Martin's axiom. Let $\langle\mathbb P,\leq\rangle$ be a ccc partially ordered set. If $\mathcal D$ is a family of dense subset of $\mathbb P$ such that $|\mathcal D|<\mathfrak c$, then there exists a $\mathcal D$-generic filter $F$ in $\mathbb P.$ it is usually abbreviated by $\mathrm{MA}$. The following theorem is well known.
Theorem: If $\mathrm{MA}$ holds then a union of less than continuum many meager subsets of $\mathbb R$ is meager in $\mathbb R$.
So, it is clear that any subset of $\mathbb R$ with cardinality less than $\mathfrak c$ is meager.
My question is Can the theorem above be hold with a weaker set-theoretical assumption ? or we need the whole $\mathrm{MA}$ to prove it ?
Any help will be appreciated greatly.
Best Answer
Probably the simplest way to build a model in which MA fails but every union of $<\mathfrak c$ meager sets is meager is to start with a model of GCH and adjoin Hechler reals in a finite-support iteration of length $\aleph_2$.
A table of facts of this sort is in Section 11 of my chapter of the Handbook of Set Theory, a preprint of which is available at http://www.math.lsa.umich.edu/~ablass/hbk.pdf .
For a lot more information about MA (and weaker forms of MA), the standard reference is David Fremlin's book, "Consequences of Martin's Axiom".