Let be a poset. A subset is said to be cofinal in if for every there is a such that . A function is said to be cofinal if is cofinal in . The least cardinality of a cofinal set of is called the cofinality of . Equivalently, the cofinality of is the least http://planetmath.org/node/2787ordinal such that there is a cofinal function . The cofinality of is written , or .
Cofinality of totally ordered sets
For any ordinal we must have , because the identity map on is cofinal. In particular, this is true for cardinals, so any cardinal either satisfies , in which case it is said to be regular, or it satisfies , in which case it is said to be singular.
The cofinality of any totally ordered set is necessarily a regular cardinal.
Cofinality of cardinals
and are regular cardinals. All other finite cardinals have cofinality and are therefore singular.
It is easy to see that , so is regular.
The smallest infinite singular cardinal is . In fact, the function given by is cofinal, so . More generally, for any nonzero limit ordinal , the function given by is cofinal, and this can be used to show that .
Let be an infinite cardinal. It can be shown that is the least cardinal such that is the sum of cardinals each of which is less than . This fact together with König’s theorem tells us that . Replacing by in this inequality we can further deduce that . In particular, , from which it follows that (this being the smallest uncountable aleph which is provably not the cardinality of the continuum).
|Date of creation||2013-03-22 12:23:55|
|Last modified on||2013-03-22 12:23:55|
|Last modified by||yark (2760)|