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TAMENESS FROM LARGE CARDINAL AXIOMS

Published online by Cambridge University Press:  12 December 2014

WILL BONEY
WILL BONEY*
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES CARNEGIE MELLON UNIVERSITY PITTSBURGH, PENNSYLVANIA, USAE-mail: [email protected]
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Abstract

We show that Shelah’s Eventual Categoricity Conjecture for successors follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with LS(K) below a strongly compact cardinal κ is < κ-tame and applying the categoricity transfer of Grossberg and VanDieren [11]. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, called type shortness, and show that it follows similarly from large cardinals.

Keywords

Abstract Elementary Classestamenesslarge cardinals
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 4 , December 2014 , pp. 1092 - 1119
Copyright
Copyright © The Association for Symbolic Logic 2014 

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References

REFERENCES

Baldwin, John, Categoricity, University Lecture Series, vol. 50, American Mathematical Society, Providence, RI, 2009.Google Scholar
Baldwin, John and Kolesnikov, Alexei, Categoricity, amalgamation, and tameness. Israel Journal of Mathematics, vol. 170 (2009), no. 1, pp. 411–443.CrossRefGoogle Scholar
Baldwin, John and Shelah, Saharon, Examples of non-locality, this Journal, vol. 73 (2008), pp. 765–782.Google Scholar
Boney, Will, Computing the number of types of infinite length, Accepted, Notre Dame Journal of Formal Logic, arXiv: 1309.4485.Google Scholar
Boney, Will and Grossberg, Rami, Forking in short and tame AECs, Submitted, arXiv: 1306.6562.Google Scholar
Boney, Will, Grossberg, Rami, Kolesnikov, Alexei, and Vasey, Sebastien, Canonical Forking in AECs, Submitted, arXiv: 1404.1494.Google Scholar
Chang, C. C., Some remarks on the model theory of infinitary languages, The Syntax and Semantics of Infinitary Languages (Barwise, J., editor), vol.72, Springer-Verlag Lecture Notes in Mathematics, 1968, pp. 36–63.Google Scholar
Ekloff, Paul and Mekler, Alan, Almost Free Modules: Set-Theoretic Methods, Elsevier, Amsterdam, 2002.CrossRefGoogle Scholar
Grossberg, Rami, Classification theory for Abstract Elementary Classes, Logic and Algebra (Zhang, Yi, editors), vol. 302, American Mathematical Society, Providence, RI, 2002, pp. 165–204.CrossRefGoogle Scholar
Grossberg, Rami, A Course in Model Theory, In Preparation, 201X.Google Scholar
Grossberg, Rami and VanDieren, Monica, Categoricity from one successor cardinal in tame Abstract Elementary Classes. Journal of Mathematical Logic, vol. 6 (2006), no. 2, pp. 181–201.Google Scholar
Grossberg, Rami, Galois-stability for tame Abstract Elementary Classes.Journal of Mathematical Logic, vol. 6 (2006), no. 1, pp. 25–49.Google Scholar
Grossberg, Rami, Shelah’s categoricity conjecture from a successor for tame Abstract Elementary Classes, this Journal, vol. 71 (2006), no. 2, pp. 553–568.Google Scholar
Grossberg, Rami, VanDieren, Monica, and Villaveces, Andres, Uniqueness of limit models in Abstract Elementary Classes, Submitted. http://www.math.cmu.edu/∼rami/GVV_1_12_2012.pdf.Google Scholar
Hart, Brad and Shelah, Saharon, Categoricity over P for first order T or categoricity for ɸ ε Lω1ωcan stop atℵkwhile holding for ℵ0,..., ℵk=1. Israel Journal of Mathematics, vol. 70 (1990), pp. 219–235.Google Scholar
Hirvonen, Åsa and Hyttinen, Tapani, Categoricity in homogeneous complete metric spaces. Archive of Mathematical Logic, vol. 48 (2009), no. 3-4, pp. 269–322.Google Scholar
Hyttinen, Tapani and Kesälä, Meeri, Superstability in simple finitary AECs. Fundamenta Mathematica, vol. 195 (2007), no. 3, pp. 221–268.CrossRefGoogle Scholar
Jech, Thomas, Categoricity transfer in simple finitary AECs. Notre Dame Journal of Formal Logic, vol. 76 (2011), no. 3, pp. 759–806.Google Scholar
Jech, Thomas, Set Theory, Third Edition, Springer, Berlin, 2006.Google Scholar
Kanamori, Akihiro, The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, Second Edition, Springer, Berlin 2008.Google Scholar
Kolman, Oren and Shelah, Saharon, Categoricity of theories in Lκω, when κ is a measureable cardinal, part 1. Fundamenta Mathematica, vol. 151 (1996), pp. 209–240.Google Scholar
Shelah, Saharon and Makkai, Michael, Categoricity of theories in L κω, with κ a compact cardinal. Annals of Pure and Applied Logic, vol. 47 (1990), pp. 41–97.Google Scholar
Magidor, Menachem and Shelah, Saharon, When does almost free imply free? (for groups, transversals, etc.). Journal of the American Mathematical Society, vol. 7 (1994), pp. 769–830.Google Scholar
Morley, M., Categoricity in power. Transactions of the American Mathematical Society, vol. 114 (1965), pp. 514–538.Google Scholar
Morley, M., Classification theory and the number of nonisomorphic models, Second Edition, vol. 92, North-Holland Publishing Co., Amsterdam, xxxiv+705 pp, 1990.Google Scholar
Morley, M., Classification Theory for Abstract Elementary Classes, vol.1 & 2, Mathematical Logic and Foundations, no. 18 & 20, College Publications, London, 2009.Google Scholar
Shelah, Saharon, Categoricity of uncountable theories, Proceedings of the Tarski Symposium, 1971, 1974, pp. 187–203.Google Scholar
Morley, M., Classification of nonelementary classes, II. Abstract Elementary Classes, Classification theory (Baldwin, John, editor), 1987, pp. 419–497.Google Scholar
Morley, M., Universal classes, Classification theory (Baldwin, John, editor), 1987, pp. 264–418.Google Scholar
Morley, M., Categoricity for abstract classes with amalgamation. Annals of Pure and Applied Logic, vol. 98 (1990), pp. 261–294.Google Scholar
Morley, M., Categoricity of theories in L κω, when κ is a measureable cardinal, part 2. Fundamenta Mathematica, vol. 170 (2001), pp. 165–196.Google Scholar
Shelah, Saharon, Categoricity in Abstract Elementary Classes: Going up inductively, math.LO/0011215.Google Scholar
Shelah, Saharon and Villaveces, Andres, Toward categoricity for classes with no maximal models. Annals of Pure and Applied Logic, vol. 97 (1999), pp. 1–25.Google Scholar
Shelah, Saharon and Villaveces, Andres, On What I Do Not Understand (And Have Something to Say), Model Theory. Mathematica Japonica, vol. 51 (2000), pp. 329–377.Google Scholar
Shelah, Saharon and Villaveces, Andres, Maximal failures of sequence locality in A. E. C., Preprint.Google Scholar
Solovay, Robert, Strongly compact cardinals and the GCH, Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics 25 (AMS), pp. 365–372.Google Scholar
VanDieren, Monica, Categoricity in Abstract Elementary Classes with no maximal models. Annals of Pure and Applied Logic, vol. 141 (2006), pp. 108–147.Google Scholar
VanDieren, Monica, Erratum to “Categoricity in Abstract Elementary Classes with no maximal models”. Annals of Pure and Applied Logic, vol. 164 (2013), pp. 131–133.Google Scholar