Home
Hostname: page-component-55597f9d44-mm7gn Total loading time: 0.392 Render date: 2022-08-15T18:45:30.054Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

# STABLY MEASURABLE CARDINALS

Part of: Set theory

Published online by Cambridge University Press:  15 June 2020

## 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.

## MSC classification

Type
Article
Information
The Journal of Symbolic Logic , June 2021 , pp. 448 - 470
© The Association for Symbolic Logic 2020

## Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

# Save article to Kindle

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

STABLY MEASURABLE CARDINALS
Available formats
×

# Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

STABLY MEASURABLE CARDINALS
Available formats
×

# Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

STABLY MEASURABLE CARDINALS
Available formats
×
×