# Difference between revisions of "Omega"

(Created page with "{{DISPLAYTITLE: $\omega$}} The smallest infinite ordinal is denoted $\omega$, having the order type of the natural numbers. As a von Neumann ordinal, $\omega$ is i...") |
|||

Line 1: | Line 1: | ||

− | {{DISPLAYTITLE: $\omega$}} | + | {{DISPLAYTITLE: Omega, $\omega$}} |

− | The smallest infinite ordinal | + | The smallest infinite ordinal, often denoted $\omega$ (omega), has the order type of the natural numbers. As a [[ordinal | von Neumann ordinal]], $\omega$ is in fact equal to the set of natural numbers. Since $\omega$ is infinite, it is not equinumerous with any smaller ordinal, and so it is an [[initial ordinal]], that is, a [[cardinal]]. When considered as a cardinal, the ordinal $\omega$ is denoted $\aleph_0$. So while these two notations are intensionally different---we use the term $\omega$ when using this number as an ordinal and $\aleph_0$ when using it as a cardinal---nevertheless in the contemporary treatment of cardinals in ZFC as initial ordinals, they are extensionally the same and refer to the same object. |

## Revision as of 16:01, 30 December 2011

The smallest infinite ordinal, often denoted $\omega$ (omega), has the order type of the natural numbers. As a von Neumann ordinal, $\omega$ is in fact equal to the set of natural numbers. Since $\omega$ is infinite, it is not equinumerous with any smaller ordinal, and so it is an initial ordinal, that is, a cardinal. When considered as a cardinal, the ordinal $\omega$ is denoted $\aleph_0$. So while these two notations are intensionally different---we use the term $\omega$ when using this number as an ordinal and $\aleph_0$ when using it as a cardinal---nevertheless in the contemporary treatment of cardinals in ZFC as initial ordinals, they are extensionally the same and refer to the same object.

## Countable sets

A set is *countable* if it can be put into bijective correspondence with a subset of $\omega$. This includes all finite sets, and a set is *countably infinite* if it is countable and also infinite.
Some famous examples of countable sets include:

- The natural numbers $\mathbb{N}=\{0,1,2,\ldots\}$.
- The integers $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$
- The rational numbers $\mathbb{Q}=\{\frac{p}{q}\mid p,q\in\mathbb{Z}, q\neq 0\}$
- The real algebraic numbers $\mathbb{A}$, consisting of all zeros of nontrivial polynomials over $\mathbb{Q}$

The union of countably many countable sets remains countable, although in the general case this fact requires the axiom of choice.

A set is *uncountable* if it is not countable.