω1-like, recursively saturated, rather classless model of PA (or ZF). Such models were constructed by Kaufmann under the combinatorial principle ♢ω1 and Shelah showed they exist in ZFC by an absoluteness argument. Kaufmann models are an important witness to the incompactness of ω1 similar to Aronszajn trees. In this paper we look at some set theoretic issues related to this motivated by the seemingly naïve question of whether such a model can be “killed” by forcing without collapsing ω1. We show that the answer to this question is independent of ZFC and closely related to similar questions about Aronszajn trees. As an application of these methods we also show that it is independent of ZFC whether or not Kaufmann models can be axiomatized in the logic Lω1,ω(Q) where Q is the quantifier “there exists uncountably many”.
Stichwort
Kaufmann ModelsStrong logicsDestructibilityMartin’s Axiom
Objekt-Typ
Sprache
Englisch [eng]
Persistent identifier
Erschienen in
Titel
Archive for Mathematical Logic
Band
61
Ausgabe
7-8
ISSN
0933-5846
Erscheinungsdatum
2022
Seitenanfang
1091
Seitenende
1111
Publication
Springer Science and Business Media LLC
Erscheinungsdatum
2022
Zugänglichkeit
Rechte
Rechteangabe
© The Author(s) 2022
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