\(\omega\)

The ordinal omega, written \(\omega\), is defined as the order type of the natural numbers \(\mathbb N\). As a von Neumann ordinal, it corresponds to the naturals themselves. Note that \(\omega\) is not to be confused with \(\Omega\), a common notation for a much larger ordinal. The existence of \(\omega\) is guaranteed by the axiom of infinity.

Properties

• It is the first infinite ordinal.
• It is the first limit ordinal.
• It is considered by some to be the first admissible ordinal.
• Using the von Neumann cardinal assignment, it is equal to \(\aleph_0\).
• It is the smallest ordinal \(\alpha\) such that \(1+\alpha=\alpha\). Every ordinal larger than it has this same property.
• It is the next ordinal after 0 that isn't a successor ordinal.
• It is additively, multiplicatively, and exponentially principal.