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STABLY MEASURABLE CARDINALS

Part of: Set theory

Published online by Cambridge University Press:  15 June 2020

PHILIP D. WELCH*
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF BRISTOLBRISTOLBS8 1TW, UKE-mail:p.welch@bristol.ac.uk

Abstract

We define a weak iterability notion that is sufficient for a number of arguments concerning $\Sigma _{1}$ -definability at uncountable regular cardinals. In particular we give its exact consistency strength first in terms of the second uniform indiscernible for bounded subsets of $\kappa $ : $u_2(\kappa )$ , and secondly to give the consistency strength of a property of Lücke’s.

TheoremThe following are equiconsistent:

  1. (i) There exists $\kappa $ which is stably measurable;

  2. (ii) for some cardinal $\kappa $ , $u_2(\kappa )=\sigma (\kappa )$ ;

  3. (iii) The $\boldsymbol {\Sigma }_{1}$ -club property holds at a cardinal $\kappa $ .

Here $\sigma (\kappa )$ is the height of the smallest $M \prec _{\Sigma _{1}} H ( \kappa ^{+} )$ containing $\kappa +1$ and all of $H ( \kappa )$ . Let $\Phi (\kappa )$ be the assertion:

TheoremAssume $\kappa $ is stably measurable. Then $\Phi (\kappa )$ .

And a form of converse:

TheoremSuppose there is no sharp for an inner model with a strong cardinal. Then in the core model K we have: $\mbox {``}\exists \kappa \Phi (\kappa ) \mbox {''}$ is (set)-generically absolute ${\,\longleftrightarrow \,}$ There are arbitrarily large stably measurable cardinals.

When $u_2(\kappa ) < \sigma (\kappa )$ we give some results on inner model reflection.

Type
Article
Copyright
© The Association for Symbolic Logic 2020

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