Finite-Support Iterations and Martin's Axiom: Examples and Counterexamples

The check names $\check{q}$ for $q\in \mathrm{Add}(\omega ,1)$ occur in the constant name $\dot{Q}$, hence lie in F1's bounded carrier $R$. They form a dense suborder of the restricted iteration: for any $(p,\dot{q})$, the forcing membership clause gives a strengthening $p^{\mathrm{\prime }}\le p$ that forces $\dot{q}=\check{q}$ for some ground $q$, and $(p^{\mathrm{\prime }},\check{q})$ extends $(p,\dot{q})$. On this dense check-name suborder, $(p,\check{q})\le (p^{\mathrm{\prime }},{\check{q}}^{\mathrm{\prime }})$ exactly when $p\le p^{\mathrm{\prime }}$ and $q\le q^{\mathrm{\prime }}$. Send this pair to the finite function $r$ on $2\times \omega$ with $r(0,n)=p(n)$ and $r(1,n)=q(n)$. Restriction is the inverse on the dense suborder, proving forcing equivalence.

F2 factors the generic into the two one-coordinate generics; F3 says their union reconstitutes the full generic and either coordinate is Cohen-generic over the extension by the other.

List the given reals as $r_{\alpha }$ for $\alpha <\lambda \le \kappa$. In $P=\mathrm{Add}(\omega ,1)$ let $E_n=\{p:(0,n)\in \mathrm{dom}p\}$ and $D_{\alpha }=\{p:\mathrm{\exists }n((0,n)\in \mathrm{dom}p\wedge p(0,n)\mathrm{\ne }{r}_{\alpha }(n))\}$. Extending at one fresh coordinate proves all these sets dense; $P$ is countable and hence ccc.

F1 supplies a filter meeting the at most $\kappa$ many $D_{\alpha }$ and countably many $E_n$. Its union is a total real $g$, and meeting $D_{\alpha }$ gives $g\mathrm{\ne }{r}_{\alpha }$. Thus no family of at most $\kappa$ reals exhausts $2^{\omega }$, so $\kappa <2^{{\mathrm{\aleph }}_0}$.

Write $X={\bigcup }_{x\in X}\{x\}$. Every singleton is closed nowhere dense and has Lebesgue measure zero.

Since the index family has size below the continuum, F1 makes its union meagre and F2 makes it null. The two conclusions are proved separately; neither is inferred from the other.

Assuming ZFC is consistent, F2 gives the consistency of a theory in which MA holds and CH fails. Thus the implication is not a theorem of ZFC, on the same metatheoretic consistency assumption under which the false statement is posed.

F1 records that reversing the arrows would confuse the valid implication with its false converse. The refutation rests on the fixed-fragment model argument of F2; it does not assume a countable transitive model of full ZFC.