## 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:

 3e+55 Large cardinals 3e+35 Consistency and independence results 3e+40 Other aspects of forcing and Boolean-valued models

### Keywords:

strongly unfoldable cardinal; forcing; indestructibility

Zbl 0915.03034
Full Text:

### References:

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