Stationary set¶

In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets and is analogous to a set of non-zero measure in measure theory.

Version: v1 · 2026-09-28 · History
Domain-specific #: 12259
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Set Theory, Model Theory → Mathematics

Core Idea¶

Stationary set is treated here as the recurring set theory identity summarized by this source-grounded definition: In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets and is analogous to a set of non-zero measure in measure theory. In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets and is analogous to a set of non-zero measure in measure.

Scope of Application¶

Classical notion. This notion should not be confused with the notion of a thin set in number theory.
Generalized notion. This notion is probably due to Magidor, Foreman and Shelah and has also been used prominently by Woodin.
Generalized notion. A set C\subseteq{\mathcal P}(X) is club (closed and unbounded) if and only if there is a function F:[X]^{ such that C={z:F[[z]^{.
Generalized notion. To see the connection with model theory, notice that if M is a structure with universe X in a countable language and F is a Skolem function for M , then a.
Classical notion. If \kappa is a cardinal of uncountable cofinality, S \subseteq \kappa, and S intersects every club set in \kappa, then S is called a stationary set.

Clarity¶

A clear use of Stationary set names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets and is analogous to a set of non-zero measure in measure theory.

Manages Complexity¶

Stationary set compresses multiple set theory details into a stable diagnostic relation. The source shows both the central mechanism—this notion is probably due to Magidor, Foreman and Shelah and has also been used prominently by Woodin.—and the practical consequence—if S is a stationary set and C is a club set, then their intersection S \cap C is also stationary.

Abstract Reasoning¶

Type the carrier. Identify the set theory entities to which the claim applies.
State the relation. Use the source-grounded identity: In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets and is analogous to a set of non-zero measure in measure theory.
Check operation and conditions.

Knowledge Transfer¶

Within the home domain. Knowledge about Stationary set transfers literally when a new case preserves the same carrier type, relation, and recognition test. This notion should not be confused with the notion of a thin set in number theory. This notion is probably due to Magidor, Foreman and Shelah and has also been used prominently by Woodin. Beyond the home domain. No canonical parent is asserted for Stationary set. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Relationships to Other Abstractions¶

Current abstraction Stationary set Domain-specific

Parents (1) — more general patterns this builds on

Stationary set is a kind of Set and Membership Prime

A stationary set is a set with a club-intersection differentia. The live eta-alpha Set endpoint is unrelated.

Hierarchy path (1) — routes to 1 parentless root

Stationary set → Set and Membership

Neighborhood in Abstraction Space¶

Stationary set sits in a moderately populated region (57^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Abstract Algebra & Category Theory (24 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08