• The cofinality of a partially ordered set with greatest element is 1 as the set consisting only of the greatest element is cofinal (and must be contained in every other cofinal subset).
  • In particular, the cofinality of any nonzero finite ordinal, or indeed any finite directed set, is 1, since such sets have a greatest element.
• Every cofinal subset of a partially ordered set must contain all maximal elements of that set. Thus the cofinality of a finite partially ordered set is equal to the number of its maximal elements.
  • In particular, let ${\displaystyle A}$  be a set of size ${\displaystyle n,}$  and consider the set of subsets of ${\displaystyle A}$  containing no more than ${\displaystyle m}$  elements. This is partially ordered under inclusion and the subsets with ${\displaystyle m}$  elements are maximal. Thus the cofinality of this poset is ${\displaystyle n}$  choose ${\displaystyle m.}$ 
• A subset of the natural numbers ${\displaystyle \mathbb {N} }$  is cofinal in ${\displaystyle \mathbb {N} }$  if and only if it is infinite, and therefore the cofinality of ${\displaystyle \aleph _{0}}$  is ${\displaystyle \aleph _{0}.}$  Thus ${\displaystyle \aleph _{0}}$  is a regular cardinal.
• The cofinality of the real numbers with their usual ordering is ${\displaystyle \aleph _{0},}$  since ${\displaystyle \mathbb {N} }$  is cofinal in ${\displaystyle \mathbb {R} .}$  The usual ordering of ${\displaystyle \mathbb {R} }$  is not order isomorphic to ${\displaystyle c,}$  the cardinality of the real numbers, which has cofinality strictly greater than ${\displaystyle \aleph _{0}.}$  This demonstrates that the cofinality depends on the order; different orders on the same set may have different cofinality.

If ${\displaystyle A}$  admits a totally ordered cofinal subset, then we can find a subset ${\displaystyle B}$  that is well-ordered and cofinal in ${\displaystyle A.}$  Any subset of ${\displaystyle B}$  is also well-ordered. Two cofinal subsets of ${\displaystyle B}$  with minimal cardinality (that is, their cardinality is the cofinality of ${\displaystyle B}$ ) need not be order isomorphic (for example if ${\displaystyle B=\omega +\omega ,}$  then both ${\displaystyle \omega +\omega }$  and ${\displaystyle \{\omega +n:n<\omega \}}$  viewed as subsets of ${\displaystyle B}$  have the countable cardinality of the cofinality of ${\displaystyle B}$  but are not order isomorphic.) But cofinal subsets of ${\displaystyle B}$  with minimal order type will be order isomorphic.

The cofinality of an ordinal ${\displaystyle \alpha }$  is the smallest ordinal ${\displaystyle \delta }$  that is the order type of a cofinal subset of ${\displaystyle \alpha .}$  The cofinality of a set of ordinals or any other well-ordered set is the cofinality of the order type of that set.

Thus for a limit ordinal ${\displaystyle \alpha ,}$  there exists a ${\displaystyle \delta }$ -indexed strictly increasing sequence with limit ${\displaystyle \alpha .}$  For example, the cofinality of ${\displaystyle \omega ^{2}}$  is ${\displaystyle \omega ,}$  because the sequence ${\displaystyle \omega \cdot m}$  (where ${\displaystyle m}$  ranges over the natural numbers) tends to ${\displaystyle \omega ^{2};}$  but, more generally, any countable limit ordinal has cofinality ${\displaystyle \omega .}$  An uncountable limit ordinal may have either cofinality ${\displaystyle \omega }$  as does ${\displaystyle \omega _{\omega }}$  or an uncountable cofinality.

The cofinality of 0 is 0. The cofinality of any successor ordinal is 1. The cofinality of any nonzero limit ordinal is an infinite regular cardinal.

A regular ordinal is an ordinal that is equal to its cofinality. A singular ordinal is any ordinal that is not regular.

Every regular ordinal is the initial ordinal of a cardinal. Any limit of regular ordinals is a limit of initial ordinals and thus is also initial but need not be regular. Assuming the axiom of choice, ${\displaystyle \omega _{\alpha +1}}$  is regular for each ${\displaystyle \alpha .}$  In this case, the ordinals ${\displaystyle 0,1,\omega ,\omega _{1},}$  and ${\displaystyle \omega _{2}}$  are regular, whereas ${\displaystyle 2,3,\omega _{\omega },}$  and ${\displaystyle \omega _{\omega \cdot 2}}$  are initial ordinals that are not regular.

The cofinality of any ordinal ${\displaystyle \alpha }$  is a regular ordinal, that is, the cofinality of the cofinality of ${\displaystyle \alpha }$  is the same as the cofinality of ${\displaystyle \alpha .}$  So the cofinality operation is idempotent.

If ${\displaystyle \kappa }$  is an infinite cardinal number, then ${\displaystyle \operatorname {cf} (\kappa )}$  is the least cardinal such that there is an unbounded function from ${\displaystyle \operatorname {cf} (\kappa )}$  to ${\displaystyle \kappa ;}$  ${\displaystyle \operatorname {cf} (\kappa )}$  is also the cardinality of the smallest set of strictly smaller cardinals whose sum is ${\displaystyle \kappa ;}$  more precisely

That the set above is nonempty comes from the fact that

Using König's theorem, one can prove ${\displaystyle \kappa <\kappa ^{\operatorname {cf} (\kappa )}}$  and ${\displaystyle \kappa <\operatorname {cf} \left(2^{\kappa }\right)}$  for any infinite cardinal ${\displaystyle \kappa .}$ 

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 ${\displaystyle \aleph _{\omega }}$  is card(ω) = ${\displaystyle \aleph _{0}.}$  (In particular, ${\displaystyle \aleph _{\omega }}$  is singular.) Therefore,

(Compare to the continuum hypothesis, which states ${\displaystyle 2^{\aleph _{0}}=\aleph _{1}.}$ )

Generalizing this argument, one can prove that for a limit ordinal ${\displaystyle \delta }$ 

On the other hand, if the axiom of choice holds, then for a successor or zero ordinal ${\displaystyle \delta }$ 

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