The omega_2 iteration forces MA and continuum aleph_2

F1 makes $P_{{\omega }_2}$ ccc, and F2 preserves ${\omega }_1,{\omega }_2$. F3 and GCH give at most ${\mathrm{\aleph }}_2^{{\mathrm{\aleph }}_0}={\mathrm{\aleph }}_2$ real names. At every selected Cohen stage, the coordinate-domain dense sets make the union a total real, and for each real in the preceding intermediate model the dense set requiring disagreement at a fresh coordinate makes the new real different from it. Hence Cohen reals added at distinct selected stages are distinct. There are cofinally, thus ${\mathrm{\aleph }}_2$, many such stages, giving the reverse inequality. Therefore the final continuum is ${\mathrm{\aleph }}_2$.

In the final extension fix $\kappa <{\mathrm{\aleph }}_2$, a ccc order $Q$, and at most $\kappa$ dense subsets. By F5 replace $Q$ by a coded dense order of size at most $\kappa$, and code it and the family by a set of ground ordinals of size at most ${\mathrm{\aleph }}_1$. F4 places that code in some intermediate $V[G_{\alpha }]$.

The earlier model sees $Q$ as ccc: otherwise it contains an ${\omega }_1$-antichain, and the ccc tail preserves both that set and its incompatibility, contradicting final ccc. Bookkeeping therefore selects the name at a later stage $\beta$. Its coordinate generic is a filter on $Q$ meeting every dense set already coded at stage $\alpha$, hence the original family. Since $\kappa <{\mathrm{\aleph }}_2$ was arbitrary, MA holds.

step 1.1 gives $2^{{\mathrm{\aleph }}_0}={\mathrm{\aleph }}_2>{\mathrm{\aleph }}_1$, so CH fails. AC is used in coding, bookkeeping, cardinal arithmetic, and the preservation/name arguments.