Skip to main content Accessibility help

Strong tree properties for small cardinals

  • Laura Fontanella (a1)


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.



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



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