First uncountable ordinal

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


 * The elements of \(\omega_1\) cannot be mapped one-to-one onto the natural numbers.
 * If we accept the, \(\omega_1\) can be mapped one-to-one onto the real numbers.
 * \(\omega_1\) is the set of all countable ordinals.
 * \(\omega_1\) is the smallest ordinal with a cardinality greater than \(\omega\): \(|\omega_1| = \aleph_1 > \aleph_0 = |\omega|\).

It is sometimes denoted \(\Omega\), which for brevity's sake is common in ordinal collapsing functions.

\(\omega_1\) has no fundamental sequence and thus marks the limit of the fast-growing hierarchy and its relatives.