Moti Gitik and Menachem Magidor. Extender based forcings. The Journal of Symbolic Logic, vol. 59 (1994), pp. 445–460. - Moti Gitik and William J. Mitchell. Indiscernible sequences for extenders, and the singular cardinal hypothesis. Annals of Pure and Applied Logic, vol. 82 (1996), pp. 273–316. - Moti Gitik. Blowing up the power of a singular cardinal. Annals of Pure and Applied Logic, vol. 80 (1996), pp. 17–33. - Moti Gitik and Carmi Merimovich. Possible values for and . Annals of Pure and Applied Logic, vol. 90 (1997), pp. 193–241. - Moti Gitik. Blowing up power of a singular cardinal—wider gaps. Annals of Pure and Applied Logic, vol. 116 (2002), pp. 1–38.

2003 ◽  
Vol 9 (2) ◽  
pp. 237-241
Author(s):  
Akihiro Kanamori
Keyword(s):  
Symbolic Logic ◽  
Singular Cardinal ◽  
Applied Logic ◽  
Singular Cardinal Hypothesis ◽  
Blowing Up

scholarly journals The tree property and the failure of the Singular Cardinal Hypothesis at ℵω2

2012 ◽  
Vol 77 (3) ◽  
pp. 934-946 ◽  
Author(s):  
Dima Sinapova
Keyword(s):  
Generic Extension ◽  
Singular Cardinal ◽  
Tree Property ◽  
Supercompact Cardinals ◽  
Singular Cardinal Hypothesis

AbstractWe show that given ω many supercompact cardinals, there is a generic extension in which the tree property holds at ℵω2+ 1 and the SCH fails at ℵω2.

A definable failure of the singular cardinal hypothesis

2012 ◽  
Vol 192 (2) ◽  
pp. 719-762 ◽  
Author(s):  
Sy-David Friedman ◽  
Radek Honzik
Keyword(s):  
Singular Cardinal ◽  
Singular Cardinal Hypothesis

scholarly journals ARONSZAJN TREES AND FAILURE OF THE SINGULAR CARDINAL HYPOTHESIS

2009 ◽  
Vol 09 (01) ◽  
pp. 139-157 ◽  
Author(s):  
ITAY NEEMAN
Keyword(s):  
Large Cardinals ◽  
Singular Cardinal ◽  
Strong Limit ◽  
Tree Property ◽  
Singular Cardinal Hypothesis ◽  
Aronszajn Trees ◽  
Strongly Compact Cardinals

The tree property at κ+ states that there are no Aronszajn trees on κ+, or, equivalently, that every κ+ tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ+ and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible above strongly compacts. In this paper, we reconcile the two. We prove from large cardinals that the tree property at κ+ is consistent with failure of the singular cardinal hypothesis at κ.

S. S. Goncharov. Autostability and computable families of constructivizations. Algebra and Logic, vol. 14 (1975), no. 6, pp. 392–409. - S. S. Goncharov. The quantity of nonautoequivalent constructivizations. Algebra and Logic, vol. 16 (1977), no. 3, pp. 169–185. - S. S. Goncharov and V. D. Dzgoev. Autostability of models. Algebra and Logic, vol. 19 (1980), no. 1, pp. 28–37. - J. B. Remmel. Recursively categorical linear orderings. Proceedings of the American Mathematical Society, vol. 83 (1981), no. 2, pp. 387–391. - Terrence Millar. Recursive categoricity and persistence. The Journal of Symbolic Logic, vol. 51 (1986), no. 2, pp. 430–434. - Peter Cholak, Segey Goncharov, Bakhadyr Khoussainov and Richard A. Shore. Computably categorical structures and expansions by constants. The Journal of Symbolic Logic, vol. 64 (1999), no. 1, pp. 13–137. - Peter Cholak, Richard A. Shore and Reed Solomon. A computably stable structure with no Scott family of finitary formulas. Archive for Mathematical Logic, vol. 45 (2006), no. 5, pp. 519–538. - Chris Ash, Julia Knight, Mark Manasse and Theodore Slaman. Generic copies of countable structures. Annals of Pure and Applied Logic, vol. 42 (1989), no. 3, pp. 195–205. - John Chisholm. Effective model theory vs. recursive model theory. The Journal of Symbolic Logic, vol. 55 (1990), no. 3, pp. 1168–1191.

2012 ◽  
Vol 18 (1) ◽  
pp. 131-134
Author(s):  
Daniel Turetsky
Keyword(s):  
Mathematical Logic ◽  
Model Theory ◽  
American Mathematical Society ◽  
Effective Model ◽  
Mathematical Society ◽  
Symbolic Logic ◽  
Applied Logic ◽  
Recursive Model ◽  
Linear Orderings ◽  
Scott Family
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