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

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

2011 ◽  
Vol 76 (2) ◽  
pp. 719-728
Author(s):  
P. D. Welch

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.


2009 ◽  
Vol 74 (1) ◽  
pp. 315-335 ◽  
Author(s):  
Ronald Jensen ◽  
Ernest Schimmerling ◽  
Ralf Schindler ◽  
John Steel

AbstractWe show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal κ ≥ ℵ such that □κ and □(κ) fail. 2) There is a cardinal κ such that κ is weakly compact in the generic extension by Col(κ, κ+). Of special interest is 1) with κ = ℵ3 since it follows from PFA by theorems of Todorcevic and Velickovic. Our main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over Kc∥κ.


2007 ◽  
Vol 72 (3) ◽  
pp. 721-737 ◽  
Author(s):  
J. R. Steel

The full-background-extender Kc -construction of [2] has the property that, if it does not break down and produces a final model , thenΉ is Woodin in V ⇒ Ή is Woodin in ,for all Ή. It is natural to ask whetherκ is strong in V ⇒ κ is λ-strong in ,for all κ, or even better,κ is λ-strong in V ⇒ κ is λ-strong in .As one might suspect, the more useful answer would be “yes”.For the Kc-construction of [2], this question is open. The problem is that the construction of [2] is not local: because of the full-background-extender demand, it may produce mice projecting to ρ at stages much greater than ρ. Because of this, there is no reason to believe that if E is a λ-strong extender of V, then The natural proof only gives that if κ is Σ2-strong, then Σ, is strong in .We do not know how to get started on this question, and suspect that in fact strong cardinals in V may fail to be strong in , if is the output of the construction of [2]. Therefore, we shall look for a modification of the construction of [2]. One might ask for a construction with output such that(1) iteration trees on can be lifted to iteration trees on V,(2) ∀δ(δ is Woodin ⇒ δ is Woodin in ), and(3) (a) ↾κ(κ is a strong cardinal ⇒ κ is strong in ), and (b) ↾κ↾λ(Lim(λ) Λ κ is λ-strong ⇒ κ is λ-strong in ).


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

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.


2003 ◽  
Vol 68 (2) ◽  
pp. 669-688 ◽  
Author(s):  
Arthur W. Apter ◽  
Joel David Hamkins

AbstractWe summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify Theorems 1 and 2 of [5], due to the first author.


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