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
×
Home
  • Home
Hostname: page-component-559fc8cf4f-dxfhg Total loading time: 0.403 Render date: 2021-03-07T10:05:30.193Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }
EnglishFrançais
The Journal of Symbolic Logic The Journal of Symbolic Logic

Article contents

Model theory without choice? Categoricity

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
Saharon Shelah
Affiliation:
The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Edmond J. Safra Campus. Givat Ram, Jerusalem 91904, Israel Department of Mathematics, Hill Center-Busch Campus, Rutgers., The State University of New Jersey, 110 Frelinghuysen Road, Piscataway. Nj 08854-8019, USA, E-mail: shlhetal@math.huji.ac.il
Corresponding
E-mail address:
shlhetal@math.huji.ac.il
Get access
Rights & Permissions[Opens in a new window]

Abstract

We prove Los conjecture = Morley theorem in ZF. with the same characterization, i.e., of first order countable theories categorical in ℵα for some (eqiuvalently for every ordinal) α > 0. Another central result here in this context is: the number of models of a countable first order T of cardinality ℵα is either ≥ ∣α∣ for every α or it has a small upper bound (independent of α close to ⊐2).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 2 , June 2009 , pp. 361 - 401
Copyright
Copyright © Association for Symbolic Logic 2009

Access options

Get access to the full version of this content by using one of the access options below.
If you should have access and can't see this content please contact technical support.

References

[BLSh:464]Baldwin, John T., Laskowski, Michael C., and Shelah, Saharon, Forcing isomorphism. this Journal, vol. 58 (1993), pp. 1291–1301, math.LO/9301208.Google Scholar
[Be84]Buechler, Steven, Kueker's conjecture for superstable theories, this Journal, vol. 49 (1984). pp. 930–934.Google Scholar
[HHLOO]Hart, Bradd, Hrushovski, Ehud, and Laskowski, Michael C., The uncountable spectra of countable theories. Annals of Mathematics, vol. 152 (2000), pp. 207–257.CrossRefGoogle Scholar
[Hr89]Hrushovski, Ehud, Kueker's conjecture for stable theories, this Journal, vol. 54 (1989), pp. 207–220.Google Scholar
[Hr89d]Hrushovski, Ehud, Unidimensional theories, Logic colloquium 88, North-Holland, 1989.Google Scholar
[Ke71a]Keisler, Jerome H., On theories categorical in their own power, vol. 36, 1971.Google Scholar
[Las88]Laskowski, Michael C., Uncountable theories that are categorical in a higher power, this Journal, vol. 53 (1988), pp. 512–530.Google Scholar
[LwSh:518]Laskowski, Michael C. and Shelah, Saharon. Forcing isomorphism 11, this Journal, vol. 61 (1996), pp. 1305–1320, math.LO/0011169.Google Scholar
[Lv71]Laver, Richard, On Fraissé's order type conjecture. Annals of Mathematics, vol. 93 (1971), pp. 89–111.CrossRefGoogle Scholar
[Mo65]Morley, Michael, Categoricity in power, Transaction of the American Mathematical Society, vol. 114 (1965), pp. 514–538.CrossRefGoogle Scholar
[Sh:3]Shelah, Saharon, Finite diagrams stable in power, Annals of Mathematical Logic, vol. 2 (1970), pp. 69–118.CrossRefGoogle Scholar
[Sh:4]Shelah, Saharon, On theories T categorical in ∣T∣, this Journal, vol. 35 (1970), pp. 73–82.Google Scholar
[Sh:12]Shelah, Saharon, The number of non-isomorphic models of an unstable first-order theory, Israel Journal of Mathematics, vol. 9 (1971), pp. 473–487.CrossRefGoogle Scholar
[Sh:31]Shelah, Saharon, Categoricity of uncountable theories, Proceedings of the Tarski symposium (University of California, Berkeley, California, 1971), Proceedings of the Symposium on Pure Mathematics, vol. XXV, American Mathematical Society, Providence, R.I., 1974, pp. 187–203.Google Scholar
[Sh: 52]Shelah, Saharon, A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals, Israel Journal of Mathematics, vol. 21 (1975), pp. 319–349.CrossRefGoogle Scholar
[Sh:54]Shelah, Saharon, The lazy model-theoretician's guide to stability, Logique et Analyse, vol. 18 (1975), pp. 241–308.Google Scholar
[Sh:E18]Shelah, Saharon, A combinatorial proof of the singular compactness theorem, 1977. Mineograph notes and lecture in a mini-conference, Berlin, 08 '77.Google Scholar
[Sh:100]Shelah, Saharon, Independence results, this Journal, vol. 45 (1980), pp. 563–573.Google Scholar
[Sh:199]Shelah, Saharon, Remarks in abstract model theory, Annals of Pure and Applied Logic, vol. 29 (1985), pp. 255–288.CrossRefGoogle Scholar
[Shx]Shelah, Saharon, Classification theory and the number of nonisomorphic models. Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990.Google Scholar
[Sh:497]Shelah, Saharon, Set theory without choice: not everything on cofinality is possible, Archive for Mathematical Logic, vol. 36 (1997), pp. 81–125, a special volume dedicated to Prof. Azriel Levy. math.L0/9512227.CrossRefGoogle Scholar
[Shx]Shelah, Saharon, Non-structure theory, Oxford University Press, accepted.Google Scholar
[Sh:300f]Shelah, Saharon, Chapter VI.Google Scholar
[Sh:750]Shelah, Saharon, On weak bethfor cofinality logic, preprint.Google Scholar
[Sh:835]Shelah, Saharon, Pcf without choice, Archive for Mathematical Logic, submitted, math. LO/0510229.Google Scholar
[Sh:938]Shelah, Saharon, Pcf arithmetic without and with choice.Google Scholar
[Sh:E38]Shelah, Saharon, Continuation of 497: Universes without choice.Google Scholar
[Sh:F701]Shelah, Saharon, More on model theory without choice.Google Scholar
[WT05]Walczak-Typke, Agatha, The first-order structure of weakly Dedekind-finite sets, this Journal, vol. 70 (2005), pp. 1161–1170.Google Scholar
[WT07]Shelah, Saharon, A model-theoretic approach to structures in set theory without the axiom of choice, Algebra, logic, set theory: Festschrift für Ulrich Feigner zum 65 Geburtstag (Loewe, B., editor), Studies in Logic. College Publication s at King s College London, to appear.Google 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: 13 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 7th March 2021. This data will be updated every 24 hours.

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.

Model theory without choice? Categoricity
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.

Model theory without choice? Categoricity
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.

Model theory without choice? Categoricity
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *