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Indestructible Strong Unfoldability

Joel David Hamkins, Thomas A. Johnstone
2010 Notre Dame Journal of Formal Logic  
Finally, we obtain global and universal forms of indestructibility for strong unfoldability, finding the exact consistency strength of universal indestructibility for strong unfoldability. 2000 Mathematics  ...  From a stronger hypothesis, however, we prove that the strong unfoldability of κ can be made indestructible by all <κ-closed forcing.  ...  Then: (1) If the (κ + 2)-strong unfoldability of κ is indestructible over V [G] by all forcing of the form Add(κ, θ), then κ was strongly unfoldable in V . (2) If the (κ + 1)-strong unfoldability of κ  ... 
doi:10.1215/00294527-2010-018 fatcat:bmwtr5jjungpde2xhdkrwlomaa

Strongly unfoldable cardinals made indestructible

Thomas A. Johnstone
2008 Journal of Symbolic Logic (JSL)  
The Main Theorem shows that any strongly unfoldable cardinal κ can be made indestructible by &lt;κ-closed, κ-proper forcing.  ...  I provide indestructibility results for large cardinals consistent with V = L, such as weakly compact, indescribable and strongly unfoldable cardinals.  ...  Let κ be strongly unfoldable. Then there is a set forcing extension in which the strong unfoldability of κ is indestructible by <κ-closed, κ-proper forcing of any size.  ... 
doi:10.2178/jsl/1230396915 fatcat:ywauowm67fhq5lfjdtlhgep6xy

Large cardinals need not be large in HOD

Yong Cheng, Sy-David Friedman, Joel David Hamkins
2015 Annals of Pure and Applied Logic  
n-superstrong, ω-superstrong, λ-extendible, almost huge, huge, n-huge, rank-into-rank (I 0 , I 1 and I 3 ); Global: unfoldable, strongly unfoldable, tall, strong, supercompact, superhuge, and many others  ...  Strong cardinals Theorem There is a class forcing notion P forcing that 1 All strong cardinals of the ground model are preserved and no new strong cardinals are created. 2 There are no strong cardinals  ... 
doi:10.1016/j.apal.2015.07.004 fatcat:gogq5gyud5alvb6xraed7t2b6m

Unfoldable cardinals and the GCH [article]

Joel David Hamkins
1999 arXiv   pre-print
the strong cardinal context.  ...  Using these techniques, I show by set-forcing over any model of ZFC that any given unfoldable cardinal kappa can be made indestructible by the forcing to add any number of Cohen subsets to kappa.  ...  Thus, the notion of strong unfoldability in the conjecture can be completely avoided.  ... 
arXiv:math/9909029v1 fatcat:kjwh4k66nbd7zfqibi46plh6ua

Page 568 of Republican Vol. 13, Issue 18 [page]

1826 Republican  
568 THE REPUBLICAN. is an integral part of all being; formed of tho same indestructible matter; the essence of all nature.  ...  I feel a strong impulse of intellectual power. in the dic- tates of wisdom to develope human energy, under the auspices of American freedom, and to profit of that universal summons to  ... 

HOD-supercompactness, Indestructibility, and Level by Level Equivalence

Arthur W. Apter, Shoshana Friedman
2014 Bulletin of the Polish Academy of Sciences Mathematics  
In an attempt to extend the property of being supercompact but not hod-supercompact to a proper class of indestructibly supercompact cardinals, a theorem is discovered about a proper class of indestructibly  ...  , the strongly compact and supercompact cardinals coincide except at measurable limit points, and level by level equivalence between strong compactness and supercompactness holds above κ 0 but fails below  ...  This is since otherwise, some cardinal γ < κ 0 is supercompact up to a strong cardinal and hence is fully supercompact.  ... 
doi:10.4064/ba62-3-1 fatcat:no44tcqvlfbafdg74t4uxspqbu

Indestructibility properties of remarkable cardinals [article]

Yong Cheng, Victoria Gitman
2015 arXiv   pre-print
Here, we study the indestructibility properties of remarkable cardinals.  ...  We show that if κ is remarkable, then there is a forcing extension in which the remarkability of κ becomes indestructible by all κ-closed ≤κ-distributive forcing and all two-step iterations of the form  ...  unfoldable cardinal κ can be made indestructible by all <κ-closed κ + -preserving forcing [HJ10] .  ... 
arXiv:1411.2551v2 fatcat:avmr4jw6jvhmlbzjwskghtw7fu

Large cardinals need not be large in HOD [article]

Yong Cheng and Sy-David Friedman and Joel David Hamkins
2015 arXiv   pre-print
Similar results hold for many other types of large cardinals, such as measurable and strong cardinals.  ...  , superstrong, n-superstrong, ω-superstrong, λ-extendible, almost huge, huge, nhuge, rank-into-rank (I 0 , I 1 and I 3 ); Global: unfoldable, strongly unfoldable, tall, strong, supercompact, superhuge,  ...  There is a class forcing notion P forcing that (1) All strong cardinals of the ground model are preserved and no new strong cardinals are created. (2) There are no strong cardinals in the HOD of the extension  ... 
arXiv:1407.6335v2 fatcat:sassxl2ltffaxlbo3uwqqdxfpq

Page 64 of New Englander and Yale Review Vol. 42, Issue 172 [page]

1883 New Englander and Yale Review  
It is, however, auxiliary to the argument for such survival, a new cord, and a strong one, in the strand of that argument as at present constituted.  ...  ., ing life shall unfold in time a form still higher. On the Dar- winian principle, the argument of the Apostle is irresistible : “Tf there is a natural body there is also a spiritual body.”  ... 

Page 620 of The Bibliotheca Sacra Vol. 35, Issue 140 [page]

1878 The Bibliotheca Sacra  
[Oct. the various elements of the concept arise in various activities of our complex manhood, and, under the pressure of strong constitutional instinct, desire, and bias, coalesce in the in¬ comparable  ...  The view was maintained that the prime and indestructible postulates of all human thought — viz. the universe is thinkable, and my thought corresponds to the reality of the thinkable universe — guarantee  ... 

Page 278 of The Christian Examiner Vol. 7, Issue 6 [page]

1830 The Christian Examiner  
Happily, human nature is too strong for the utilitarian. It cannot satisfy itself with the convenient. No passion unfolds itself sooner than the love of the ornamental.  ...  Still, it teaches, even in its excesses, that the idea of beauty is an indestructible principle of our nature, and this single truth is enough to put us on our guard against vulgar notions of utility.  ... 

The proper and semi-proper forcing axioms for forcing notions that preserve $\aleph _2$ or $\aleph _3$

Joel David Hamkins, Thomas A. Johnstone
2008 Proceedings of the American Mathematical Society  
We prove that the PFA lottery preparation of a strongly unfoldable cardinal κ under ¬0 forces PFA(ℵ 2 -preserving), PFA(ℵ 3 -preserving) and PFA ℵ 2 , with 2 ω = κ = ℵ 2 .  ...  It follows by a result of Miyamoto that the existence of a strongly unfoldable cardinal is equiconsistent with the conjunction SPFA(ℵ 2 -preserving)+ SPFA(ℵ 3 -preserving) + SPFA ℵ 2 +2 ω = ℵ 2 .  ...  For example, Johnstone [Joh07, Joh] , fulfilling earlier hints that a Laver-style indestructibility phenomenon may be possible, proved that any strongly unfoldable cardinal κ can be made indestructible  ... 
doi:10.1090/s0002-9939-08-09727-x fatcat:s2qtkvr2tzbgjox4hurxn6a3xu

Page 171 of Andover Review; A Religious and Theological Monthly Vol. 10, Issue 56 [page]

1888 Andover Review; A Religious and Theological Monthly  
And the mind in this stage, while it keeps the notion of thing well in view as the type of reality, also has a strong teleo- logical bias.  ...  When to this is added a hear. say knowledge of the indestructibility of matter, it is hard to be- lieve that we have not in matter an absolute and eternal existence.  ... 

Page 522 of Bookman; A Review of Books and Life Vol. 73, Issue 5 [page]

1931 Bookman; A Review of Books and Life  
In the unfolding of this tearful history M. Bernstein has provided each of his three principals with sure-fire scenes. Mr.  ...  The original meaning of this word was drama accompanied by music; but it has come to mean with us popular drama of strong emotions, and at last sentimental drama in which the emotions are in bad taste.  ... 

Indestructibility properties of remarkable cardinals

Yong Cheng, Victoria Gitman
2015 Archive for Mathematical Logic  
Here, we study the indestructibility properties of remarkable cardinals.  ...  We show that if κ is remarkable, then there is a forcing extension in which the remarkability of κ becomes indestructible by all <κ-closed ≤κ-distributive forcing and all twostep iterations of the form  ...  unfoldable cardinal κ can be made indestructible by all <κ-closed κ + -preserving forcing [HJ10] .  ... 
doi:10.1007/s00153-015-0453-8 fatcat:ftcp4twdaba6rhbk3nnuiag4ha
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