Journal of Symbolic Logic

$P_\kappa\lambda$ Combinatorics II: The RK Ordering Beneath a Supercompact Measure

William S. Zwicker

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We characterize some large cardinal properties, such as $\mu$-measurability and $P^2(\kappa)$-measurability, in terms of ultrafilters, and then explore the Rudin-Keisler (RK) relations between these ultrafilters and supercompact measures on $P_\kappa(2^\kappa)$. This leads to the characterization of $2^\kappa$-supercompactness in terms of a measure on measure sequences, and also to the study of a certain natural subset, $\mathrm{Full}_\kappa$, of $P_\kappa(2^\kappa)$, whose elements code measures on cardinals less than $\kappa$. The hypothesis that $\mathrm{Full}_\kappa$ is stationary (a weaker assumption than $2^\kappa$-supercompactness) is equivalent to a higher order Lowenheim-Skolem property, and settles a question about directed versus chain-type unions on $P_\kappa\lambda$.

Article information

J. Symbolic Logic, Volume 51, Issue 3 (1986), 604-616.

First available in Project Euclid: 6 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03E55: Large cardinals

$P_\kappa\lambda$ hypermeasurable supercompact Lowehneim-Skolem theorem


Zwicker, William S. $P_\kappa\lambda$ Combinatorics II: The RK Ordering Beneath a Supercompact Measure. J. Symbolic Logic 51 (1986), no. 3, 604--616.

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See also

  • Part I: William S. Zwicker. $P_\kappa\lambda$ Combinatorics. I: Stationary coding sets rationalize the club filter.