Cardinality, Cardinal Numbers
Cardinal Numbers
A cardinal number, or cardinal, is an ordinal that cannot map 1-1 into any lower ordinal.
These cardinals represent cardinal classes of sets, as described earlier.
Every finite ordinal is a cardinal,
since a finite ordinal cannot embed in a lesser ordinal.
The set ω, which contains all the finite ordinals,
is also a cardinal,
since this infinite set cannot embed in a finite ordinal.
If S is infinite and T is the successor of S, T is not a cardinal.
Build a map from T onto S as follows.
Map S to 0, 0 to 1, 1 to 2, n to n+1, and all infinite ordinals other than S to themselves.
Therefore all infinite cardinals are limit ordinals.