First uncountable ordinal

\(\omega_1\) (called omega-one or the first uncountable ordinal) is the smallest uncountable ordinal. It has several equivalent definitions:


 * It is the smallest ordinal that, treated as a set, cannot be mapped one-to-one into the natural numbers.
 * \(\omega_1\) is the set of all countable ordinals.
 * \(\omega_1\) is the smallest ordinal with a cardinality greater than \(\omega\): \(|\omega_1| = \aleph_1 > \aleph_0 = |\omega|\).
 * \(\omega_1 = \Omega\)

The first uncountable ordinal is used in ordinal collapsing functions because 1) it is by default larger than any ordinal constructible in these notations, 2) we can conveniently use the word "countable." In such contexts it is usually denoted with a capital omega \(\Omega\), as in \(\vartheta(\Omega^\Omega)\).

\(\omega_1\) has no fundamental sequence and thus marks the limit of the fast-growing hierarchy and its relatives. This is because every fundamental sequence of countable ordinals is still countable ordinal.

The states that \(\omega_1\) has the same cardinality as the real numbers; that is, the countable ordinals can be mapped one-to-one onto the real numbers. The continuum hypothesis is independent of ZFC.