Questions: Stationary Sets and Club Filters

Question 1

A set S ⊆ ω₁ contains only successor ordinals and is unbounded in ω₁. Is S stationary in ω₁?

• A. Yes — any unbounded subset of ω₁ is automatically stationary
• B. No — the set of all limit ordinals below ω₁ is a club, and S (containing only successors) has empty intersection with it
• C. Yes — S is dense enough to intersect every club because it is unbounded in ω₁
• D. It depends on whether S has cardinality ω₁

Answer: (B)

The set of all limit ordinals below ω₁ is a club: it is clearly unbounded (limit ordinals occur cofinally) and closed (the limit of a sequence of limit ordinals is itself a limit ordinal). Since S contains only successor ordinals, S ∩ {limit ordinals} = ∅, so S fails to intersect this club. Therefore S is not stationary, despite being unbounded. This shows the key distinction: unbounded means you can find elements arbitrarily high; stationary means you must hit every club, which is a much stronger requirement.

Question 2

A function f: S → ω₁ is defined on a stationary set S ⊆ ω₁, with f(α) < α for every α ∈ S. Fodor's lemma concludes:

• A. f is eventually constant on all of S — there exists γ such that f(α) = γ for all sufficiently large α ∈ S
• B. f is constant on a stationary subset of S — there exists γ < ω₁ such that {α ∈ S : f(α) = γ} is stationary
• C. f is bounded — there exists β < ω₁ such that f(α) < β for all α ∈ S
• D. The range of f contains a club subset of ω₁

Answer: (B)

Fodor's lemma (the pressing-down lemma) says: any regressive function on a stationary set is constant on a stationary subset. The full set S need not be sent to a single value — only a stationary portion of it. Option (a) would require constancy on a tail, which is stronger than what the lemma gives. Option (c), boundedness, follows from a simpler argument and does not require stationarity. The power of Fodor's lemma is that it forces a stationary concentration at a single value, which is what makes it useful in combinatorial arguments.

Question 3

A set that is stationary in κ remains stationary in any larger regular cardinal λ > κ.

• T. True
• F. False

Answer: False

Stationarity is κ-dependent. A set S ⊆ κ is stationary 'in κ' relative to the clubs of κ. The clubs of λ are subsets of λ, and a club in λ restricted to κ need not be a club in κ, and vice versa. A set can intersect every club in κ while missing some club in λ entirely. This is why one must always specify the ambient cardinal when speaking of stationarity: 'S is stationary in κ' is a well-defined statement, but stationarity does not automatically transfer upward.

Question 4

If S is a stationary subset of κ and C is any club in κ, then S ∩ C is non-empty.

• T. True
• F. False

Answer: True

This is the definition of stationarity: S ⊆ κ is stationary if and only if S ∩ C ≠ ∅ for every club C in κ. Equivalently, S cannot be 'avoided' by any club — it is spread throughout κ in a way that no closed unbounded set can dodge it. Non-stationary sets, by contrast, are contained in the complement of some club, meaning some club entirely avoids them.

Question 5

State Fodor's lemma (the pressing-down lemma) and explain why it is named a 'pressing-down' lemma.

Model answer: Fodor's lemma: if κ is a regular uncountable cardinal, S ⊆ κ is stationary, and f: S → κ is a regressive function (f(α) < α for all α ∈ S with α > 0), then there exists a value γ < κ such that f⁻¹(γ) = {α ∈ S : f(α) = γ} is stationary in κ. The name 'pressing-down' comes from the image of mapping each ordinal α in S to some strictly smaller value f(α) < α — pressing the set downward. The lemma says that this downward pressure must concentrate a stationary portion of S on a single value. It is a pigeonhole principle for the ordinal hierarchy: you cannot disperse a stationary set over many distinct small values without some value receiving stationary mass.

Fodor's lemma is one of the central tools in infinitary combinatorics. It is used to prove properties of stationary sets, to establish regularity of large cardinals, and in forcing arguments. The key application pattern: define a regressive function encoding some combinatorial property, invoke Fodor to get a stationary homogeneous fiber, and use that fiber to build the desired object.