Uncountable Set - Characterizations

Characterizations

There are many equivalent characterizations of uncountability. A set X is uncountable if and only if any of the following conditions hold:

• There is no injective function from X to the set of natural numbers.
• X is nonempty and every ω-sequence of elements of X fails to include at least one element of X. That is, X is nonempty and there is no surjective function from the natural numbers to X.
• The cardinality of X is neither finite nor equal to (aleph-null, the cardinality of the natural numbers).
• The set X has cardinality strictly greater than .

The first three of these characterizations can be proven equivalent in Zermelo–Fraenkel set theory without the axiom of choice, but the equivalence of the third and fourth cannot be proved without additional choice principles.