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Indestructible remarkable cardinals

Friday 28th August 2015 - 16:00 to 17:00
INI Seminar Room 1
In 2000, Schindler introduced remarkable cardinals and showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of $L(\mathbb R)$ is absolute for proper forcing. Remarkable cardinals can be thought of either as a miniature version of strong cardinals or as having aspects of generic supercompactness, but they are relatively low in the large cardinal hierarchy. They are downward absolute to $L$ and lie (consistency-wise) between the 1-iterable and 2-iterable cardinals of the $\alpha$-iterable cardinals hierarchy (below Ramsey cardinals). I will discuss the indestructibility properties of remarkable cardinals, which are similar to those of strong cardinals. I will show that a remarkable cardinal $\kappa$ can be made simultaneously indestructible by all $\lt\kappa$-closed $\leq\kappa$-distributive forcing and by all forcing of the form ${\rm Add}(\kappa,\theta)*\mathbb R$, where $\mathbb R$ is forced to be $\lt\kappa$-closed and $\leq\kappa$-distributive. For this argument, I will introduce the notion of a remarkable Laver function and show that every remarkable cardinal has one. Although, the existence of Laver-like functions can be forced for most large cardinals, few, such as strong, supercompact, and extendible cardinals, have them outright. The established indestructibility can be used to show, for instance, that any consistent continuum pattern on the regular cardinals can be realized above a remarkable cardinal and that a remarkable cardinal need not be even weakly compact in ${\rm HOD}$. This is joint work with Yong Cheng.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons