Cofinality - Cofinality of Cardinals

Cofinality of Cardinals

If κ is an infinite cardinal number, then cf(κ) is the least cardinal such that there is an unbounded function from it to κ; and cf(κ) = the cardinality of the smallest collection of sets of strictly smaller cardinals such that their sum is κ; more precisely

That the set above is nonempty comes from the fact that

i.e. the disjoint union of κ singleton sets. This implies immediately that cf(κ) ≤ κ. The cofinality of any totally ordered set is regular, so one has cf(κ) = cf(cf(κ)).

Using König's theorem, one can prove κ < κcf(κ) and κ < cf(2κ) for any infinite cardinal κ.

The last inequality implies that the cofinality of the cardinality of the continuum must be uncountable. On the other hand,

.

the ordinal number ω being the first infinite ordinal, so that the cofinality of is card(ω) = . (In particular, is singular.) Therefore,

(Compare to the continuum hypothesis, which states .)

Generalizing this argument, one can prove that for a limit ordinal δ

.