Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-26T00:09:40.149Z Has data issue: false hasContentIssue false

Semiproper forcing axiom implies Martin maximum but not PFA+

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah*
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109 Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
Institute of Mathematics, The Hebrew University, Jerusalem, Israel


We prove that MM (Martin maximum) is equivalent (in ZFC) to the older axiom SPFA (semiproper forcing axiom). We also prove that SPFA does not imply SPFA+ or even PFA+ (using the consistency of a large cardinal).

Research Article
Copyright © Association for Symbolic Logic 1987

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.)



[1]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. 913959.CrossRefGoogle Scholar
[2]Foreman, M., Magidor, M. and Shelah, S., Martin maximum, saturated ideals and nonregular ultrafilters. I, Annals of Mathematics (to appear).Google Scholar
[3]Laver, R., Making the supercompactness of κ: indestructible under κ-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), pp. 385388.CrossRefGoogle Scholar
[4]Shelah, S., Iterated forcing and changing cofinalities, Israel Journal of Mathematics, vol. 40 (1981), pp. 132.CrossRefGoogle Scholar
[5]Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar
[6]Todorčević, S., Forcing positive partition relations, Transactions of the American Mathematical Society, vol. 280 (1983), pp. 703720.CrossRefGoogle Scholar