Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

The Largest Suslin Axiom

We develop the basic fine structure theory of the minimal model of the Largest Suslin Axiom. In particular, we prove that that the minimal model of the Largest Suslin Axiom satisfies the Mouse Set Conjecture, and that the Proper Forcing Axiom implies the minimal model of the Largest Suslin Axiom exists

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