15. ÖMG-Kongress
Jahrestagung der Deutschen Mathematikervereinigung

16. bis 22. September 2001 in Wien

Sektion 4 - Mathematische Logik, Theoretische Informatik
Donnerstag, 20. September 2001, 15.00, Hörsaal 46

 

Forcing absoluteness for $ \Sigma_1$ formulas and the continuum problem

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 $ \aleph_2$. Bounded forcing axioms ([5]) are weak forms of these axioms which, being characterizable as principles of forcing absoluteness for $ \Sigma_1$-formulas with parameters in the initial segment of the universe $ H(\omega_2)$ ([3]), are arguably natural axioms extending $ ZFC$ 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 ($ BMM$), which is the bounded form of the maximal forcing axiom Martin's Maximum ([4]), and the continuum problem. More precisely, $ BMM$ is known to imply, in each of the following situations, that the size of the continuum is $ \aleph_2$ ([1], [2], [6]):

(a)
If the sharp of some set does not exist,
(b)
if there exists some $ \omega_1$-Erdös cardinal (and actually something slightly weaker than that suffices),
(c)
if the nonstationary ideal over $ \omega_1$ is precipitous, and
(d)
if that same ideal has a very weak degree of precipitousness and the second uniform indiscernible is $ \omega_2$.

The question whether $ BMM$ 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 $ \psi_{AC}$. 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

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