## Journal of the Mathematical Society of Japan

### Some consequences from Proper Forcing Axiom together with large continuum and the negation of Martin's Axiom

Teruyuki YORIOKA

#### Abstract

Recently, David Asperó and Miguel Angel Mota discovered a new method of iterated forcing using models as side conditions. The side condition method with models was introduced by Stevo Todorčević in the 1980s. The Asperó–Mota iteration enables us to force some $\Pi_2$-statements over $H(\aleph_2)$ with the continuum greater than $\aleph_2$. In this article, by using the Asperó–Mota iteration, we prove that it is consistent that $\mho$ fails, there are no weak club guessing ladder systems, $\mathfrak{p}= {\mathrm{add}}(\mathcal{N}) = 2^{\aleph_0}>\aleph_2$ and ${MA}_{\aleph_1}$ fails.

#### Article information

Source
J. Math. Soc. Japan, Volume 69, Number 3 (2017), 913-943.

Dates
First available in Project Euclid: 12 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1499846513

Digital Object Identifier
doi:10.2969/jmsj/06930913

Mathematical Reviews number (MathSciNet)
MR3685031

Zentralblatt MATH identifier
06786984

#### Citation

YORIOKA, Teruyuki. Some consequences from Proper Forcing Axiom together with large continuum and the negation of Martin's Axiom. J. Math. Soc. Japan 69 (2017), no. 3, 913--943. doi:10.2969/jmsj/06930913. https://projecteuclid.org/euclid.jmsj/1499846513

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