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) (Erods) 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" dichotmoy, it would be reasonable to have statements that with "continuum/smaller than continuum" dichotmoty.

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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 Mar 25 '18 at 16:54

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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