Skip to main content Accessibility help
×
×
Home

Model theory without choice? Categoricity

  • Saharon Shelah (a1) (a2)

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).

Copyright

References

Hide All
[BLSh:464]Baldwin, John T., Laskowski, Michael C., and Shelah, Saharon, Forcing isomorphism. this Journal, vol. 58 (1993), pp. 12911301, math.LO/9301208.
[Be84]Buechler, Steven, Kueker's conjecture for superstable theories, this Journal, vol. 49 (1984). pp. 930934.
[HHLOO]Hart, Bradd, Hrushovski, Ehud, and Laskowski, Michael C., The uncountable spectra of countable theories. Annals of Mathematics, vol. 152 (2000), pp. 207257.
[Hr89]Hrushovski, Ehud, Kueker's conjecture for stable theories, this Journal, vol. 54 (1989), pp. 207220.
[Hr89d]Hrushovski, Ehud, Unidimensional theories, Logic colloquium 88, North-Holland, 1989.
[Ke71a]Keisler, Jerome H., On theories categorical in their own power, vol. 36, 1971.
[Las88]Laskowski, Michael C., Uncountable theories that are categorical in a higher power, this Journal, vol. 53 (1988), pp. 512530.
[LwSh:518]Laskowski, Michael C. and Shelah, Saharon. Forcing isomorphism 11, this Journal, vol. 61 (1996), pp. 13051320, math.LO/0011169.
[Lv71]Laver, Richard, On Fraissé's order type conjecture. Annals of Mathematics, vol. 93 (1971), pp. 89111.
[Mo65]Morley, Michael, Categoricity in power, Transaction of the American Mathematical Society, vol. 114 (1965), pp. 514538.
[Sh:3]Shelah, Saharon, Finite diagrams stable in power, Annals of Mathematical Logic, vol. 2 (1970), pp. 69118.
[Sh:4]Shelah, Saharon, On theories T categorical in ∣T, this Journal, vol. 35 (1970), pp. 7382.
[Sh:12]Shelah, Saharon, The number of non-isomorphic models of an unstable first-order theory, Israel Journal of Mathematics, vol. 9 (1971), pp. 473487.
[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. 187203.
[Sh: 52]Shelah, Saharon, A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals, Israel Journal of Mathematics, vol. 21 (1975), pp. 319349.
[Sh:54]Shelah, Saharon, The lazy model-theoretician's guide to stability, Logique et Analyse, vol. 18 (1975), pp. 241308.
[Sh:E18]Shelah, Saharon, A combinatorial proof of the singular compactness theorem, 1977. Mineograph notes and lecture in a mini-conference, Berlin, 08 '77.
[Sh:100]Shelah, Saharon, Independence results, this Journal, vol. 45 (1980), pp. 563573.
[Sh:199]Shelah, Saharon, Remarks in abstract model theory, Annals of Pure and Applied Logic, vol. 29 (1985), pp. 255288.
[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.
[Sh:497]Shelah, Saharon, Set theory without choice: not everything on cofinality is possible, Archive for Mathematical Logic, vol. 36 (1997), pp. 81125, a special volume dedicated to Prof. Azriel Levy. math.L0/9512227.
[Shx]Shelah, Saharon, Non-structure theory, Oxford University Press, accepted.
[Sh:300f]Shelah, Saharon, Chapter VI.
[Sh:750]Shelah, Saharon, On weak bethfor cofinality logic, preprint.
[Sh:835]Shelah, Saharon, Pcf without choice, Archive for Mathematical Logic, submitted, math. LO/0510229.
[Sh:938]Shelah, Saharon, Pcf arithmetic without and with choice.
[Sh:E38]Shelah, Saharon, Continuation of 497: Universes without choice.
[Sh:F701]Shelah, Saharon, More on model theory without choice.
[WT05]Walczak-Typke, Agatha, The first-order structure of weakly Dedekind-finite sets, this Journal, vol. 70 (2005), pp. 11611170.
[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.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed