Universal Indestructibility
Arthur W. Apter, Joel David Hamkins
TL;DR
This paper constructs a model in set theory where all supercompact and partially supercompact cardinals are indestructible by certain forcing, based on a large cardinal hypothesis, highlighting limitations of indestructibility with multiple supercompact cardinals.
Contribution
It introduces a large cardinal hypothesis that yields a model with universal indestructibility of supercompact cardinals, which was previously impossible with multiple such cardinals.
Findings
A model with a supercompact cardinal exhibiting universal indestructibility.
Indestructibility holds for all supercompact and partially supercompact cardinals in the model.
Such a state is impossible with two supercompact cardinals or beyond a measurable cardinal.
Abstract
From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal kappa is fully indestructible by kappa-directed closed forcing. Such a state of affairs is impossible with two supercompact cardinals or even with a cardinal which is supercompact beyond a measurable cardinal.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Computability, Logic, AI Algorithms · Mathematical and Theoretical Analysis