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 \(V=L\).


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


Zbl 0915.03034
Full Text: DOI


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