PP-2006-22: Notions of Strong Compactness without the Axiom of Choice

PP-2006-22: Kieftenbeld, Vincent (2006) Notions of Strong Compactness without the Axiom of Choice. [Report]

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Abstract

The study of large cardinal axioms is an active part of contemporary set theory. For a large cardinal notion there are often several definitions possible. For example, two common ways to define a large cardinal notion is as a critical point of an elementary embedding with certain properties, or in terms of ultrafilters. Many other types of definitions exist. With the axiom of choice these definitions are often equivalent. Without the axiom of choice, these definitions may not be equivalent anymore. Moreover the consistency strength of the large cardinal axiom may change with the ambient set theory, depending on which definition you choose. In this thesis we study several different definitions related to the notion of a compact cardinal. We will be guided by two main questions: What is the structure of implications between different definitions? And: What is the relative consistency strength of these definitions? In both cases the answers may depend on the presence or absence of the axiom of choice.

Item Type: Report
Report Nr: PP-2006-22
Series Name: Prepublication (PP) Series
Year: 2006
Uncontrolled Keywords: strongly compact cardinals; axiom of choice; elementary embedding; infinitary language
Subjects: Language
Depositing User: Benedikt
Date Deposited: 12 Oct 2016 14:36
Last Modified: 12 Oct 2016 14:36
URI: https://eprints.illc.uva.nl/id/eprint/197

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