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The set of numbers that can represent the number of elements in a set.

Finite cardinal numbers:

0 = The number of elements in the empty set: {}

1 = For example, the number of elements in the set of subsets of the empty set: { {} }

2 = Examples: The number of elements in the set of subsets of a one-element set. Or, the number of elements in this set: { {}, { {} } } = the set which contains the empty set along with the set containing the empty set.

etc....

Infinite cardinal numbers:

Aleph-0 = "Countable" = the number of elements in the set of finite cardinal numbers = the number of elements in the set of natural numbers { 1, 2, 3, 4, .... } = the number of integers = the number of rational numbers = the number of algebraic numbers = the number of lattice points on a grid .....

Aleph-1 = The smallest cardinal number greater than Aleph-0.

The number of subsets of a countable set = the number of real numbers = the number of transcendental numbers = the number of irrational numbers = the number of points in a line or plane...... The continuum problem of whether this is in fact equal to Aleph-1 is unprovable using the usual ZFC axioms of set theory. Either the continuum hypothesis (the statement that they ARE equal) or its negation can therefore be used as additional axioms.

There are infinite infinite cardinal numbers since a greater infinity can always be generated by taking the cardinal number of the set of subsets (or power set) of a set with a lesser infinity of elements.