## 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 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
. 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]):

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

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. |

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