Questions: Cofinality and Regular Cardinals

Question 1

What is the cofinality of ℵ_ω?

• A. cf(ℵ_ω) = ℵ_ω, because ℵ_ω is a well-defined limit cardinal and therefore regular
• B. cf(ℵ_ω) = ω, because ℵ_ω = sup{ℵ₀, ℵ₁, ℵ₂, ...} and this sequence has length ω
• C. cf(ℵ_ω) = ℵ₁, because ℵ_ω is an uncountable cardinal
• D. cf(ℵ_ω) = ℵ_ω − 1, by definition of limit ordinals

Answer: (B)

ℵ_ω is the supremum of the sequence ℵ₀, ℵ₁, ℵ₂, ... — a sequence of length ω. No finite subsequence (or shorter sequence) has supremum ℵ_ω, so the minimum cofinal sequence length is ω, giving cf(ℵ_ω) = ω. Since ω < ℵ_ω, this means ℵ_ω is singular — it can be 'approached from below' by a mere countable sequence, despite being uncountable itself. This is the key contrast with ℵ₁, which is a successor cardinal: no countable sequence of countable ordinals can have supremum ℵ₁.

Question 2

A set theorist proposes that the continuum 2^ℵ₀ might equal ℵ_ω. König's theorem rules this out. The correct argument is:

• A. ℵ_ω is a singular cardinal, and the continuum must be a regular cardinal
• B. cf(ℵ_ω) = ω = ℵ₀, but König's theorem requires cf(2^ℵ₀) > ℵ₀, which is a contradiction
• C. ℵ_ω is too small to be the continuum — by Cantor's theorem, 2^ℵ₀ > ℵ_ω
• D. König's theorem states that 2^ℵ₀ > ℵ_ω, which directly excludes this value

Answer: (B)

König's theorem states cf(2^κ) > κ for all infinite κ. Taking κ = ℵ₀: cf(2^ℵ₀) > ℵ₀. If 2^ℵ₀ = ℵ_ω, then cf(2^ℵ₀) = cf(ℵ_ω) = ω = ℵ₀. But cf(2^ℵ₀) > ℵ₀ requires cf(2^ℵ₀) ≥ ℵ₁ > ω. Contradiction. This argument holds regardless of additional axioms — it is an unconditional constraint from König's theorem alone, not contingent on the Continuum Hypothesis or forcing.

Question 3

Nearly every infinite cardinal is regular, since regularity is a property shared by most alephs.

• T. True
• F. False

Answer: False

Regularity fails for many limit cardinals. A cardinal κ is regular if cf(κ) = κ — it cannot be approached by a shorter sequence. Every successor cardinal (ℵ₁, ℵ₂, etc.) is regular. But limit cardinals like ℵ_ω, ℵ_{ω₁}, and many others are singular: cf(ℵ_ω) = ω < ℵ_ω. The regular/singular distinction is one of the most important in cardinal arithmetic, and singular cardinals are not pathological edge cases — they are the norm among uncountable limit cardinals.

Question 4

König's theorem places an unconditional constraint on the continuum: regardless of what additional set-theoretic axioms are assumed, 2^ℵ₀ cannot equal ℵ_ω.

• T. True
• F. False

Answer: True

König's theorem (cf(2^κ) > κ) is provable in ZFC without additional axioms. Its conclusion that 2^ℵ₀ ≠ ℵ_ω does not depend on the Continuum Hypothesis, forcing, or large cardinal axioms — it holds in every model of ZFC. This makes it one of the rare unconditional constraints on the continuum function, ruling out specific values of 2^ℵ₀ (any ℵ_α with cf(ℵ_α) ≤ ℵ₀) without needing to settle the question of what 2^ℵ₀ actually equals.

Question 5

Explain the concept of cofinality by contrasting cf(ω) = ω with cf(ℵ_ω) = ω. What does it mean that these two very different cardinals have the same cofinality?

Model answer: Cofinality of a limit ordinal is the length of the shortest cofinal sequence — one whose supremum equals the ordinal. For ω, the sequence 0, 1, 2, ... is cofinal with length ω, and no finite sequence suffices, so cf(ω) = ω. For ℵ_ω, the sequence ℵ₀, ℵ₁, ℵ₂, ... is cofinal with length ω, and again no shorter sequence suffices. Having the same cofinality ω means both can be 'snuck up on' by a countable sequence. For ω, this is expected — it is a countable ordinal. For ℵ_ω, this is remarkable: despite being a vastly larger (uncountable) cardinal, it is just as 'approachable from below' as ω, which is exactly what makes it singular.

The cofinality tells you how 'inaccessible' a limit cardinal is from below. Regular cardinals (like ℵ₁) require sequences of their own length to approach them — you can't sneak up on ℵ₁ with countably many steps. Singular cardinals like ℵ_ω can be approached much more efficiently than their size suggests. This distinction has deep consequences in combinatorics, cardinal arithmetic, and the independence results of set theory.