Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-18T19:38:38.385Z Has data issue: false hasContentIssue false

The proper forcing axiom and the singular cardinal hypothesis

Published online by Cambridge University Press:  12 March 2014

Matteo Viale*
Affiliation:
Dipartimento di Matematica, Universitá di Torino, Torino, Italy Equipe de Logique Mathematique, Université Paris7, Paris, France. E-mail: viale@dm.unito.it

Abstract

We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof uses the reflection principle MRP introduced by Moore in [11].

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2006

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]Caicedo, A. and Veličković, B., Bounded proper forcing axiom and well orderings of the reals, Mathematical Research Letters, to appear, 18 pages.Google Scholar
[2]Cummings, J. and Schimmerling, E., Indexed squares, Israel Journal of Mathematics, vol. 131 (2002). pp. 6199.CrossRefGoogle Scholar
[3]Easton, W. B., Powers of regular cardinals, Annals of Pure and Applied Logic, vol. 1 (1970). pp. 139178.Google Scholar
[4]Foreman, M., Magidor, M., and Shelah, S.. Martin's maximum, saturated ideals, and nonregular ultrafilters., Annals of Mathematics. Second Series, vol. 127 (1988). no. 1, pp. 147.CrossRefGoogle Scholar
[5]Gitik, M., Blowing up power of a singular cardinal—wider gaps. Annals of Pure and Applied Logic, vol. 116 (2002), pp. 138.CrossRefGoogle Scholar
[6]Gitik, M.Prikry type forcings, Handbook of set theory (Foreman, , Kanamori, , and Magidor, , editors), to appear.Google Scholar
[7]Jech, T., Set theory: the millennium edition, Springer-Verlag, Berlin, 2003.Google Scholar
[8]Köng, B. and Yoshinobu, Y., Fragments of Martin's maximum in generic extensions, Mathematical Logic Quarterly, vol. 50 (2004). no. 3, pp. 297302.CrossRefGoogle Scholar
[9]Magidor, M., Lectures on weak square principles and forcing axioms, unpublished notes of the course held in the Jerusalem Logic Seminar, summer 1995.Google Scholar
[10]Moore, J. T., The proper forcing axiom. Prikry forcing, and the singular cardinals hypothesis, Annals of Pure and Applied Logic, to appear.Google Scholar
[11]Moore, J. T.. Set mapping reflection, Journal of Mathematical Logic, vol. 5 (2005), no. 1. pp. 8797.CrossRefGoogle Scholar
[12]Silver, J. H.. On the singular cardinals problem, Proceedings of the international congress of mathematicians (Vancouver, B. C., 1974), vol. 1, Canadian Mathematical Congress, Montreal. Que., 1975, pp. 265268.Google Scholar
[13]Solovay, R. M., Strongly compact cardinals and the GCH, Proceedings of the Tarski Symposium (Proceedings of the Symposium for Pure Mathematics, Vol. XXV, Univ. California, Berkeley, California, 1971) (Henkin, L.et al., editors), American Mathematical Society, Providence, R.I., 1974, pp. 365372.Google Scholar
[14]Veličković, B.. Forcing axioms and stationary sets, Advances in Mathematics, vol. 94 (1992). no. 2, pp. 256284.CrossRefGoogle Scholar