Fodor's lemma

Fodor's lemma states that every regressive function on a stationary subset of a regular uncountable cardinal is constant on a stationary subset.

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Stationary set: A subset of a regular uncountable cardinal that intersects every closed unbounded subset of that cardinal. The lemma begins with a stationary domain and guarantees a stationary subset among its fibers.
Pressing down lemma: Another name for Fodor's lemma, emphasizing that regressive values are bounded on a stationary subset. The name captures the proof's central move: press values below the ordinals in their domain.
Stationary set reflection: The phenomenon that a stationary set remains stationary when restricted to a smaller ordinal of suitable cofinality. Pressing down often helps identify stationary subsets that reflect at a common level.
Ulam's theorem: A theorem that partitions certain stationary sets into many disjoint stationary subsets. It shows that stationary sets can split extensively, while Fodor constrains regressive fiber partitions.
Closed unbounded set: A subset of an ordinal that is unbounded and closed under limits below that ordinal. Stationarity is defined by meeting every such set, and the lemma's proof uses diagonal intersections of clubs.
Diagonal intersection: For a sequence of sets indexed by an ordinal, the set of points belonging to every earlier indexed set. Diagonal intersections of clubs underpin the contradiction used in a standard proof.
Normal ideal: An ideal closed under diagonal unions, often defined on a regular uncountable cardinal. Fodor's lemma is closely tied to normality and yields key properties of the nonstationary ideal.
Nonstationary set: A subset of a regular uncountable cardinal that misses some closed unbounded subset. Without stationarity, a regressive function need not have a stationary constant fiber.
Regressive function: A function on nonzero ordinals that maps each ordinal to a smaller ordinal. This is the map whose values Fodor's lemma makes constant on a stationary subset.
Club filter: The filter on a regular uncountable cardinal generated by its closed unbounded subsets. Stationary sets are precisely those that remain outside the dual ideal of this filter.