# [FOM] Huge cardinals

Robert Solovay solovay at gmail.com
Thu Apr 19 19:53:31 EDT 2018

This is in partial response to the following question of Jan Pax: Is it
consistent with ZFC to have a cardinal kappa with the set of HUGE Cardinals
below it forming a stationary set.

We work in Kelly-Morse class-set theory.

Assume that kappa is 2-huge. Then kappa is measurable and there is a normal
measure mu on kappa such that the set of HUGE Cardinals less than kappa has
mu measure one. (A fortiori it is stationary.)

Hence if KM + "there is a 2-huge cardinal" is consistent, so is ZFC plus
"there is a strongly inaccessible cardinal kappa such that the set of HUGE
Cardinals less than kappa is stationary in kappa".

-- Bob Solovay

On Fri, Apr 13, 2018, 9:12 AM <pax0 at seznam.cz> wrote:

> Dear FOMers,
>
> is it consistent with ZFC to have a cardinal \kappa with
> HUGE cardinals below it being a stationary set?
>
> Thank you, Jan Pax
>
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