Skip to main content Accessibility help
×
Home

Identity crises and strong compactness

Published online by Cambridge University Press:  12 March 2014

Arthur W. Apter
Affiliation:
Department of Mathematics, Baruch College of Cuny, New York, New York 10010, USA, E-mail:awabb@cunyvm.cuny.edu
James Cummings
Affiliation:
Department of Mathematics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, USA, E-mail:jcumming+@andrew.cmu. edu
Corresponding

Abstract

Combining techniques of the first author and Shelah with ideas of Magidor, we show how to get a model in which, for fixed but arbitrary finite n, the first n strongly compact cardinals k1..…kn are so that ki; for i = 1..…n is both the ith measurable cardinal and supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2000

Access options

Get access to the full version of this content by using one of the access options below.

References

[A83]Apter, Arthur W., Some results on consecutive large cardinals, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 117.CrossRefGoogle Scholar
[A95]Apter, Arthur W., On the first n strongly compact cardinals, Proceedings of the American Mathematical Society, vol. 123 (1995), pp. 22292235.Google Scholar
[A97a]Apter, Arthur W., More on the least strongly compact cardinal, Mathematical Logic Quarterly, vol. 43 (1997), pp. 427430.CrossRefGoogle Scholar
[A97b]Apter, Arthur W., Patterns of compact cardinals, Annals of Pure and Applied Logic, vol. 89 (1997), pp. 101115.CrossRefGoogle Scholar
[A98]Apter, Arthur W., Laver indestructibility and the class of compact cardinals, this Journal, vol. 63 (1998), pp. 149157.Google Scholar
[A99]Apter, Arthur W., On measurable limits of compact cardinals, this Journal, vol. 64 (1999), pp. 16751688.Google Scholar
[AS97b]After, Arthur W. and Shelah, Saharon, Menas' result is best possible, Transactions of the American Mathematical Society, vol. 349 (1997), pp. 20072034.Google Scholar
[AS97a]Apter, Arthur W., On the strong equality between supercompactness and strong compactness, Transactions of the American Mathematical Society, vol. 349 (1997), pp. 103128.CrossRefGoogle Scholar
[B]Burgess, J., Forcing, Handbook of Mathematical Logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 403452.CrossRefGoogle Scholar
[CFM]Cummings, J., Foreman, M., and Magidor, M., Squares, scales, and stationary reflection, to appear in the Journal of Mathematical Logic.Google Scholar
[CW]Cummings, J. and Woodin, W.H., Generalised Prikry forcings, circulated manuscript of a forthcoming book.Google Scholar
[C]Cummings, James, A model in which GCH holds at successors but fails at limits, Transactions of the American Mathematical Society, vol. 329 (1992), pp. 139.CrossRefGoogle Scholar
[DH]Di Prisco, C. A. and Henle, J., On the compactness of ℵ1 and ℵ1, this Journal, vol. 43 (1978), pp. 394401.Google Scholar
[H]Hamkins, J., Lifting and extending measures; fragile measurability, Ph.D. thesis, University of California, Berkeley, 1994.Google Scholar
[J]Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[Ka]Kanamori, A., The higher infinite, Springer-Verlag, Berlin, 1994.Google Scholar
[KaM]Kanamori, A. and Magidor, M., The evolution of large cardinal axioms in set theory, Lecture notes in mathematics 669, Springer-Verlag, Berlin, 1978, pp. 99275.Google Scholar
[KiM]Kimchi, Y. and Magidor, M., The independence between the concepts of compactness and supercompactness, circulated manuscript.Google Scholar
[L]Laver, Richard, Making the supercompactness of K indestructible under n-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), pp. 385388.CrossRefGoogle Scholar
[LS]LÉvy, A. and Solovay, R. M., Measurable cardinals and the continuum hypothesis, Israel Journal of Mathematics, vol. 5 (1967), pp. 234248.CrossRefGoogle Scholar
[Ma]Magidor, Menachem, How large is the first strongly compact cardinal?, Annals of Mathematical Logic, vol. 10 (1976), pp. 3357.CrossRefGoogle Scholar
[MS]Mekler, A. and Shelah, S., When does k-free imply strongly k-Free?, Proceedings of the Third Conference on Abelian Group Theory, Gordon and Breach, Salzburg, 1987, pp. 137148.Google Scholar
[Me]Menas, Telis K., On strong compactness and supercompactness, Annals of Mathematical Logic, vol. 7 (1974), pp. 327359.CrossRefGoogle Scholar
[SRK]Solovay, Robert M., Reinhardt, William N., and Kanamori, Akihiro, Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73116.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 6 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 24th January 2021. This data will be updated every 24 hours.

Hostname: page-component-76cb886bbf-gtgjg Total loading time: 1.645 Render date: 2021-01-24T00:04:57.070Z Query parameters: { "hasAccess": "0", "openAccess": "0", "isLogged": "0", "lang": "en" } Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false }

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@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 sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent 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.

Identity crises and strong compactness
Available formats
×

Send article to Dropbox

To send 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 use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Identity crises and strong compactness
Available formats
×

Send article to Google Drive

To send 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 use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Identity crises and strong compactness
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *