Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-05T21:41:20.691Z Has data issue: false hasContentIssue false

Cardinal preserving elementary embeddings

Published online by Cambridge University Press:  01 March 2011

By
Andrés Eduardo Caicedo
Edited by
Françoise Delon ,
Ulrich Kohlenbach ,
Penelope Maddy  and
Frank Stephan
Françoise Delon
Affiliation:
UFR de Mathématiques
Ulrich Kohlenbach
Affiliation:
Technische Universität, Darmstadt, Germany
Penelope Maddy
Affiliation:
University of California, Irvine
Frank Stephan
Affiliation:
National University of Singapore
Get access

Summary

Abstract. Say that an elementary embedding j : N → M is cardinal preserving if CARM = CARN = CAR. We show that if PFA holds then there are no cardinal preserving elementary embeddings j : M → V. We also show that no ultrapower embedding j : V → M induced by a set extender is cardinal preserving, and present some results on the large cardinal strength of the assumption that there is a cardinal preserving j : V → M.

Introduction. This paper is the first of a series attempting to investigate the structure of (not necessarily fine structural) inner models of the set theoretic universe under assumptions of two kinds:

  1. Forcing axioms, holding either in the universe ∨ of all sets or in both ∨ and the inner model under study, and

  2. Agreement between (some of) the cardinals of ∨ and the cardinals of the inner model.

I try to be as self-contained as is reasonably possible, given the technical nature of the problems under consideration. The notation is standard, as in Jech. I assume familiarity with inner model theory; for fine structural background and notation, the reader is urged to consult Steel and Mitchell.

In the remainder of this introduction, I include some general observations on large cardinal theory, forcing axioms, and fine structure, and state the main results of the paper.

Type
Chapter
Information
Logic Colloquium 2007 , pp. 14 - 31
Publisher: Cambridge University Press
Print publication year: 2010

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] A., Andretta, I., Neeman, and J., Steel, The domestic levels of Kc are iterable, Israel Journal of Mathematics, vol. 125 (2001), pp. 157–201.Google Scholar
[2] J., Baumgartner, Applications of the proper forcing axiom, Handbook of Set-Theoretic Topology (K., Kunen and J., Vaughan, editors), North-Holland, Amsterdam, 1984, pp. 913–959.Google Scholar
[3] M., Bekkali, Topics in Set Theory, Lecture Notes in Mathematics, vol. 1476, Springer-Verlag, Berlin, 1991, Lebesgue measurability, large cardinals, forcing axioms, rho-functions, Notes on lectures by Stevo Todorčević.Google Scholar
[4] A., Caicedo and B., Veličković, Properness and Reflection Principles in Set Theory, in preparation.
[5] M., Foreman, M., Magidor, and S., Shelah, Martin's maximum, saturated ideals, and nonregular ultrafilters. I, Annals of Mathematics. Second Series, vol. 127 (1988), no. 1, pp. 1–47.Google Scholar
[6] J., Hamkins, Forcing and Large Cardinals, in preparation.
[7] M., Holz, K., Steffens, and E., Weitz, Introduction to Cardinal Arithmetic, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1999.Google Scholar
[8] T., Jech, Set Theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003, The third millennium edition, revised and expanded.Google Scholar
[9] R., Jensen, E., Schimmerling, R., Schindler, and J., Steel, Stacking mice, The Journal of Symbolic Logic, vol. 74 (2009), no. 1, pp. 315–335.Google Scholar
[10] A., Kanamori, The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, second ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.Google Scholar
[11] A., Kanamori and M., Magidor, The evolution of large cardinal axioms in set theory, Higher Set Theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1977), Lecture Notes in Mathematics, vol. 669, Springer, Berlin, 1978, pp. 99–275.Google Scholar
[12] J., Ketonen, Strong compactness and other cardinal sins, Annals of Pure and Applied Logic, vol. 5 (1972/73), pp. 47–76.Google Scholar
[13] P., Larson, The Stationary Tower: Notes on a Course by W. Hugh Woodin, University Lecture Series, vol. 32, American Mathematical Society, Providence, RI, 2004.Google Scholar
[14] D. A., Martin and R. M., Solovay, Internal Cohen extensions, Annals of Pure and Applied Logic, vol. 2, (1970), no. 2, pp. 143–178.Google Scholar
[15] W., Mitchell, The Covering Lemma, to appear in Handbook of Set Theory, Foreman and Kanamori (Editors).
[16] J., Moore, Set mapping reflection, Journal of Mathematical Logic, vol. 5 (2005), no. 1, pp. 87–97.Google Scholar
[17] S., Shelah, Cardinal Arithmetic, Oxford Logic Guides, vol. 29, The Clarendon Press Oxford University Press, New York, 1994.Google Scholar
[18] S., Shelah, Proper and Improper Forcing, second ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998.Google Scholar
[19] J., Steel, An Outline of Inner Model Theory, to appear in Handbook of Set Theory, Foreman and Kanamori (Editors).
[20] S., Todorčević, A note on the proper forcing axiom, Axiomatic Set Theory (Boulder, Colorado, 1983) (J., Baumgartner and D., Martin, editors), Contemporary Mathematics, vol. 31, American Mathematical Society, Providence, RI, 1984, pp. 209–218.Google Scholar
[21] M., Viale, Applications of the proper forcing axiom to cardinal arithmetic, Ph.D. Dissertation, Université Paris 7, Denis Diderot, Paris, 2006.
[22] M., Viale, The proper forcing axiom and the singular cardinal hypothesis, The Journal of Symbolic Logic, vol. 71 (2006), no. 2, pp. 473–479.Google Scholar
[23] M., Viale, A family of covering properties, Mathematical Research Letters, vol. 15 (2008), no. 2, pp. 221–238.Google Scholar
[24] J., Vickers and P. D., Welch, On elementary embeddings from an inner model to the universe, The Journal of Symbolic Logic, vol. 66 (2001), no. 3, pp. 1090–1116.Google Scholar
[25] J., Zapletal, A new proof of Kunen's inconsistency, Proceedings of the American Mathematical Society, vol. 124 (1996), no. 7, pp. 2203–2204.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] 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.

Available formats
×

Save book to Dropbox

To save content items to your account, please 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 account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please 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 account. Find out more about saving content to Google Drive.

Available formats
×