Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-31T19:35:23.950Z Has data issue: false hasContentIssue false
  • English
  • Français

Forcing Axioms, Supercompact Cardinals, Singular Cardinal Combinatorics

Published online by Cambridge University Press:  15 January 2014

Matteo Viale
Matteo Viale*
Affiliation:
Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Strasse 25, A-1090, Wien, AustriaE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the results so obtained in suitable large cardinals properties.

The first example I will treat is the proof that the proper forcing axiom PFA implies the singular cardinal hypothesis SCH, this will easily lead to a new proof of Solovay's theorem that SCH holds above a strongly compact cardinal. I will also outline how some of the ideas involved in these proofs can be used as means to evaluate the “saturation” properties of models of strong forcing axioms like MM or PFA.

The second example aims to show that the transfer principle (ℵω+1, ℵω) ↠ (ℵ2, ℵ1) fails assuming Martin's Maximum MM. Also in this case the result can be translated in a large cardinal property, however this requires a familiarity with a rather large fragment of Shelah's pcf-theory.

Only sketchy arguments will be given, the reader is referred to the forthcoming [25] and [38] for a thorough analysis of these problems and for detailed proofs.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 14 , Issue 1 , March 2008 , pp. 99 - 113
Copyright
Copyright © Association for Symbolic Logic 2008

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. and Magidor, M., Cardinal arithmetic, Handbook of set theory (Foreman, M., Kanamori, A., and Magidor, M., editors), North Holland, to appear.Google Scholar
[2] Abraham, U. and Todorčević, S., Partition Properties of ω1 Compatible with CH, Fundamenta Mathematicae, vol. 152 (1997), pp. 165–180.Google Scholar
[3] Balcar, B., Jech, T., and Pazák, T., Complete ccc Boolean algebras, the order sequential topology, and a problem of von Neumann, Bulletin of the London Mathematical Society, vol. 37 (2005), pp. 885–898.Google Scholar
[4] Baumgartner, J., Applications of the proper forcing axiom, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J. E., editors), North-Holland, Amsterdam, 1984, pp. 913–959.Google Scholar
[5] Caicedo, A. and Veličković, B., Bounded proper forcing axiom and well orderings of the reals, Mathematical Research Letters, vol. 13 (2006), no. 2–3, pp. 393–408.CrossRefGoogle Scholar
[6] Cummings, J., Collapsing successors of singulars, Proceedings of the American Mathematical Society, vol. 125 (1997), no. 9, pp. 2703–2709.Google Scholar
[7] Cummings, J. and Schimmerling, E., Indexed squares, Israel Journal of Mathematics, vol. 131 (2002), pp. 61–99.CrossRefGoogle Scholar
[8] Devlin, K. and Jensen, R. B., Marginalia to a theorem of Silver, ⊨ ISILC Logic Conference (G. H.Müller, Oberschelp, A., and Potthoff, K., editors), Lecture Notes in Mathematics, vol. 499, Springer, 1975, pp. 115–142.Google Scholar
[9] Easton, W. B., Powers of regular cardinals, Annals of Mathematical Logic, vol. 1 (1970), pp. 139–178.CrossRefGoogle Scholar
[10] Foreman, M., Ideals and Generic Elementary Embeddings, Handbook of set theory (Foreman, M., Kanamori, A., and Magidor, M., editors), North Holland, to appear.CrossRefGoogle Scholar
[11] Foreman, M. and Magidor, M., Avery weak square principle, The Journal of Symbolic Logic, vol. 1 (1997), pp. 175–196.CrossRefGoogle Scholar
[12] Foreman, M., Magidor, M, and Shelah, S., Martin's Maximum, saturated ideals and nonregular ultrafilters, Annals of Mathematics (2), vol. 127 (1988), no. 1, pp. 1–47.Google Scholar
[13] Gitik, M., Changing cofinalities and the non-stationary ideal, Israel Journal of Mathematics, vol. 56 (1986), no. 3, pp. 280–314.Google Scholar
[14] Gitik, M., The strength of the failure of SCH, Annals of Pure and Applied Logics, vol. 51 (1991), no. 3, pp. 215–240.CrossRefGoogle Scholar
[15] Jech, T., Set theory, The Third Millennium Edition ed., Springer, 2002, revised and expanded.Google Scholar
[16] König, B., Forcing indestructibility of set theoretic axioms, The Journal of Symbolic Logic, vol. 72 (2007), pp. 349–360.CrossRefGoogle Scholar
[17] König, B. and Yoshinobu, Y., Fragments of Martin's Maximum in generic extensions, Mathematical Logic Quarterly, vol. 50 (2004), no. 3, pp. 297–302.Google Scholar
[18] Larson, P. B., The Stationary Tower: Notes on a Course by W. Hugh Woodin, AMS, 2004.Google Scholar
[19] Levinski, J., Magidor, M, and Shelah, S., Chang's conjecture for ℵω , Israel Journal of Mathematics, vol. 69 (1990), no. 2, pp. 161–172.Google Scholar
[20] Magidor, M., On the singular cardinal problem I, Israel Journal of Mathematics, vol. 28 (1977), pp. 1–31.Google Scholar
[21] Magidor, M., Reflecting stationary sets, The Journal of Symbolic Logic, vol. 4 (1982), pp. 755–771.Google Scholar
[22] Moore, J. T., Proper forcing, the continuum, and uncountable linear orders, this Bulletin, vol. 11 (2005), no. 1, pp. 51–60.Google Scholar
[23] Moore, J. T., A five element basis for the uncountable linear orders, Annals of Mathematics, vol. 163 (2006), no. 2, pp. 669–688.Google Scholar
[24] Moore, J. T., The proper forcing axiom, Prikry forcing, and the singular cardinals hypothesis, Annals of Pure and Applied Logic, vol. 140 (2006), no. 1–3, pp. 128–132.CrossRefGoogle Scholar
[25] Sharon, A. and Viale, M., A mild reflection principle on ℵ implies that all points in are approachable, in preparation.Google Scholar
[26] Shelah, S., Proper Forcing, Springer, 1982.CrossRefGoogle Scholar
[27] Shelah, S., Cardinal Arithmetic, Oxford University Press, 1994.Google Scholar
[28] Silver, J. H., On the singular cardinal problem, Proceedings of the International Congress of Mathematicians, Vancouver, B.C., 1974, vol. 1, 1975, pp. 265–268.Google Scholar
[29] Solovay, R. M., Strongly compact cardinals and the GCH, Proceedings of the Symposium in Pure Mathematics (Henkin, L. et al., editors), vol. XXV, University of California, Berkeley, CA, 1971, pp. 365–372.Google Scholar
[30] Solovay, R. M. and Tennebaum, S., Iterated Cohen extensions and Souslin's problem, Annals of Mathematics, vol. 94 (1971), no. 2, pp. 201–245.Google Scholar
[31] Todorčević, S., A note on the proper forcing axiom, Axiomatic set theory (Boulder, Colorado, 1983), Contemporary Mathematics, vol. 31, American Mathematical Society, 1984, pp. 209–218.Google Scholar
[32] Todorčević, S., Partitions problems in topology, Contemporary mathematics, vol. 84, American Mathematical Society, 1989.Google Scholar
[33] Todorčević, S., A dichotomy for P-ideals of countable sets, Fundamenta Mathematicae, vol. 166 (2000), no. 3, pp. 251–267.CrossRefGoogle Scholar
[34] Veličković, B., Forcing axioms and stationary sets, Advances in Mathematics, vol. 94 (1992), no. 2, pp. 256–284.Google Scholar
[35] Veličković, B., CCC Forcing and Splitting Reals, Israel Journal of Mathematics, vol. 147 (2005), pp. 209–220.Google Scholar
[36] Viale, M., Application of the proper forcing axiom to cardinal arithmetic, Ph.D. thesis, Université Paris 7-Denis Diderot, 2006.Google Scholar
[37] Viale, M., The Proper Forcing Axiom and the Singular Cardinal Hypothesis, The Journal of Symbolic Logic, vol. 71 (2006), no. 2, pp. 473–479.CrossRefGoogle Scholar
[38] Viale, M., A family of covering properties, Mathematical Research Letters, to appear.Google Scholar