Editing Cofinality (section)

==Cofinality of cardinals==

If <math>\kappa</math> is an infinite cardinal number, then <math>\operatorname{cf}(\kappa)</math> is the least cardinal such that there is an [[bounded (set theory)|unbounded]] function from <math>\operatorname{cf}(\kappa)</math> to <math>\kappa;</math> <math>\operatorname{cf}(\kappa)</math> is also the cardinality of the smallest set of strictly smaller cardinals whose sum is <math>\kappa;</math> more precisely
<math display=block>\operatorname{cf}(\kappa) = \min \left\{ |I|\ :\ \kappa = \sum_{i \in I} \lambda_i\ \land \forall i \in I \colon \lambda_i < \kappa\right\}.</math>

That the set above is nonempty comes from the fact that
<math display=block>\kappa = \bigcup_{i \in \kappa} \{i\}</math>
that is, the [[disjoint union]] of <math>\kappa</math> singleton sets. This implies immediately that <math>\operatorname{cf}(\kappa) \leq \kappa.</math> 
The cofinality of any totally ordered set is regular, so <math>\operatorname{cf}(\kappa) = \operatorname{cf}(\operatorname{cf}(\kappa)).</math>

Using [[König's theorem (set theory)|König's theorem]], one can prove <math>\kappa < \kappa^{\operatorname{cf}(\kappa)}</math> and <math>\kappa < \operatorname{cf}\left(2^\kappa\right)</math> for any infinite cardinal <math>\kappa.</math>

The last inequality implies that the cofinality of the cardinality of the continuum must be uncountable. On the other hand,
<math display=block>\aleph_\omega = \bigcup_{n < \omega} \aleph_n,</math>
the ordinal number ω being the first infinite ordinal, so that the cofinality of <math>\aleph_\omega</math> is card(ω) = <math>\aleph_0.</math>  (In particular, <math>\aleph_\omega</math> is singular.) Therefore,
<math display=block>2^{\aleph_0} \neq \aleph_\omega.</math>

(Compare to the [[continuum hypothesis]], which states <math>2^{\aleph_0} = \aleph_1.</math>)

Generalizing this argument, one can prove that for a limit ordinal <math>\delta</math>
<math display=block>\operatorname{cf} (\aleph_\delta) = \operatorname{cf} (\delta).</math>

On the other hand, if the [[axiom of choice]] holds, then for a successor or zero ordinal <math>\delta</math>
<math display=block>\operatorname{cf} (\aleph_\delta) = \aleph_\delta.</math>