Worked solution: Independence of Suslin's Hypothesis from ZFC (Jensen, Solovay–Tennenbaum, 1971)

Frequently asked questions

Why length

\aleph_2

specifically, rather than

\omega_1

\aleph_3

$\aleph_2$ gives exactly enough room for the bookkeeping to revisit every ccc poset of size at most $\aleph_1$ that could appear at any intermediate stage, while simultaneously being the value that ends up forced for the continuum; using a longer length works too and simply forces $2^{\aleph_0}$ to that larger value instead.

Source

Ronald Jensen, Robert M. Solovay, and Stanley Tennenbaum, 1971

Iterated Cohen Extensions and Souslin's Problem

Condensed summary

Step 6 of 8: Solovay–Tennenbaum: iterating away every Suslin tree

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In plain words

Forcing with one specific Suslin tree, using the tree's own order (reversed) as the poset, adds a branch through it and thereby destroys it — like finally completing that one endless-yet-thin family line so it stops being a counterexample. But this single step could, in principle, spawn a brand-new Suslin tree elsewhere in the freshly extended universe.

Solovay and Tennenbaum's trick was to repeat the destruction step in a long relay race of length $\aleph_2$ , using a bookkeeping device that eventually schedules every tree that could ever appear, at any point along the way, for destruction — while proving that chaining together countably many ccc steps like this never itself breaks the ccc property, so no cardinals ever collapse.

\langle \mathbb{P}_\alpha : \alpha \le \aleph_2 \rangle \text{ finite-support ccc iteration} \implies V[G_{\aleph_2}] \models \mathrm{MA} + \neg\mathrm{CH}

Detailed analysis

Solovay and Tennenbaum (1971) constructed a finite-support iteration $\langle \mathbb{P}_\alpha, \dot{\mathbb{Q}}_\beta : \alpha \le \aleph_2, \beta < \aleph_2 \rangle$ : at each stage $\beta$ , $\dot{\mathbb{Q}}_\beta$ is (a name for) some ccc poset chosen by a bookkeeping function so that, by the end, every ccc poset (in particular every tree-killing forcing and every instance needed to secure Martin's Axiom) that could possibly arise gets addressed at some stage. "Finite support" means each condition in $\mathbb{P}_{\aleph_2}$ only makes nontrivial demands at finitely many coordinates $\beta$ .

Their key preservation theorem: a finite-support iteration of ccc forcings is again ccc. This is the technical heart of the construction, since it guarantees that no cardinal collapses at any stage of this very long iteration, so $\aleph_1$ and $\aleph_2$ from the ground model survive intact all the way to the final model $V[G_{\aleph_2}]$ , where $2^{\aleph_0} = \aleph_2$ and $\mathrm{MA}$ holds.

Because the bookkeeping scheduled every Suslin tree that could arise for destruction at some stage, no Suslin tree survives into $V[G_{\aleph_2}]$ : $\mathrm{SH}$ holds there. This gives $\mathrm{Con}(\mathrm{ZFC}) \implies \mathrm{Con}(\mathrm{ZFC} + \mathrm{SH})$ , the second half of the independence proof — and their finite-support iteration technique itself became the founding method of modern iterated forcing.

Terms in this step

Finite-support iterated forcing: A method of chaining together a transfinite sequence of forcing notions, one after another, where each single condition in the final poset only makes a nontrivial demand at finitely many stages of the sequence.

Knowledge used in this step

The ZFC axioms