# Difference between revisions of "Zero"

From Cantor's Attic

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Trixie Wolf (Talk | contribs) (Added some detail to definition, and removed a couple of mistaken statements (the collection of all ordinals is not a set, for one).) |
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== Definition == | == Definition == | ||

− | Zero is the smallest [[ordinal]] number. It is represented by | + | Zero, or $0$, is the smallest [[ordinal]] (and [[cardinal]]) number. It is usually represented by the [[empty set]], $\varnothing$ (or $\{\}$). |

− | Given an ordinal $x$ | + | Given an ordinal (or cardinal) $x$: |

* $x + 0 = 0 + x = x$. | * $x + 0 = 0 + x = x$. | ||

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* $x \cdot 0 = 0 \cdot x = 0$ | * $x \cdot 0 = 0 \cdot x = 0$ | ||

− | + | * $x^0 = 1$ | |

+ | |||

+ | * $0^x = 0$ (if $x > 0$) | ||

+ | |||

+ | Most set theorists classify zero as the only ordinal which is neither a [[limit ordinal]] nor the [[successor ordinal|successor]] of any ordinal. | ||

+ | |||

+ | Some set theorists (e.g. Thomas Jech) instead classify zero as a [[limit ordinal]], because like other limit ordinals, it is the limit of all ordinals less than it (of which there are none). |

## Revision as of 18:18, 10 January 2017

## Definition

Zero, or $0$, is the smallest ordinal (and cardinal) number. It is usually represented by the empty set, $\varnothing$ (or $\{\}$).

Given an ordinal (or cardinal) $x$:

- $x + 0 = 0 + x = x$.

- $x \cdot 0 = 0 \cdot x = 0$

- $x^0 = 1$

- $0^x = 0$ (if $x > 0$)

Most set theorists classify zero as the only ordinal which is neither a limit ordinal nor the successor of any ordinal.

Some set theorists (e.g. Thomas Jech) instead classify zero as a limit ordinal, because like other limit ordinals, it is the limit of all ordinals less than it (of which there are none).