\(\omega_1\) (also commonly denoted \(\Omega\)), called omega-one or the first uncountable ordinal, is the smallest uncountable ordinal. It has several equivalent definitions: * Call an ordinal \(\alpha\) countable if there exists a surjection from the natural numbers onto \(\alpha\). \(\omega_1\) is the smallest ordinal that is not countable. * \(\omega_1\) is the second smallest infinite ordinal whose cofinality is equal to itself. * \(\omega_1\) is the supremum of all ordinals that can be mapped one-to-one onto the natural numbers. * \(\omega_1\) is the set of all countable ordinals. As every ordinal, it is the set of all ordinals less than it. * \(\omega_1\) is the smallest ordinal with a cardinality greater than \(\omega\): \(|\omega_1| = \aleph_1 > \aleph_0 = |\omega|\).