A premouse inheriting strong cardinals from V

2015/06/12 by Schlutzenberg, Farmer · 1 citation
#03E45 #03E55 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1506.04116

Abstract

We identify a premouse inner model L[𝔼], such that for any coarsely iterable background universe R modelling ZFC, L[𝔼]^R is a proper class premouse of R inheriting all strong and Woodin cardinals from R. Moreover, for each α\inOR, L[𝔼]^R|α is (ω,α)-iterable, via iteration trees which lift to coarse iteration trees on R. We prove that (k+1)-condensation follows from (k+1)-solidity together with (k,ω₁+1)-iterability (that is, roughly, iterability with respect to normal trees). We also prove that a slight weakening of (k+1)-condensation follows from (k,ω₁+1)-iterability (without the (k+1)-solidity hypothesis). The results depend on the theory of generalizations of bicephali, which we also develop.

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