The Ostaszewski club principle

In ZFC, the Ostaszewski club principle $\mathrm{♣}$ asserts a sequence $(C_{\alpha })$ indexed by the nonzero limit ordinals $\alpha <{\omega }_1$, where $C_{\alpha }\subseteq \alpha$ is cofinal in $\alpha$, such that for every uncountable $X\subseteq {\omega }_1$ the set

$\{\alpha <{\omega }_1:\alpha \text{ is a nonzero limit and }{C}_{\alpha }\subseteq X\}$

is stationary. Use The first uncountable ordinal ${\omega }_1:=\mathrm{\aleph }(\omega )$, the cofinality convention of Cofinal subset of an ordinal, and stationarity from The club filter and nonstationary ideal. The sequence is an extra principle; no existence is asserted in ZFC alone. Each $C_{\alpha }$ is bounded in ${\omega }_1$, so it is not a club of ${\omega }_1$.

One may equivalently require each $C_{\alpha }$ to have order type $\omega$. Here is the thinning argument, including its choice use. By The Axiom of Choice, fix a surjection $e_{\alpha }:\omega \to \alpha$ for every nonzero countable limit $\alpha$ simultaneously. For any cofinal $C\subseteq \alpha$, set $c_0=\min \{c\in C:c>e_{\alpha }(0)\}$ and recursively set $c_{n+1}=\min \{c\in C:c>\max (c_n,e_{\alpha }(n+1))\}$. Cofinality and limitness ensure every minimum exists. The sequence is strictly increasing, and every $\beta <\alpha$ occurs in the enumeration and hence is below some $c_n$. Its range therefore has order type $\omega$ and is cofinal. This recursion is an instance of The recursion theorem (store the stage and last value in the state). Apply it to each supplied $C_{\alpha }$. The new ladder is a subset of the old, so every old containment guess is preserved and the stationary requirement persists. Conversely an order-type-$\omega$ witness already meets the original cofinal-set definition.

Zero and successor ordinals are excluded as ladder indices. Empty or countable targets $X$ carry no guessing requirement; the quantified targets are uncountable subsets of ${\omega }_1$.