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Model theory without choice? Categoricity

Published online by Cambridge University Press:  12 March 2014

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

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
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[BLSh:464]Baldwin, John T., Laskowski, Michael C., and Shelah, Saharon, Forcing isomorphism. this Journal, vol. 58 (1993), pp. 12911301, math.LO/9301208.Google Scholar
[Be84]Buechler, Steven, Kueker's conjecture for superstable theories, this Journal, vol. 49 (1984). pp. 930934.Google Scholar
[HHLOO]Hart, Bradd, Hrushovski, Ehud, and Laskowski, Michael C., The uncountable spectra of countable theories. Annals of Mathematics, vol. 152 (2000), pp. 207257.CrossRefGoogle Scholar
[Hr89]Hrushovski, Ehud, Kueker's conjecture for stable theories, this Journal, vol. 54 (1989), pp. 207220.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. 512530.Google Scholar
[LwSh:518]Laskowski, Michael C. and Shelah, Saharon. Forcing isomorphism 11, this Journal, vol. 61 (1996), pp. 13051320, math.LO/0011169.Google Scholar
[Lv71]Laver, Richard, On Fraissé's order type conjecture. Annals of Mathematics, vol. 93 (1971), pp. 89111.CrossRefGoogle Scholar
[Mo65]Morley, Michael, Categoricity in power, Transaction of the American Mathematical Society, vol. 114 (1965), pp. 514538.CrossRefGoogle Scholar
[Sh:3]Shelah, Saharon, Finite diagrams stable in power, Annals of Mathematical Logic, vol. 2 (1970), pp. 69118.CrossRefGoogle Scholar
[Sh:4]Shelah, Saharon, On theories T categorical in ∣T, this Journal, vol. 35 (1970), pp. 7382.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. 473487.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. 187203.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. 319349.CrossRefGoogle Scholar
[Sh:54]Shelah, Saharon, The lazy model-theoretician's guide to stability, Logique et Analyse, vol. 18 (1975), pp. 241308.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. 563573.Google Scholar
[Sh:199]Shelah, Saharon, Remarks in abstract model theory, Annals of Pure and Applied Logic, vol. 29 (1985), pp. 255288.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. 81125, 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. 11611170.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