Skip to main content Accessibility help
×
Home
Hostname: page-component-56f9d74cfd-s5lbf Total loading time: 0.211 Render date: 2022-06-25T11:11:59.464Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

On inverse γ-systems and the number of L∞λ-equivalent, non-isomorphic models for λ singular

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, Israel Rutgers University, Hill Ctr-Busch, New Brunswick, New Jersey 08903, USA, E-mail: shelah@math.rutgers.edu
Pauli Väisänen
Affiliation:
Department of Mathematics, P. O. Box 4, 00014, University of Helsinki, Finland, E-mail: pauli.vaisanen@helsinki.fi

Abstract

Suppose λ is a singular cardinal of uncountable cofinality κ. For a model of cardinality λ, let No() denote the number of isomorphism types of models of cardinality λ which are L∞λ-equivalent to . In [7] Shelah considered inverse κ-systems of abelian groups and their certain kind of quotient limits Gr()/ Fact(). In particular Shelah proved in [7, Fact 3.10] that for every cardinal Μ there exists an inverse κ-system such that consists of abelian groups having cardinality at most Μκ and card(Gr()/ Fact()) = Μ. Later in [8, Theorem 3.3] Shelah showed a strict connection between inverse κ-systems and possible values of No (under the assumption that θκ < λ for every θ < λ): if is an inverse κ-system of abelian groups having cardinality < λ, then there is a model such that card() = λ and No() = card(Gr()/ Fact()). The following was an immediate consequence (when θκ < λ for every θ < λ): for every nonzero Μ < λ or Μ = λκ there is a model , of cardinality λ with No() = Μ. In this paper we show: for every nonzero Μ ≤ λκ there is an inverse κ-system of abelian groups having cardinality < λ such that card(Gr()/ Fact()) = Μ (under the assumptions 2κ < λ and θ < λ for all θ < λ when Μ > λ), with the obvious new consequence concerning the possible value of No. Specifically, the case No() = λ is possible when θκ > λ for every λ < λ.

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. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Chang, C. C., Some remarks on the model theory ofinfinitary languages, The syntax and semantics of inflnitary languages (Barwise, J., editor), Lecture Notes in Mathematics, no. 72, Springer-Verlag, Berlin, 1968, pp. 3663.CrossRefGoogle Scholar
[2]Palyutin, E. A., Number of models in L ∞,ω theories, II, Algebra i Logika, vol. 16 (1977), no. 4, pp. 443456, English translation in [3].Google Scholar
[3]Palyutin, E. A., Number of models in theories, II, Algebra and Logic, vol. 16 (1977), no. 4, pp. 299309.CrossRefGoogle Scholar
[4]Scott, Dana, Logic with denumerably long formulas and finite strings of quantifiers, Theory of models (Proceedings of the 1963 International Symposium, Berkeley) (Addison, J. W., Henkin, Leon, and Tarski, Alfred, editors), North-Holland, Amsterdam, 1965, pp. 329334.Google Scholar
[5]Shelah, Saharon, On the number of nonisomorphic models of cardinality λ L∞,λ-equivalent to a fixed model, Notre Dame Journal of Formed Logic, vol. 22 (1981), no. 1, pp. 510.CrossRefGoogle Scholar
[6]Shelah, Saharon, On the number of nonisomorphic models in L∞,λ when κ is weakly compact, Notre Dame Journal of Formal Logic, vol. 23 (1982), no. 1, pp. 2126.CrossRefGoogle Scholar
[7]Shelah, Saharon, On the possible number no(M) = the number of nonisomorphic models L∞,λ-equivalent to M of power λ, for λ singular, Notre Dame Journal of Formal Logic, vol. 26 (1985), no. 1, pp. 3650.CrossRefGoogle Scholar
[8]Shelah, Saharon, On the no(M) for M of singular power, Around classification theory of models, Lecture Notes in Mathematics, no. 1182, Springer-Verlag, Berlin, 1986, pp. 120134.CrossRefGoogle Scholar
[9]Shelah, Saharon, The number ofpairwisenon-elementarily-embeddable models, this Journal, vol. 54 (1989), no. 4, pp. 14311455.Google Scholar
[10]Shelah, Saharon, Cardinal arithmetic, Oxford Logic Guides, no. 29, The Clarendon Press Oxford University Press, New York, 1994, Oxford Science Publications.Google Scholar
[11]Shelah, Saharon and VÄisÄnen, Pauli, On the number of L∞,λ-equivalent, non-isomorphic models, to appear in Transactions of the American Mathematical Society.Google Scholar

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.

On inverse γ-systems and the number of L∞λ-equivalent, non-isomorphic models for λ singular
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.

On inverse γ-systems and the number of L∞λ-equivalent, non-isomorphic models for λ singular
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.

On inverse γ-systems and the number of L∞λ-equivalent, non-isomorphic models for λ singular
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? *