Cofinality and cardinal arithmetic

For an ordinal α, a subset is cofinal in the order sense when every element of α is at most some member of that subset. Cofinality is the least ordinal length of an increasing cofinal map into α. Set cf(0)=0; a nonzero successor has cofinality one because its greatest element is cofinal. For a nonzero limit ordinal, cofinality can equivalently be computed by the least size of a subset with supremum α. Using the supremum formulation indiscriminately for successors would give the wrong convention.

In ZFC, the cofinality of a nonzero limit ordinal is an infinite regular cardinal. If a supposedly shortest cofinal index itself had a shorter cofinal sequence, composition would shorten the original map. An infinite cardinal κ is regular when cf(κ)=κ and singular otherwise. Regularity says that fewer than κ many ordinals below κ have a supremum still below κ. It does not say every increasing sequence stabilizes, and it does not forbid cofinal sequences of length exactly κ.

For infinite cardinals κ and λ in ZFC, their sum and product equal their maximum, provided both are nonzero in the product statement. These are theorems about well-ordered cardinalities, not identities for ordinal concatenation. Exponentiation counts functions: κ to the power λ is the number of maps from λ into κ. In particular 2 to the power κ counts subsets of κ via characteristic functions. Cantor's diagonal argument shows that no set surjects onto its power set, so this exponential is strictly larger than κ.

Monotonicity gives one constraint on exponentiation, but it leaves substantial freedom. König's theorem supplies another: if κ_i<λ_i for every member of a set-indexed family of cardinals, then the cardinal sum of the κ_i is strictly below the product of the λ_i. A diagonal proof defeats any proposed enumeration by choosing, in each coordinate, a value missed by the corresponding smaller block. Its familiar consequence λ to the power cf(λ) is larger than λ for every infinite cardinal λ is particularly useful at singular limits.

Let μ=2 to the power κ, with κ infinite. If cf(μ) were at most κ, exponentiation rules would give μ to the power cf(μ) equal to μ: write μ as 2 to the power κ and use κ multiplied by a nonzero cardinal at most κ equals κ. König's consequence says that same power is strictly larger than μ. Therefore cf(2 to the power κ)>κ. The argument imposes a restriction without deciding a particular value for the exponential.

For the continuum this excludes aleph-omega, whose cofinality is ω, even though aleph-omega is larger than every finite-index aleph. It does not prove the Continuum Hypothesis: many uncountable cardinals satisfy the cofinality restriction. Nor can the calculation be transferred unchanged to ZF without Choice. A proposed independence result must respect these compulsory inequalities before one asks which remaining values can occur in a model. Later forcing constructions will need both a count of new subsets and preservation of the cardinals used in that count.