A club of correctly coded maximal-antichain restrictions

Every level $T_{\alpha }$ is nonempty: height ${\omega }_1$ supplies a node of height at least $\alpha$, and F1 supplies its predecessor at $\alpha$ if necessary. AC chooses an injection of each countable level into $\omega$. The map sending a node to its height and its chosen level index injects $T$ into ${\omega }_1\times \omega$, of cardinality ${\mathrm{\aleph }}_1$ by F5. AC also chooses a node at every level, injecting ${\omega }_1$ into $T$. Thus $\mathrm{\mid }T\mathrm{\mid }={\mathrm{\aleph }}_1$ and a bijection $b$ exists. Fix any such $b$.

Define $h(\xi )=\mathrm{ht}(b^{-1}(\xi ))$ and $g(\alpha )=\sup \{b(t)+1:t\in T_{\alpha }\}$. Each $g(\alpha )<{\omega }_1$ by countability of the level and F4. Maximality of $A$ gives a member comparable with each node: otherwise that node could be adjoined to $A$. Let $w(\xi )$ be the least code of a member of $A$ comparable with $b^{-1}(\xi )$. This minimum exists. Thus $h,g,w$ are self-maps of ${\omega }_1$.

By F4 and A1, ${\omega }_1$ is regular uncountable: every smaller cardinal is countable and cannot be cofinal. Apply F2 to $h,g,w$ and intersect their three closure clubs with the club $L$ of nonzero limit ordinals, using F3. The set $L$ is closed; it is unbounded because $\beta +\omega$ is a countable nonzero limit above any countable $\beta$. Call the resulting club $C$. For $\delta \in C$, if $b(t)<\delta$, then $\mathrm{ht}(t)=h(b(t))<\delta$. Conversely if $\mathrm{ht}(t)=\alpha <\delta$, then $b(t)<g(\alpha )<\delta$. These prove $b^{-1}[\delta ]=T_{<\delta }$.

For $t\in T_{<\delta }$, step 3.1 gives $b(t)<\delta$ and closure under $w$ gives $w(b(t))<\delta$. The corresponding node lies in $A\cap T_{<\delta }$ and is comparable with $t$. The restriction of $A$ is still an antichain, so this comparability with every restricted node proves maximality: no further node can be adjoined. This proves the assertion for every $\delta \in C$.