Certain very large cardinals are not created in small forcing extensions
نویسندگان
چکیده
منابع مشابه
Certain very large cardinals are not created in small forcing extensions
The large cardinal axioms of the title assert, respectively, the existence of a nontrivial elementary embedding j : Vλ → Vλ, the existence of such a j which is moreover Σn, and the existence of such a j which extends to an elementary j : Vλ+1 → Vλ+1. It is known that these axioms are preserved in passing from a ground model to a small forcing extension. In this paper the reverse directions of t...
متن کاملLarge Cardinals with Forcing
This chapter describes, following the historical development, the investigation of large cardinal hypotheses using the method of forcing. Large cardinal hypotheses, also regarded as strong axioms of infinity, have stimulated a vast mainstream of modern set theory, and William Mitchell’s chapter in this volume deals with their investigation through inner models, Menachem Kojman’s chapter with th...
متن کاملSuccessor Large Cardinals in Symmetric Extensions ∗
We give an exposition in modern language (and using partial orders) of Jech’s method for obtaining models where successor cardinals have large cardinal properties. In such models, the axiom of choice must necessarily fail. In particular, we show how, given any regular cardinal and a large cardinal of the requisite type above it, there is a symmetric extension of the universe in which the axiom ...
متن کاملVery Large Cardinals and Combinatorics
Large cardinals are currently one of the main areas of investigation in Set Theory. They are possible new axioms for mathematics, and they have been proven essential in the analysis of the relative consistency of mathematical propositions. It is particularly convenient the fact that these hypotheses are neatly well-ordered by consistency strength, therefore giving a meaningful tool of compariso...
متن کاملSmall Forcing Creates Neither Strong nor Woodin Cardinals
After small forcing, almost every strongness embedding is the lift of a strongness embedding in the ground model. Consequently, small forcing creates neither strong nor Woodin cardinals. The widely known Levy-Solovay Theorem [LevSol67] asserts that small forcing does not affect the measurability of any cardinal. If a forcing notion P has size less than κ, then κ is measurable in V P if and only...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Annals of Pure and Applied Logic
سال: 2007
ISSN: 0168-0072
DOI: 10.1016/j.apal.2007.07.002