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

Frequently asked questions

Why did two different methods (Jensen's and Jech/Tennenbaum's) both settle the same consistency result at nearly the same time?

The late 1960s were a period of rapid development for both constructibility theory and forcing, and Suslin's problem was a natural target for each; the coincidence highlights how the two techniques, though very different in spirit, are complementary tools for building models with tailored properties.

Source

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

Iterated Cohen Extensions and Souslin's Problem

Condensed summary

Step 4 of 8: Two independent routes build a Suslin tree:

\neg\mathrm{SH}

is consistent

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In plain words

Using $\diamondsuit$ as a fortune-teller, Jensen built a Suslin tree level by level: whenever the guess at stage $\alpha$ correctly predicts a maximal antichain-in-progress, the construction takes evasive action right there, so no maximal antichain ever grows uncountable and no branch is ever forced to run forever.

Around the same time, working independently and without assuming $V = L$ at all, Thomas Jech (1967) and Stanley Tennenbaum (1968) forced a Suslin tree into existence directly, adding it piece by piece with finite conditions — a second, completely different route to the very same consistency result.

V=L \ (\text{Jensen}) \ \text{or forcing (Jech 1967, Tennenbaum 1968)} \implies \exists \text{ Suslin tree} \implies \neg\mathrm{SH}

Detailed analysis

Jensen's construction: assuming $\diamondsuit$ , build a tree $T$ of height $\omega_1$ by transfinite recursion, adding one level $T_\alpha$ at a time; whenever the diamond sequence's guess $A_\alpha$ happens to code a maximal antichain of the tree built so far, extend every node not already in that antichain in a way that defeats it (ensures it cannot remain maximal). Since $\diamondsuit$ guarantees this guess is correct on a stationary set of stages, every potential maximal antichain gets caught and defeated at some point, so the resulting tree has no uncountable antichain (and, by a companion argument, no uncountable branch): a genuine $\omega_1$ -Suslin tree.

Independently, Jech (1967) and Tennenbaum (1968) showed a Suslin tree can be added by forcing with a poset of finite approximations to the tree's order relation, without any appeal to $V = L$ or $\diamondsuit$ ; genericity again ensures no uncountable branch or antichain can be completed. Both routes yield: $\mathrm{Con}(\mathrm{ZFC}) \implies \mathrm{Con}(\mathrm{ZFC} + \neg\mathrm{SH})$ .

This settles one half of Suslin's Hypothesis's independence: $\mathrm{SH}$ can never be proved from $\mathrm{ZFC}$ alone. The remaining steps build the opposite kind of model, where every possible Suslin tree is systematically destroyed.

Terms in this step

Suslin tree: An $\omega_1$ -tree (height $\omega_1$ , $\aleph_1$ nodes total) in which every branch and every antichain is countable; its existence is equivalent to the existence of a Suslin line.

Knowledge used in this step

The ZFC axioms