Supercompactness and measurable limits of strong cardinals

2001 ◽  
Vol 66 (2) ◽  
pp. 629-639 ◽  
Author(s):  
Arthur W. Apter
Keyword(s):  
Strong Cardinals

AbstractIn this paper, two theorems concerning measurable limits of strong cardinals and supercompactness are proven. This generalizes earlier work, both individual and joint with Shelah.

scholarly journals Laver sequences for extendible and super-almost-huge cardinals

1999 ◽  
Vol 64 (3) ◽  
pp. 963-983 ◽  
Author(s):  
Paul Corazza
Keyword(s):  
Critical Point ◽  
Upper Bound ◽  
Large Cardinal ◽  
Strong Cardinal ◽  
Regular Class ◽  
Strong Cardinals ◽  
Extendible Cardinals

AbstractVersions of Laver sequences are known to exist for supercompact and strong cardinals. Assuming very strong axioms of infinity, Laver sequences can be constructed for virtually any globally defined large cardinal not weaker than a strong cardinal; indeed, under strong hypotheses. Laver sequences can be constructed for virtually any regular class of embeddings. We show here that if there is a regular class of embeddings with critical point κ, and there is an inaccessible above κ, then it is consistent for there to be a regular class that admits no Laver sequence. We also show that extendible cardinals are Laver-generating, i.e., that assuming only that κ is extendible, there is an extendible Laver sequence at κ. We use the method of proof to answer a question about Laver-closure of extendible cardinals at inaccessibles. Finally, we consider Laver sequences for super-almost-huge cardinals. Assuming slightly more than super-almost-hugeness, we show that there are super-almost-huge Laver sequences, improving the previously known upper bound for such Laver sequences. We also describe conditions under which the canonical construction of a Laver sequence fails for super-almost-huge cardinals.

scholarly journals A premouse inheriting strong cardinals from V

2020 ◽  
Vol 171 (9) ◽  
pp. 102826
Author(s):  
Farmer Schlutzenberg
Keyword(s):  
Strong Cardinals

scholarly journals Indestructibility of generically strong cardinals

2016 ◽  
Vol 232 (2) ◽  
pp. 131-149 ◽  
Author(s):  
Brent Cody ◽  
Sean Cox
Keyword(s):  
Strong Cardinals

scholarly journals Strong cardinals in the core model

1997 ◽  
Vol 83 (2) ◽  
pp. 165-198 ◽  
Author(s):  
Kai Hauser ◽  
Greg Hjorth
Keyword(s):  
Core Model ◽  
The Core ◽  
Strong Cardinals

Universally Baire sets and definable well-orderings of the reals

2003 ◽  
Vol 68 (4) ◽  
pp. 1065-1081
Author(s):  
SY D. Friedman ◽  
Ralf Schindler
Keyword(s):  
Inner Models ◽  
Strong Cardinals

AbstractLet n ≥ 3 be an integer. We show that it is consistent (relative to the consistency of n − 2 strong cardinals) that every Σ1n-set of reals is universally Baire yet there is a (lightface) projective well-ordering of the reals. The proof uses “David's trick” in the presence of inner models with strong cardinals.

scholarly journals Determinacy in strong cardinal models

2011 ◽  
Vol 76 (2) ◽  
pp. 719-728
Author(s):  
P. D. Welch
Keyword(s):  
Strong Cardinal ◽  
Inner Models ◽  
Inner Model ◽  
Strong Cardinals

AbstractWe give limits defined in terms of abstract pointclasses of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example:Theorem A. Det(-IND) ⇒ there exists an inner model with a strong cardinal.Theorem B. Det(AQI) ⇒ there exist type-l mice and hence inner models with proper classes of strong cardinals.where -IND(AQI) is the pointclass of boldface -inductive (respectively arithmetically quasi-inductive) sets of reals.

Between strong and superstrong

1986 ◽  
Vol 51 (3) ◽  
pp. 547-559 ◽  
Author(s):  
Stewart Baldwin
Keyword(s):  
Critical Point ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Elementary Embedding ◽  
Proper Class ◽  
Consistency Strength ◽  
Strong Cardinal ◽  
Definition Of ◽  
Strong Cardinals

Definition. A cardinal κ is strong iff for every x there is an elementary embedding j:V → M with critical point κ such that x ∈ M.κ is superstrong iff ∃j:V → M with critical point κ such that Vj(κ) ∈ M.These definitions are natural weakenings of supercompactness and hugeness respectively and display some of the same relations. For example, if κ is superstrong then Vκ ⊨ “∃ proper class of strong cardinals”, but the smallest superstrong cardinal is less than the smallest strong cardinal (if both types exist). (See [SRK] and [Mo] for the arguments involving supercompact and huge, which translate routinely to strong and superstrong.)Given any two types of large cardinals, a typical vague question which is often asked is “How large is the gap in consistency strength?” In one sense the gap might be considered relatively small, since the “higher degree” strong cardinals described below (a standard trick that is nearly always available) and the Shelah and Woodin hierarchies of cardinals (see [St] for a definition of these) seem to be (at least at this point in time) the only “natural” large cardinal properties lying between strong cardinals and superstrong cardinals in consistency strength.

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