David Asperó, Universität Wien
Strong forcing axioms, like the Proper Forcing Axiom, are known, since the 1980s, to decide the size of the continuum and to give it the value . Bounded forcing axioms ([5]) are weak forms of these axioms which, being characterizable as principles of forcing absoluteness for -formulas with parameters in the initial segment of the universe ([3]), are arguably natural axioms extending set theory. Therefore, a natural question is whether these principles are strong enough to decide the size of the continuum. In this talk I will give a survey of recent results concerning the Bounded Martin's Maximum (), which is the bounded form of the maximal forcing axiom Martin's Maximum ([4]), and the continuum problem. More precisely, is known to imply, in each of the following situations, that the size of the continuum is ([1], [2], [6]):
The question whether decides, whithout any additional hypotheses, the size of the continuum remains open.
[1] | D. Asperó, Bounded forcing axioms and the size of the continuum. Submitted. |
[2] | D. Asperó, The Bounded Martin's Maximum, Erdös cardinals and . Submitted. |
[3] | J. Bagaria, Bounded forcing axioms as principles of generic absoluteness, Archive for Mathematical Logic, vol. 39 (2000), 393-401. |
[4] | M. Foreman, M. Magidor, S. Shelah, Martin's Maximum, saturated ideals, and non-regular ultrafilters. Part I, Annals of Mathematics, vol. 127 (1988), 1-47. |
[5] | M. Goldstern, S. Shelah, The Bounded Proper Forcing Axiom, J. Symbolic Logic, vol. 60 (1995), 58-73. |
[6] | H. Woodin, The axiom of Determinacy, Forcing Axioms, and the Nonstationary ideal, De Gruyter Series in Logic and its Applications. Number 1. Berlin, New York, 1999. |
E-Mail: | aspero@logic.univie.ac.at |