Results 241 to 250 of about 1,318 (281)

Intensionality, Reflection, and Large Cardinals

Siberian Mathematical Journal, 2002
The authors study some axiomatic systems of set theory in which the reflection principle is postulated for all formulas without class quantifiers. These systems are interesting because in some of them the existence of large cardinals can be proven. Some aspects of interpretability of these theories in classical systems and in each other are considered.
Belyakin, N. V., Ganov, V. A.
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Inner Models and Large Cardinals

Bulletin of Symbolic Logic, 1995
In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic
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Large Cardinal Properties of Small Cardinals

1998
The fact that small cardinals (for example N1 and N2) can consistently have properties similar to those of large cardinals (for example measurable or supercompact cardinals) is a recurring theme in set theory. In these notes I discuss three examples of this phenomenon; stationary reflection, saturated ideals and the tree property.
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Weak Covering at Large Cardinals

Mathematical Logic Quarterly, 1997
AbstractWe show that weakly compact cardinals are the smallest large cardinals k where k+ < k+ is impossible provided 0# does not exist. We also show that if k+Kc < k+ for some k being weakly compact (where Kc is the countably complete core model below one strong cardinal), then there is a transitive set M with M ⊨ ZFC + “there is a strong ...
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Sequentially Large Cardinals

1983
Publisher Summary This chapter discusses sequentially large cardinals. Large cardinals defined via elementary embeddings are the largest cardinals and the strongest in terms of relative consistency strength. The chapter defines the sequentially large cardinals are defined, a spectrum of large cardinals that are defined via elementary embeddings, in ...
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Large cardinals and large dilators

Journal of Symbolic Logic, 1998
AbstractApplying Woodin's non-stationary tower notion of forcing, I prove that the existence of a supercompact cardinal κ in V and a Ramsey dilator in some small forcing extension V[G] implies the existence in V of a measurable dilator of size κ, measurable by κ-complete measures.
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Other Large Cardinals

1997
In Section 32 we studied weakly compact cardinals, which are inaccessible cardinals satisfying the weak compactness theorem for the infinitary language L k,ω. If we remove the restriction on the size of sets of sentences in the model theoretic characterization of weakly compact cardinals, we obtain a considerably stronger notion of large cardinals ...
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Large Cardinals from Determinacy

2009
This chapter gives an account of Woodin’s general technique for deriving large cardinal strength from determinacy hypotheses. These results appear here for the first time and the treatment is self-contained.
Peter Koellner, W. Hugh Woodin
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Combinatorics on large cardinals

Journal of Symbolic Logic, 1992
Our framework is ZFC, and we view cardinals as initial ordinals. Baumgartner ([Bal] and [Ba2]) studied properties of large cardinals by considering these properties as properties of normal ideals and not as properties of cardinals alone. In this paper we study these combinatorial properties by defining operations which take as input one or more ideals ...
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Small Definably-large Cardinals

2006
We study the definably-Mahlo, definably-weakly-compact, and the definably-indescribable cardinals, which are the definable versions of, respectively, Mahlo, weakly-compact, and indescribable cardinals. We study their strength as large cardinals and we show that the relationship between them is almost the same as the relationship between Mahlo, weakly ...
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