Omega: Difference between revisions
Jump to navigation
Jump to search
Content added Content deleted
OfficialURL (talk | contribs) (Added properties) |
OfficialURL (talk | contribs) No edit summary |
||
Line 10: | Line 10: | ||
* It is the smallest ordinal \(\alpha\) such that \(1+\alpha=\alpha\). Every ordinal larger than it has this same property. |
* 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 the next ordinal after [[0]] that isn't a [[successor ordinal]]. |
||
* It is additively, multiplicatively, and exponentially [[principal]]. |
Revision as of 02:11, 6 September 2022
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.
Properties
- It is the first infinite ordinal.
- It is the first limit ordinal.
- It is 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.