Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T03:04:46.505Z Has data issue: false hasContentIssue false

Strong tree properties for small cardinals

Published online by Cambridge University Press:  12 March 2014

Laura Fontanella*
Affiliation:
IMJ, Equipe de Logique Mathématique, Université Paris Diderot7, UFR de Mathematiques Case 7012, Site Chevaleret 75205, Paris Cedex 13, France, E-mail: [email protected]

Abstract

An inaccessible cardinal κ is supercompact when (κ, λ)-ITP holds for all λκ. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where for every n ≥ 2 and μ ≥ ℕn, we have (ℕn, μ)-ITP.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2013

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

REFERENCES

[1]Abraham, U., Aronszajn trees on ℕ2 and ℕ3, Annals of Pure and Applied Logic, vol. 24 (1983), pp. 213230.CrossRefGoogle Scholar
[2]Cummings, J. and Foreman, M., The tree property, Advances in Mathematics, vol. 133 (1998), pp. 132.CrossRefGoogle Scholar
[3]Fontanella, L., Strong tree properties for two successive cardinals, Archive for Mathematical Logic, vol. 51 (2012), no. 5–6, pp. 601620.CrossRefGoogle Scholar
[4]Jech, T., Some combinatorial problems concerning uncountable cardinals, Annals of Mathematical Logic, vol. 5 (1973), pp. 165198.CrossRefGoogle Scholar
[5]Kanamori, A., The higher infinite. Large cardinals in set theory from their beginnings, Perspectives in Mathematical Logic, Springer, 1994.Google Scholar
[6]Kunen, K., Set theory. An introduction to independence proofs, North-Holland, 1980.Google Scholar
[7]Laver, R., Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), pp. 385388.CrossRefGoogle Scholar
[8]Magidor, M., Combinatorial characterization of supercompact cardinals, Proceedings of the American Mathematical Society, vol. 42 (1974), pp. 279285.CrossRefGoogle Scholar
[9]Mitchell, W. J., Aronszajn trees and the independence of the transfer property, Annals of Mathematical Logic, vol. 5 (1972), pp. 2146.CrossRefGoogle Scholar
[10]Mitchell, W. J., On the Hamkins approximation property, Annals of Pure and Applied Logic, vol. 144 (2006), no. 1–3, pp. 126129.CrossRefGoogle Scholar
[11]Unger, S., The ineffable tree property, in preparation.Google Scholar
[12]Viale, M., Guessing models and generalized Laver diamond, Annals of Pure and Applied Logic, vol. 163 (2012), no. 11, pp. 16601678.CrossRefGoogle Scholar
[13]Viale, M. and Weiss, C., On the consistency strength of the proper forcing axiom, Advances in Mathematics, vol. 228 (2011), no. 5, pp. 26722687.CrossRefGoogle Scholar
[14]Weiss, C., Subtle and ineffable tree properties, Ph.D. thesis, Ludwig-Maximilians-Universität München, 2010.Google Scholar