\(\omega_1\) (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 \(\omega\): \(|\omega_1| = \aleph_1 > \aleph_0 = |\omega|\).
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. Any fundamental sequence of countable ordinals is limited by a countable ordinal. Note that there are ordinals beyond \(\omega_1\) that do have fundamental sequences; e.g. \(\omega_1 \times \omega\) and \(\omega_\omega\).
The continuum hypothesis 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 was shown to be independent of ZFC, which means it cannot be proven nor disproven, and a satisfying resolution to the hypothesis has yet to be found.