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At a correctly guessing limit stage of a normal tree construction, make every new level node extend a member of the guessed maximal tree antichain. It follows that the antichain has no later member. This produces a Suslin tree from the stationary diamond principle.

A subset of a forcing order whose distinct members have no common stronger extension. A maximal forcing antichain has a compatible member for every condition. For forcing by nodes of a set-theoretic tree, ordered by extension, this coincides with a tree antichain.

An Aronszajn tree is a set-theoretic tree of height ${\omega }_1$, with countable levels and no uncountable chain in a partial order, equivalently no cofinal branch. The strict predecessors of a node are well-ordered, and its height is their order type. Such trees exist in ZFC, as the following rationally labelled Aronszajn tree construction shows.

We build countable nonempty levels $T_{\alpha }$ for all $\alpha <{\omega }_1$. A node $t\in T_{\alpha }$ is a strictly increasing function $t:\alpha \to \mathbb{Q}\cap (0,1)$, and the tree order is proper initial-segment extension. Every restriction $t\upharpoonright \beta$ must lie in $T_{\beta }$. Give nonempty $t$ the rational labeland give the empty root label $0$. We require labels to increase strictly along the tree. We also maintain the bounded rational extension property:Keeping room below every rational cap is what makes the limit step possible.

Start with $T_0=\{\varnothing \}$. At a successor stage, for every $s\in T_{\alpha }$ and every rational $r$ with $\ell (s)<r<1$, includein $T_{\alpha +1}$. Its label is $r$, strictly larger than its predecessor's. This level is countable. To extend an earlier node below a prescribed cap $q$, first extend to $T_{\alpha }$ below $q$ by the inductive property, if necessary, then choose a new rational label between that label and $q$. Thus the extension property is preserved.

Now let $\lambda <{\omega }_1$ be a nonzero limit. The whole tree below $\lambda$ is countable. For each pair $(s,q)$ with $s\in T_{\beta }$, $\beta <\lambda$, and $\ell (s)<q<1$, construct one node at height $\lambda$ extending $s$ below $q$. There are only countably many such pairs.

Choose a rational $r$ with $\ell (s)<r<q$ and an increasing sequenceSet $s_0=s$. Given $s_n\in T_{{\alpha }_n}$ with $\ell (s_n)<r$, setThis is rational and satisfies $\max \{\ell (s_n),r-1/(n+1)\}<r_n<r$. The successor rule supplies $u_n=s_n^{⌢}{r}_n$ at height ${\alpha }_n+1$. If ${\alpha }_{n+1}={\alpha }_n+1$, take $s_{n+1}=u_n$; otherwise use the already established extension property below $\lambda$ to extend $u_n$ to $s_{n+1}\in T_{{\alpha }_{n+1}}$ with label below $r$.

The union $t={\bigcup }_n{s}_n$ has domain $\lambda$ and extends $s$. It is strictly increasing, and every shorter restriction is already in its prescribed level. All its values are below $r$, while it contains the values $r_n>r-1/(n+1)$. HenceEvery predecessor's label is below the label of some $s_n$, and hence below $r$. Thus strict increase of labels survives at the limit. Include one such $t$ for each pair $(s,q)$ in $T_{\lambda }$, discarding duplicates. This gives a countable level and preserves every bounded extension requirement. In particular applying it to the root makes the level nonempty. The choices here are permitted by the ambient axiom of choice; transfinite recursion through the set ${\omega }_1$ completes the construction.

Put $T={\bigcup }_{\alpha <{\omega }_1}{T}_{\alpha }$. A node of domain $\alpha$ has exactly the restrictions of domains $\beta <\alpha$ as predecessors, so its height is $\alpha$. Therefore $T$ has height ${\omega }_1$ and countable levels. Since $\ell :T\to \mathbb{Q}$ is strictly increasing on comparable nodes, an uncountable chain would inject into the countable set $\mathbb{Q}$, which is impossible. Each label fiber is a tree antichain, so the tree is even a special Aronszajn tree:

Prune an ${\aleph }_1$-Suslin tree as in part (a), then use its nodes as forcing conditions, with extensions stronger. Two conditions are compatible exactly when comparable, so the absence of uncountable tree antichains is the forcing countable chain condition for forcing. For each $\alpha <{\omega }_1$, the set $D_{\alpha }$ of nodes of height at least $\alpha$ is dense, by well-pruned set-theoretic tree.

If $2^{{\aleph }_0}>{\aleph }_1$, full Martin's axiom includes ${\mathrm{MA}}_{{\aleph }_1}$. It would provide a filter in an ordered set meeting all these dense subsets of a forcing order. Directedness makes that filter in an ordered set a chain in a partial order, and meeting every $D_{\alpha }$ makes its heights unbounded, producing an uncountable branch. This contradicts the Suslin-tree property. A Suslin set-theoretic tree together with failure of Continuum hypothesis therefore implies failure of Martin's axiom. This is the Suslin-tree obstruction to Martin's axiom.

Fix a diamond principle sequence $D_{\alpha }\subseteq \alpha$. Construct a normal splitting set-theoretic tree of height ${\omega }_1$ with countable levels. At successors give every node two successors. At a countable limit stage $\alpha$, the set-theoretic tree below $\alpha$ is countable. Choose countably many cofinal branches through it covering all its nodes, and put one node at level $\alpha$ above each distinct chosen branch. This preserves extension to all higher levels and tree with unique limits.

Arrange a coding of each level into the ordinal block $[\omega \alpha ,\omega (\alpha +1))$. On the club set of limit fixed points of $\alpha \mapsto \omega \alpha$, the nodes coded below $\alpha$ are exactly the nodes of height below $\alpha$. At a limit stage, if $D_{\alpha }$ codes a maximal tree antichain of the current set-theoretic tree below $\alpha$, require every chosen branch to meet it. This is possible: for any starting node $t$, maximality provides a comparable tree antichain member; if above $t$, first extend to it, and if below $t$, it has already been met. Then extend along a sequence of heights cofinal in $\alpha$. If the prediction is not a maximal tree antichain, use the ordinary covering branches. Thus every level is countable and the construction remains normal.

Here is the full chain-condition verification. Let $A$ be a maximal tree antichain in the final set-theoretic tree. For every node $t$, choose a witness $a_t\in A$ comparable with $t$. There is a club set of countable limit stages $\alpha$ closed under these witness choices: starting from any bound, repeatedly bound the heights of witnesses for all the countably many nodes below the current stage, and take the supremum after countably many steps. At such an $\alpha$, $A\cap T_{<\alpha }$ is already maximal in $T_{<\alpha }$.

View $A$ as a subset of ${\omega }_1$ through the coding. Diamond gives stationarily many stages with $D_{\alpha }=A\cap \alpha$. Choose one also in the witness-closure club set and the coding club set. The construction at that stage seals this very tree antichain: every node of level $\alpha$ extends one of its members below $\alpha$, and so does every node at a later level. No such node can itself belong to $A$, since it is comparable with an earlier member of $A$. HenceEvery tree antichain extends to a maximal one, so the set-theoretic tree has no uncountable tree antichain. Its normal splitting also excludes uncountable branches by part (ii). It is therefore a Suslin tree. By the standard Suslin-tree characterization of Suslin hypothesis, diamond implies failure of Suslin hypothesis. The decisive step is antichain sealing by diamond, with maximality below a correctly guessed club set stage verified explicitly.

Use the standard normal set-theoretic tree convention: a unique root, extensions at every higher level, splitting into at least two successors, and tree with unique limits. The small-level and height assumptions already give an ${\omega }_1$-tree, while the given tree antichain condition gives the countable chain condition for forcing. We only need to exclude an uncountable branch.

If such a branch existed, its heights would be unbounded, since each initial segment contains only countably many nodes. Fill in predecessors to obtain its node $b_{\alpha }$ at every level. At each successor step choose a successor $s_{\alpha }$ of $b_{\alpha }$ different from $b_{\alpha +1}$. For $\alpha <\beta$, the node $s_{\beta }$ extends the branch successor $b_{\alpha +1}$, and so is incompatible with $s_{\alpha }$. Thus $\{s_{\alpha }:\alpha <{\omega }_1\}$ is an uncountable tree antichain, a contradiction.

Therefore the set-theoretic tree is ${\aleph }_1$-Suslin. The splitting part of normality matters: a single chain in a partial order of height ${\omega }_1$ would satisfy the tree antichain condition but not the conclusion if one used a weakened definition of normality allowing no splitting.

Let $⟨D_{\alpha }:\alpha \in S⟩$ witness the stationary diamond principle: for every $A\subseteq {\omega }_1$, the set of $\alpha \in S$ with $D_{\alpha }=A\cap \alpha$ is stationary. We construct a normal splitting Suslin tree; this will be a nonspecial Aronszajn tree.

Construct its levels by recursion. Start with one root, and give every node two immediate successors. At a countable limit $\alpha$, the constructed portion $T\upharpoonright \alpha$ is countable. Through each of its nodes choose a cofinal branch of that portion, and put one new node above each chosen branch at level $\alpha$. This keeps the level countable and gives every earlier node an extension. Branches are identified by their predecessor chains, so nodes at a limit level are uniquely determined by their predecessors.

At a limit $\alpha \in S$, decode $D_{\alpha }$ as a candidate tree antichain $B$ of $T\upharpoonright \alpha$. If it is maximal, choose the branches just described to meet $B$. This is possible: for any node, maximality supplies a comparable member of $B$, and normality of the already constructed portion extends the larger of those two nodes to a cofinal branch up to $\alpha$. Then every node at level $\alpha$, and every later node, lies above a member of $B$. This is antichain sealing by diamond.

Here is a precise way to handle the coding. Give the countable level $\xi$ node codes in $[\omega \xi ,\omega \xi +\omega )$. There is a club set of countable limit $\alpha$ with $\omega \alpha =\alpha$, and on this club the nodes below $\alpha$ have exactly the relevant codes below $\alpha$. Empty unused codes are ignored. Thus any subset of the entire tree has an ordinal code set to which the stationary diamond principle applies.

Let $B$ now be any maximal tree antichain of the completed tree. There is a club set of $\alpha$ such that $B\cap (T\upharpoonright \alpha )$ is maximal in $T\upharpoonright \alpha$. Indeed, choose a comparable member of $B$ for each node; closure under the heights of these witnesses gives that club. Intersect it with the coding club. Stationary correct guessing supplies an $\alpha \in S$ on this intersection at which $B$ is sealed. A member of $B$ at or above level $\alpha$ would extend a member of $B$ below $\alpha$, contradicting the tree antichain property. So $B$ is contained in the countable portion below $\alpha$. Every tree antichain extends to a maximal one, hence every tree antichain is countable.

There is no cofinal branch of length ${\omega }_1$. Otherwise, choosing at each successor level the other successor of its branch node would give an uncountable tree antichain. Thus the resulting tree is a Suslin tree. A special Aronszajn tree is a union of countably many tree antichains; here those would all be countable and could not cover the ${\aleph }_1$ nodes. Therefore

The finite binary-function Cohen forcing $\mathrm{Fn}(\omega ,2,<\omega )$ is countable in $M$. It preserves ${\omega }_1$, and the countable levels, height, and normal extensions of the ground-model tree remain unchanged.

If the extension contained an uncountable tree antichain, apply the ground-model uncountable subset lemma for countable forcing to obtain an uncountable $X\in M$ contained in it. Incomparability in the fixed ground-model tree is absolute, so $M$ already regards $X$ as an uncountable tree antichain, contradicting that $T$ is Suslin in $M$. Similarly, a new cofinal branch has an uncountable ground-model subset. Comparability is absolute, so this would be an uncountable chain in the ground-model tree, again impossible.

Therefore countable forcing preserves Suslin trees, and in particular

Construct countable levels of bounded increasing rational sequences, maintaining the bounded rational extension property. At successor stages append every allowed rational. At a countable limit, handle each earlier-node/rational-cap pair by a cofinal sequence of extensions with values approaching a rational strictly below the cap. Its union has that rational supremum. Countably many requirements give a countable new level. Strictly increasing labels exclude uncountable chains, and label fibers are tree antichains, so the result is a special Aronszajn tree.

An Aronszajn tree that is a union of countably many tree antichains. Equivalently, it admits a map into a countable set that is injective on each chain.

An Aronszajn tree with no uncountable tree antichain. A normal splitting Suslin tree yields a Suslin line by a lexicographic ordering followed by Dedekind completion.

A set of pairwise incomparable nodes in a set-theoretic tree. A maximal tree antichain has a comparable member for every node of the tree.