Skip to main content Accessibility help
×
Home
Hostname: page-component-5c569c448b-4wdfl Total loading time: 0.225 Render date: 2022-07-01T09:26:04.075Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Elementary embeddings and infinitary combinatorics

Published online by Cambridge University Press:  12 March 2014

Kenneth Kunen*
Affiliation:
University of Wisconsin, Madison, Wisconsin

Extract

One of the standard ways of postulating large cardinal axioms is to consider elementary embeddings, j, from the universe, V, into some transitive submodel, M. See Reinhardt–Solovay [7] for more details. If j is not the identity, and κ is the first ordinal moved by j, then κ is a measurable cardinal. Conversely, Scott [8] showed that whenever κ is measurable, there is such j and M. If we had assumed, in addition, that , then κ would be the κth measurable cardinal; in general, the wider we assume M to be, the larger κ must be.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1972

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.)

References

[1]Erdös, P. and Hajnal, A., On a problem of B. Jónsson, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 14 (1966), pp. 1923.Google Scholar
[2]Gaifman, H., Pushing up the measurable cardinal, Lecture notes 1967 Institute on Axiomatic Set Theory (University of California, Los Angeles, Calif., 1967), American Mathematical Society, Providence, R.I., 1967, pp. IV R 1–16. (Mimeographed).Google Scholar
[3]Gödel, K., The consistency of the continuum hypothesis, Princeton Univ. Press, Princeton, 1940.Google Scholar
[4]Kelley, J. L., General topology, D. Van Nostrand Co., Inc., Princeton, 1965.Google Scholar
[5]Kunen, K., Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179227.CrossRefGoogle Scholar
[6]Kunen, K., On the GCH at measurable cardinals, Proceedings of 1969 Logic Summer School at Manchester, pp. 107110.Google Scholar
[7]Reinhardt, W. and Solovay, R., Strong axioms of infinity and elementary embeddings, to appear.Google Scholar
[8]Scott, D., Measurable cardinals and constructible sets, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 9 (1961) pp. 521524.Google Scholar
78
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your 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.

Elementary embeddings and infinitary combinatorics
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.

Elementary embeddings and infinitary combinatorics
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.

Elementary embeddings and infinitary combinatorics
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *