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  • Print publication year: 2010
  • Online publication date: March 2011

Cardinal preserving elementary embeddings

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Summary

Abstract. Say that an elementary embedding j : N → M is cardinal preserving if CARM = CARN = CAR. We show that if PFA holds then there are no cardinal preserving elementary embeddings j : M → V. We also show that no ultrapower embedding j : V → M induced by a set extender is cardinal preserving, and present some results on the large cardinal strength of the assumption that there is a cardinal preserving j : V → M.

Introduction. This paper is the first of a series attempting to investigate the structure of (not necessarily fine structural) inner models of the set theoretic universe under assumptions of two kinds:

Forcing axioms, holding either in the universe ∨ of all sets or in both ∨ and the inner model under study, and

Agreement between (some of) the cardinals of ∨ and the cardinals of the inner model.

I try to be as self-contained as is reasonably possible, given the technical nature of the problems under consideration. The notation is standard, as in Jech. I assume familiarity with inner model theory; for fine structural background and notation, the reader is urged to consult Steel and Mitchell.

In the remainder of this introduction, I include some general observations on large cardinal theory, forcing axioms, and fine structure, and state the main results of the paper.

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Logic Colloquium 2007
  • Online ISBN: 9780511778421
  • Book DOI: https://doi.org/10.1017/CBO9780511778421
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