Large Cardinals and Definable Counterexamples to the Continuum Hypothesis
نویسندگان
چکیده
In this paper we consider whether L(R) has “enough information” to contain a counterexample to the continuum hypothesis. We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for L(R) not to contain such a counterexample. Along the way we establish many results about non-stationary towers, non-reflecting stationary sets, generalizations of proper and semi-proper forcing and Chang’s conjecture.
منابع مشابه
Large Cardinals and Lightface Definable Well-Orders, without the GCH
This paper deals with the question whether the assumption that for every inaccessible cardinal κ there is a well-order of H(κ+) definable over the structure 〈H(κ+),∈〉 by a formula without parameters is consistent with the existence of (large) large cardinals and failures of the GCH. We work under the assumption that the SCH holds at every singular fixed point of the i-function and construct a c...
متن کاملLarge Cardinals and L-like Universes
There are many different ways to extend the axioms of ZFC. One way is to adjoin the axiom V = L, asserting that every set is constructible. This axiom has many attractive consequences, such as the generalised continuum hypothesis (GCH), the existence of a definable wellordering of the class of all sets, as well as strong combinatorial principles, such as ♦, and the existence of morasses. Howeve...
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 کاملViolating the Singular Cardinals Hypothesis Without Large Cardinals
Easton proved that the behavior of the exponential function 2 at regular cardinals κ is independent of the axioms of set theory except for some simple classical laws. The Singular Cardinals Hypothesis SCH implies that the Generalized Continuum Hypothesis GCH 2 = κ holds at a singular cardinal κ if GCH holds below κ. Gitik and Mitchell have determined the consistency strength of the negation of ...
متن کاملLarge Cardinals and Definable Well-Orderings of the Universe
We use a reverse Easton forcing iteration to obtain a universe with a definable wellordering, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle ♦∗ κ+ at a proper class of cardinals κ. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardin...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Ann. Pure Appl. Logic
دوره 76 شماره
صفحات -
تاریخ انتشار 1995