×

Strongly unfoldable cardinals made indestructible. (English) Zbl 1168.03039

In the paper, the author considers indestructibility for large cardinals consistent with \(V=L\), particularly strongly unfoldable cardinals. In [“Chains of end elementary extensions of models of set theory”, J. Symb. Log. 63, No. 3, 1116–1136 (1998; Zbl 0915.03034)], A. Villaveces defined a strongly unfoldable cardinal as follows: For an inaccessible cardinal \(\kappa\), we say that a transitive structure \(M\) of size \(\kappa\) is a \(\kappa\)-model if and only if \(M\) satisfies ZFC\(^-\), \(\kappa\in M\), and \(M\) is closed under \(<\kappa\)-sequences. For an ordinal \(\theta\), we say that a cardinal \(\kappa\) is \(\theta\)-strongly unfoldable if \(\kappa\) is inaccessible and for every \(\kappa\)-model \(M\) there is an elementary embedding \(j:M\rightarrow N\) with critical point \(\kappa\) such that \(\theta<j(\kappa)\) and \(V_\theta\subseteq N\). \(\kappa\) is called strongly unfoldable if and only if it is \(\theta\)-strongly unfoldable for every ordinal \(\theta\). Every strongly unfoldable cardinal is weakly compact and totally indescribable. This large cardinal is consistent with \(V=L\) if it is consistent with ZFC.
The main result is that if \(\kappa\) is a strongly unfoldable cardinal, then there is a forcing extension in which the strong unfoldability of \(\kappa\) is indestructible by \({<}\kappa\)-closed, \(\kappa\)-proper forcing. The author gives two fairly different proofs to this theorem. Note that in the witnessing model, the weak compactness and total indescribability of \(\kappa\) are also indestructible by \(<\kappa\)-closed, \(\kappa\)-proper forcing.
He also proves the global version: there is a class forcing extension \(V[G]\) such that (1) every strongly unfoldable cardinal of \(V\) remains strongly unfoldable in \(V[G]\), (2) in \(V[G]\), every strongly unfoldable cardinal \(\kappa\) is indestructible by \(<\kappa\)-closed, \(\kappa\)-proper set forcing, and (3) no new strongly unfoldable cardinals are created. In addition, he considers a wide range of issues about indestructibility and destructibility of large cardinals consistent with \(V=L\).

MSC:

03E55 Large cardinals
03E35 Consistency and independence results
03E40 Other aspects of forcing and Boolean-valued models

Citations:

Zbl 0915.03034
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Shelah’s search for properness for iterations with uncountable supports (2005)
[2] Independence results 45 pp 563– (1980)
[3] DOI: 10.1016/S0168-0072(99)00010-X · Zbl 0949.03045
[4] Small forcing makes any cardinal superdestructible 63 pp 51– (1998)
[5] DOI: 10.1007/BF01624081 · Zbl 0663.03041
[6] DOI: 10.4064/fm179-3-4 · Zbl 1066.03051
[7] DOI: 10.1016/j.apal.2006.05.001 · Zbl 1110.03032
[8] Constructibility (1984) · Zbl 0542.03029
[9] DOI: 10.1142/9789812812940_0011
[10] DOI: 10.1007/BF02761175 · Zbl 0381.03039
[11] Set theory, An introduction to independence proofs (1980) · Zbl 0443.03021
[12] Set theory (2003)
[13] The canary tree revisited 66 pp 1677– (2001) · Zbl 0993.03061
[14] Indescribable cardinals and elementary embeddings 56 pp 439– (1991)
[15] DOI: 10.4064/fm180-3-4 · Zbl 1066.03052
[16] Chains of end elementary extensions of models of set theory 63 pp 1116– (1998) · Zbl 0915.03034
[17] Proper and improper forcing (1998) · Zbl 0889.03041
[18] Unfoldable cardinals and the GCH 66 pp 1186– (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.