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I am seeking "quotable equivalents" for MA (Martin's axiom). For the continuum hypothesis, examples of such statements are as follows.

(a) (Sierpinski) The (xy) plane can be covered by countably many $x \mapsto y$ and $y \mapsto x$ functions.

(b) (Zoli) The set of transcendental reals is a union of countably many transcendence bases for $\mathbb{R}$.

(c) (Erdős) There is an uncountable family of analytic functions on $\mathbb{C}$ that takes only countably many values at each complex number.

(d) (Freiling) There is a function $F$ from $\mathbb{R}$ to the family of countable subsets of $\mathbb{R}$ such that for every $x, y \in \mathbb{R}$, either $x \in F(y)$ or $y \in F(x)$.

Since each one of (a)-(d) refers to a "countable/uncountable" dichotomy, it would be reasonable to have statements with a "continuum/smaller than continuum" dichotomy.

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    $\begingroup$ Section 13 of Consequences of Martin's axiom by David Fremlin should cover the most important equivalences. $\endgroup$
    – Michael Greinecker
    Commented Mar 25, 2018 at 16:54
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    $\begingroup$ By "quotable", I'm guessing you mean something you could easily say to a friend while on a hike. $\endgroup$
    – Todd Trimble
    Commented May 11, 2020 at 12:37
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    $\begingroup$ @ToddTrimble: Not sure what sort of hikes you like, but MA is equivalent to the statement that Compact Hausdorff spaces without uncountable disjoint families of open subsets cannot be written as the union of fewer than continuum many closed nowhere dense sets. $\endgroup$
    – Todd Eisworth
    Commented May 11, 2020 at 14:30
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    $\begingroup$ The comment from @ToddEisworth is also what I would suggest, but you can shorten the hike a little (eliminate two negations) by taking the contrapositive: If a compact Hausdorff space can be covered by fewer than $\mathfrak c$ closed sets with empty interiors, then it has uncountably many pairwise disjoint open sets. $\endgroup$
    – Andreas Blass
    Commented May 11, 2020 at 14:38
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    $\begingroup$ Todd and Andreas: this is all interesting, and thanks, but mainly I was trying to clarify the intent of the question (and so my comment was directed to the OP). $\endgroup$
    – Todd Trimble
    Commented May 11, 2020 at 15:40

1 Answer 1

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Martin's axiom is equivalent to the assertion that $$H_{\frak{c}}\prec_{\Sigma_1} V[G]$$ for all c.c.c. forcing extensions $V[G]$. In other words, $H_{\frak{c}}$ is existentially closed in all c.c.c. forcing extensions.

This is proved in Bagaria, Joan, A characterization of Martin’s axiom in terms of absoluteness, J. Symb. Log. 62, No. 2, 366-372 (1997). ZBL0883.03039. The characterization also appears independently, attributed to J. Stavi, in Stavi, Jonathan; Väänänen, Jouko, Reflection principles for the continuum, Zhang, Yi (ed.), Logic and algebra. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 302, 59-84 (2002). ZBL1013.03059.

This characterization leads naturally to the resurrection axioms, which I explored with Thomas Johnstone in several articles. The resurrection axiom for a class of forcing $\Gamma$ is the assertion that for every $\Gamma$ extension $V[g]$ there is further $\Gamma$ forcing $V[g][h]$ such that $$H_{\frak{c}}\prec H_{\frak{c}}^{V[g][h]}.$$ Thus, one attains full elementarity, at the cost of further forcing.

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