Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T01:59:50.926Z Has data issue: false hasContentIssue false
  • English
  • Français

FORKING AND SUPERSTABILITY IN TAME AECS

Published online by Cambridge University Press:  09 March 2016

SEBASTIEN VASEY
SEBASTIEN VASEY*
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES CARNEGIE MELLON UNIVERSITY PITTSBURGH, PENNSYLVANIA, USAE-mail: [email protected]: http://math.cmu.edu/∼svasey/
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that any tame abstract elementary class categorical in a suitable cardinal has an eventually global good frame: a forking-like notion defined on all types of single elements. This gives the first known general construction of a good frame in ZFC. We show that we already obtain a well-behaved independence relation assuming only a superstability-like hypothesis instead of categoricity. These methods are applied to obtain an upward stability transfer theorem from categoricity and tameness, as well as new conditions for uniqueness of limit models.

Keywords

Primary: 03C48Secondary: 03C4503C5203C55Abstract elementary classesforkingindependenceclassification theorygood framescategoricitysuperstabilitystabilitystability spectrumtameness
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 1 , March 2016 , pp. 357 - 383
Copyright
Copyright © The Association for Symbolic Logic 2016 

References

REFERENCES

Baldwin, John T., Categoricity, University Lecture Series, vol. 50, American Mathematical Society, Providence, RI, 2009.Google Scholar
Baldwin, John T., Categoricity: Errata, Available online. Last accessed on June 28, 2014, URL: http://www.math.uic.edu/∼jbaldwin/pub/bookerrata.pdf.Google Scholar
Boney, Will and Grossberg, Rami, Forking in short and tame AECs, preprint, URL: http://arxiv.org/abs/1306.6562v9.Google Scholar
Boney, Will, Grossberg, Rami, Kolesnikov, Alexei, and Vasey, Sebastien, Canonical forking in AECs, preprint, URL: http://arxiv.org/abs/1404.1494v2.Google Scholar
Baldwin, John T., Kueker, David, and VanDieren, Monica, Upward stability transfer for tame abstract elementary classes. Notre Dame Journal of Formal Logic, vol. 47 (2006), no. 2, pp. 291–298.Google Scholar
Boney, Will, Tameness and extending frames. Journal of Mathematical Logic, vol. 14 (2014), no. 2, pp. 1450007.CrossRefGoogle Scholar
Boney, Will, Tameness from large cardinal axioms, this Journal, vol. 79 (2014), no. 4, pp. 1092–1119.Google Scholar
Boney, Will, Computing the number of types of infinite length, preprint, URL: http://arxiv.org/abs/1309.4485v2.Google Scholar
Boney, Will and Vasey, Sebastien, Chains of saturated models in AECs, preprint, URL: http://arxiv.org/abs/1503.08781v3.Google Scholar
Boney, Will and Vasey, Sebastien, Tameness and frames revisited, preprint, URL: http://arxiv.org/abs/1406.5980v4.Google Scholar
Grossberg, Rami, Classification theory for abstract elementary classes. Contemporary Mathematics, vol. 302 (2002), pp. 165–204.Google Scholar
Grossberg, Rami and VanDieren, Monica, Categoricity from one successor cardinal in tame abstract elementary classes. Journal of Mathematical Logic, vol. 6 (2006), pp. 181–201.CrossRefGoogle Scholar
Grossberg, Rami and VanDieren, Monica, Galois-stability for tame abstract elementary classes. Journal of Mathematical Logic, vol. 6 (2006), no. 1, pp. 25–49.Google Scholar
Grossberg, Rami, VanDieren, Monica, and Villaveces, Andrés, Uniqueness of limit models in classes with amalgamation, preprint, URL: http://arxiv.org/abs/math/0509338v3.Google Scholar
Hyttinen, Tapani and Lessmann, Olivier, A rank for the class of elementary submodels of a superstable homogeneous model, this Journal, vol. 67 (2002), no. 4, pp. 1469–1482.Google Scholar
Jarden, Adi, Primeness triples in non-forking frames, preprint, available online from http://ariel.academia.edu/AdiJarden.Google Scholar
Jarden, Adi and Shelah, Saharon, Non-forking frames in abstract elementary classes. Annals of Pure and Applied Logic, vol. 164 (2013), pp. 135–191.Google Scholar
Jarden, Adi and Shelah, Saharon, Non forking good frames without local character, preprint, URL: http://arxiv.org/abs/1105.3674v1.Google Scholar
Jarden, Adi and Sitton, Alon, Independence, dimension and continuity in non-forking frames, this Journal, vol. 78 (2012), no. 2, pp. 602–632.Google Scholar
Shelah, Saharon, Categoricity in ℵ of sentences in Lω 1,ω (Q). Israel Journal of Mathematics, vol. 20 (1975), no. 2, pp. 127–148.Google Scholar
Shelah, Saharon, Classification theory and the number of non-isomorphic models, second edition, Studies in logic and the foundations of mathematics, vol. 92, North-Holland, Amsterdam, 1990.Google Scholar
Shelah, Saharon, Categoricity for abstract classes with amalgamation. Annals of Pure and Applied Logic, vol. 98 (1999), no. 1, pp. 261–294.Google Scholar
Shelah, Saharon, Categoricity of an abstract elementary class in two successive cardinals. Israel Journal of Mathematics, vol. 126 (2001), pp. 29–128.Google Scholar
Shelah, Saharon, Classification theory for abstract elementary classes, Studies in Logic: Mathematical logic and foundations, vol. 18, College Publications, London, 2009.Google Scholar
Shelah, Saharon, Classification theory for abstract elementary classes 2, Studies in Logic: Mathematical logic and foundations, vol. 20, College Publications, London, 2009.Google Scholar
Shelah, Saharon, Categoricity for abstract classes with amalgamation (updated), Oct. 29, 2004 version, URL: http://shelah.logic.at/files/394.pdf.Google Scholar
Shelah, Saharon and Villaveces, Andrés, Toward categoricity for classes with no maximal models. Annals of Pure and Applied Logic, vol. 97 (1999), pp. 1–25.CrossRefGoogle Scholar
VanDieren, Monica, Categoricity and stability in abstract elementary classes, Ph.D. Thesis, 2002.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.CrossRefGoogle Scholar
VanDieren, Monica, Erratum to “Categoricity in abstract elementary classes with no maximal models” [Ann. Pure Appl. Logic 141 (2006) 108-147]. Annals of Pure and Applied Logic, vol. 164 (2013), no. 2, pp. 131–133.Google Scholar
Vasey, Sebastien, Independence in abstract elementary classes, preprint, URL: http://arxiv.org/abs/1503.01366v4.Google Scholar
Vasey, Sebastien, Infinitary stability theory, preprint, URL: http://arxiv.org/abs/1412.3313v3.Google Scholar