The Bounded Martin’s Maximum and the Size of the Continuum

David Asperó

Bounded forms of forcing axioms were introduced in [G-S]. For a class of partially ordered sets Γ, the Bounded Forcing Axiom for Γ, BF A(Γ), is the following statement:

Let ℙ be a poset in Γ and let A be a set of size at most ℵ_{1} consisting of maximal antichains of ℙ, all of size at most ℵ_{1}. Then there is a A–generic filter G ⊆ ℙ.

Bounded forcing axioms are arguably natural axioms, since they can be characterized in terms of a principle of generic absoluteness in the following way ([B], [M]):

Fact 1 Let Γ be a class of complete Boolean algebras. Then the following conditions are equivalent:

(a) BF A(Γ).

(b) For every a ∈ H(ω_{2}) and every Σ_{1} formula φ(x), if there is a poset ℙ in Γ forcing φ(ǎ), then H(ω_{2}) ⊧ φ(a).

The Bounded Martin’s Maximum (BMM) is BF A(Γ), where Γ is the class of all stationary set preserving posets. Forcing axioms are known to decide the size of the continuum. It is therefore a natural problem to find out whether bounded forcing axioms also do. We prove the following result:

Theorem 2 Assume BMM and suppose X^{♯} does not exist for some X ∈ H(ω_{2}). Then 2^{ℵ_{0}} = ℵ_{2}.

This should be compared with the following result, due to Woodin ([W]):

Theorem 3 Assume BMM and suppose that there is a measurable cardinal or that the nonstationary ideal over ω_{1} is precipitous. Then 2^{ℵ_{0}} = ℵ_{2}.

Also, Theorem 2 can be improved by considering natural strengthenings of the absoluteness principle in (b) of Fact 1.

[B ] J. Bagaria, Bounded forcing axioms as principles of generic absoluteness, Archive for Mathematical Logic. To appear.

[G-S ] M. Goldstern, S. Shelah, The Bounded Proper Forcing Axiom, J. Symbolic Logic, vol. 60 (1995), 58–73.

[W ] 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.

Address: Departament de Lògica, Història i Filosofia de la Ciència, Universitat de Barcelona, Baldiri i Reixach, s/n, 08028 Barcelona, Catalonia, Spain

E-mail: daspero@hotmail.com