Diamonds on large cardinals
نویسندگان
چکیده
Acknowledgements I want to express my sincere gratitude to my supervisor Professor Jouko Väänänen for supporting me during all these years of getting acquainted with the intriguing field of set theory. I am also grateful to all other members of the Helsinki Logic Group for interesting discussions and guidance. Especially I wish to thank Do-cent Tapani Hyttinen who patiently has worked with all graduate students regardless of whether they study under his supervision or not. Professor Saharon Shelah deserve special thanks for all collaboration and sharing of his insight in the subject. The officially appointed readers Professor Boban Veličkovi´c and Docent Kerkko Luosto have done a careful and minute job in reading the thesis. In effect they have served as referees for the second paper and provided many valuable comments. Finally I direct my warmest gratitude and love to my family to whom I also wish to dedicate this work. My wife Carmela and my daughters Jolanda and Vendela have patiently endured the process and have always stood me by.
منابع مشابه
ar X iv : 1 70 8 . 02 14 5 v 1 [ m at h . L O ] 7 A ug 2 01 7 JOINT DIAMONDS AND LAVER DIAMONDS
We study the concept of jointness for guessing principles, such as ♦κ and various Laver diamonds. A family of guessing sequences is joint if the elements of any given sequence of targets may be simultaneously guessed by the members of the family. We show that, while equivalent in the case of ♦κ, joint Laver diamonds are nontrivial new objects. We give equiconsistency results for most of the lar...
متن کاملTopics in Set Theory
Axiomatics. The formal axiomatic system of ordinary set theory (ZFC). Models of set theory. Absoluteness. Simple independence results. Transfinite recursion. Ranks. Reflection principles. Constructibility. [4] Infinitary combinatorics. Cofinality. Stationary sets. Fodor’s lemma. Solovay’s theorem. Cardinal exponentiation. Beth and Gimel functions. Generalized Continuum Hypothesis. Singular Card...
متن کاملIndescribable cardinals without diamonds
where E is a stationary subset of some cardinal ~. Jensen (cf. I-D2]) showed that if K is regular and uncountable and V=L then O~,E holds. Following his work, a number of applications of this principle and its modifications have been developed which are wide ranging and not restricted to set theory (cf. [-D1]). Jensen had also discovered that various large cardinals carry diamond sequences with...
متن کاملSouslin Trees Which Are Hard to Specialise
We construct some +-Souslin trees which cannot be specialised by any forcing which preserves cardinals and coonalities. For a regular cardinal we use the principle, for singular we use squares and diamonds.
متن کاملLarge cardinals and the Continuum Hypothesis
This is a survey paper which discusses the impact of large cardinals on provability of the Continuum Hypothesis (CH). It was Gödel who first suggested that perhaps “strong axioms of infinity” (large cardinals) could decide interesting set-theoretical statements independent over ZFC, such as CH. This hope proved largely unfounded for CH – one can show that virtually all large cardinals defined s...
متن کاملOn Iterated Forcing for Successors of Regular Cardinals
We investigate the problem of when ≤ λ–support iterations of < λ–complete notions of forcing preserve λ. We isolate a property — properness over diamonds — that implies λ is preserved and show that this property is preserved by λ–support iterations. Our condition is a relative of that presented in [1]; it is not clear if the two conditions are equivalent. We close with an application of our tec...
متن کامل