Difference between revisions of "Zero"

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(Definition)
(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 $0$, and, in the set of ordinals, $0$ is the empty set $\varnothing$.
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Zero, or $0$, is the smallest [[ordinal]] (and [[cardinal]]) number.  It is usually represented by the [[empty set]], $\varnothing$ (or $\{\}$).
  
Given an ordinal $x$
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Given an ordinal (or cardinal) $x$:
  
 
* $x + 0 = 0 + x = x$.
 
* $x + 0 = 0 + x = x$.
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* $x \cdot 0 = 0 \cdot x = 0$
 
* $x \cdot 0 = 0 \cdot x = 0$
  
Zero is the only ordinal which is neither a [[limit ordinal]] nor a [[successor ordinal|successor]] of any other ordinal.
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* $x^0 = 1$
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* $0^x = 0$ (if $x > 0$)
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Most set theorists classify zero as the only ordinal which is neither a [[limit ordinal]] nor the [[successor ordinal|successor]] of any ordinal.
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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).