Stationary set ($S\cap C\ne \varnothing \text{ for every club }C$)

A subset of a regular uncountable cardinal number $\kappa$ is stationary if it meets every club set in $\kappa$. Intersecting a stationary set with a club set preserves stationarity. Every member of the club filter is stationary. At a regular infinite $\theta <\kappa$, the stationarity of ordinals of prescribed cofinality supplies important examples.

Every stationary subset of a successor cardinal ${\lambda }^{+}$ can be partitioned into ${\lambda }^{+}$ disjoint stationary subsets. For each $\beta \in [\lambda ,{\lambda }^{+})$ choose a bijection $e_{\beta }:\lambda \to \beta$. For each $\xi <{\lambda }^{+}$, the function $\beta \mapsto e_{\beta }^{-1}(\xi )$ is regressive on the stationary tail with $\beta >\xi$. The Fodor lemma gives a stationary fiber with one value $i_{\xi }<\lambda$. Some value occurs for ${\lambda }^{+}$ many indices, whose fibers are pairwise disjoint. Assign any leftover part of $S$ to one fiber.

A subset $X\subseteq \lambda$ is non-reflecting in the indicated sense if it is not stationary below any $\delta <\lambda$ of uncountable cofinality. The subset itself need not be stationary. Nonreflection is inherited by subsets. With this definition every subset of ${\omega }_1$ is non-reflecting, since all smaller ordinals have countable cofinality.

For a coherent club set sequence, the limit points of $C_{\delta }$ form a club set whenever $\mathrm{cf}(\delta )>\omega$. Along these points, $\mathrm{otp}(C_{\gamma })=\mathrm{otp}(C_{\delta }\cap \gamma )$ strictly increases, so a fixed fiber $E_{\xi }$ meets this a club set at most once and cannot reflect at $\delta$. Under the square principle order-type bound, the fibers with $\xi \le \kappa$ partition all limit ordinals below ${\kappa }^{+}$ into $\kappa$ non-reflecting pieces.

For any nonzero limit ordinal $\delta$, a club set is an unbounded subset closed under limit points below $\delta$. A subset is stationary if it meets every such club set. Statements concerning the completeness of the club filter or Fodor lemma require regular uncountable height; they should not be transferred to arbitrary singular or countable limit ordinals without checking their hypotheses.

For an uncountable regular cardinal $\kappa$, choose cofinal sequences $c_{\alpha }:\omega \to \alpha$ for the stationary set of ordinals of cofinality $\omega$. Above any bound, some fixed coordinate exceeds the bound on a stationary subset; Fodor lemma makes that coordinate constant on a stationary subset. Thus the stationary constant fibers, over all coordinates, have unboundedly many values. Regularity makes one coordinate have $\kappa$ such values. Its fibers are disjoint stationary sets; adding all leftover ordinals to one piece partitions $\kappa$.

A family $(A_{\alpha ,n}{)}_{\alpha <{\omega }_1,n<\omega }$ whose row at $\alpha$ partitions $(\alpha ,{\omega }_1)$ and whose fixed-column cells are pairwise disjoint. Choose injections $e_{\beta }:\beta \to \omega$ and put $A_{\alpha ,n}=\{\beta >\alpha :e_{\beta }(\alpha )=n\}$. This gives a matrix used to split stationary sets into many stationary pieces.

There are ${\aleph }_1$ pairwise disjoint stationary subsets of the nonzero countable limit ordinals. In a Ulam matrix on omega-one, each row partitions a stationary tail into countably many cells, so some cell is stationary. One column contains stationary cells in uncountably many rows, and the column cells are disjoint. Unions indexed by different subsets of ${\omega }_1$ then differ on a stationary set.

A sequence $D_{\alpha }\subseteq \alpha$ guesses every $X\subseteq {\omega }_1$ stationarily often: $\{\alpha :D_{\alpha }=X\cap \alpha \}$ is stationary.

For each index, collect the ground subsets which some condition forces to be the corresponding diamond guess. Distinct forced values have incompatible witnessing conditions, so the countable chain condition for forcing makes this a countable family. Every ground subset is guessed by these families on a ground stationary set, by testing each ground club set. The countable-family diamond equivalence then yields diamond in the ground model. A generic guess need not itself be a ground subset; only values forced equal to one are collected.

For a stationary $S\subseteq {\omega }_1$, a sequence $D_{\alpha }\subseteq \alpha$ indexed by $S$ guesses every $X\subseteq {\omega }_1$ on a stationary subset of $S$.

A sequence $D_{\alpha }\subseteq \alpha$, indexed by a stationary subset $S$ of a regular uncountable cardinal $\lambda$, guesses every $X\subseteq \lambda$ stationarily often: $\{\alpha \in S:D_{\alpha }=X\cap \alpha \}$ is stationary. This extends the usual stationary diamond principle at ${\omega }_1$.

At each $\alpha \in S$, a countable family $E_{\alpha }\subseteq \mathcal{P}(\alpha )$ is supplied. Every $X\subseteq \lambda$ has $X\cap \alpha \in E_{\alpha }$ on a stationary subset of $S$. The countable-family diamond equivalence shows that this apparently weaker prediction is equivalent to a single diamond guess at each index.

Enumerate each countable guessing family and use a bijection $\pi :\lambda \times \omega \to \lambda$ with a club set of prefix-closure points. The $n$th candidate sequence decodes the $n$th component of the $n$th family entry. If all candidates fail, choose counterexample sets $X_n$ and club sets witnessing failure, then code the $X_n$ together using $\pi$. A correct family guess on the common closure club set and all failure club sets decodes to one of the forbidden correct guesses. The singleton-family implication supplies the reverse direction.

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.

There are cofinal sequences $A_{\delta }\subseteq \delta$ of order type $\omega$ for the countable limit ordinals, such that every uncountable $X\subseteq {\omega }_1$ contains some $A_{\delta }$.

For a stationary $S\subseteq {\omega }_1$, there are cofinal ladders $A_{\delta }\subset \delta$ of order type $\omega$, indexed by limit ordinals in $S$, such that every uncountable $X\subseteq {\omega }_1$ contains one of these ladders. This predicts a contained ladder, whereas stationary diamond principle predicts a whole initial segment.

A function on a set of ordinals with $f(\alpha )<\alpha$ at every nonzero point of its domain. On a stationary subset of a regular uncountable cardinal, Fodor lemma makes it constant on a stationary subset.

A regressive function on a stationary subset of a regular uncountable cardinal is constant on a stationary subset.

If $S\subseteq \kappa$ is stationary and $f(\alpha )\in A_{\alpha }$ for a kappa-filtration, then $f$ is constant on a stationary subset. Restrict to limit indices and use continuity to find a smaller stage containing each value. Fodor lemma fixes that stage on a stationary subset. Its size is less than $\kappa$, so club filter completeness makes one value fiber stationary.

For a regular infinite $\theta <\kappa$, with $\kappa$ a regular uncountable cardinal number, $S_{\theta }^{\kappa }=\{\delta <\kappa :\mathrm{cf}(\delta )=\theta \}$ is stationary. Build a strictly increasing continuous $\theta$-sequence in any club set, and take its supremum. Closure places it in that club and the cofinality of an increasing ordinal supremum gives cofinality $\theta$. In particular the two disjoint sets $S_{\omega }^{{\omega }_2}$ and $S_{{\omega }_1}^{{\omega }_2}$ show that the club filter on ${\omega }_2$ is not an ultrafilter.