61 research outputs found
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
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Set Theory
This workshop included selected talks on pure set theory and its applications, simultaneously showing diversity and coherence of the subject
Guessing axioms, invariance and suslin trees
In this thesis we investigate the properties of a group of axioms known as 'Guessing Axioms,' which extend the standard axiomatisation of set theory, ZFC. In particular, we focus on the axioms called 'diamond' and 'club,' and ask to what extent properties of the former hold of the latter. A question of 1. Juhasz, of whether club implies the existence of a Suslin tree, remains unanswered at the time of writing and motivates a large part of our in- vestigation into diamond and club. We give a positive partial answer to Juhasz's question by defining the principle Superclub and proving that it implies the exis- tence of a Suslin tree, and that it is weaker than diamond and stronger than club (though these implications are not necessarily strict). Conversely, we specify some conditions that a forcing would have to meet if it were to be used to provide a negative answer, or partial answer, to Juhasz's question, and prove several results related to this. We also investigate the extent to which club shares the invariance property of diamond: the property of being formally equivalent to many of its natural strength- enings and weakenings. We show that when certain cardinal arithmetic statements hold, we can always find different variations on club t.hat will be provably equiv- alent. Some of these hold in ZFC. But, in the absence of the required cardinal arithmetic, we develop a general method for proving that most variants of club are pairwise inequivalent in ZFC.EThOS - Electronic Theses Online ServiceGBUnited Kingdo
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Mini-Workshop: Feinstrukturtheorie und Innere Modelle
The main aim of fine structure theory and inner model theory can be summarized as the construction of models which have a canonical inner structure (a fine structure), making it possible to analyze them in great detail, and which at the same time reflect important aspects of the surrounding mathematical universe, in that they satisfy certain strong axioms of infinity, or contain complicated sets of reals. Applications range from obtaining lower bounds on the consistency strength of all sorts of set theoretic principles in terms of large cardinals, to proving the consistency of certain combinatorial properties, their compatibility with strong axioms of infinity, or outright proving results in descriptive set theory (for which no proofs avoiding fine structure and inner models are in sight). Fine structure theory and inner model theory has become a sophisticated and powerful apparatus which yields results that are among the deepest in set theory
Club guessing and the universal models
We survey the use of club guessing and other pcf constructs in the context of
showing that a given partially ordered class of objects does not have a
largest, or a universal element
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