We define the $\aleph_{1.5}$ chain condition. The corresponding forcing axiom
is a generalization of Martin's Axiom and implies certain uniform failures of
club--guessing on $\omega_1$ that don't seem to have been considered in the
literature before.Comment: 36 page

D. H. Fremlin

D. Martin

David Asperó

I. Juhasz

I. Neeman

J. Bagaria

J. D. Hamkins

J. T. Moore

K. Kunen

Miguel Angel Mota

R. M. Solovay

S. Shelah

T. Jech

T. Miyamoto

English

arXiv

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Israel Journal of Mathematics

A generalization of Martin’s Axiom

We define the $\aleph_{1.5}$-chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom; in fact, $MA^{1.5}_{<\kappa}$ implies $MA_{<\kappa}$. Also, $MA^{1.5}_{<\kappa}$ implies certain uniform failures of club-guessing on $\omega_1$ that do not seem to have been considered in the literature before. We show, assuming CH and given any regular cardinal $\kappa\geq\omega_2$ such that $\mu^{\aleph_0}< \kappa$ for all $\mu < \kappa$ and such that $\diamondsuit(\{\alpha<\kappa\,:\, cf(\alpha)\geq\omega_1\})$ holds, that there is a proper $\aleph_2$-c.c. partial order of size $\kappa$ forcing $2^{\aleph_0}=\kappa$ together with $MA^{1.5}_{<\kappa}$

A Generalization of Martin's Axiom

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