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ordinal number (Definition)

An ordinal number is a well ordered set $ S$ such that, for every $ x \in S$,

$\displaystyle x = \{z \in S \mid z < x\} $
(where $ <$ is the ordering relation on $ S$).

There is a theory of ordinal arithmetic which allows construction of various ordinals; for example, all the numbers 0, $ 1$, $ 2$, ...have natural interpretations as ordinals, as does the set of natural numbers itself (often denoted $ \omega$ in this context).



"ordinal number" is owned by djao. [ full author list (2) ]
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See Also: von Neumann ordinal

Other names:  ordinal
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Cross-references: natural numbers, interpretations, ordinal arithmetic, theory, ordering relation, well ordered set
There are 38 references to this entry.

This is version 3 of ordinal number, born on 2002-01-05, modified 2004-03-17.
Object id is 1300, canonical name is OrdinalNumber.
Accessed 19853 times total.

Classification:
AMS MSC: 03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)
Pending Errata and Addenda
None.
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problem with ``ordinal'' in html mode by Wkbj79 on 2006-09-02 01:36:00
I preferred not to file this as a correction since the problem that I see does not seem to be a problem with the actual content of this entry.

When I view the entry ``ordinal'' in html mode, the first line reads:

Proposition: The sets of real numbers $[0,1]$, $[0,1)$, $(0,1]$, and $(0,1)$ all have the same cardinality.

...and the rest of the article is a proof of this proposition. Despite reloading the page, clearing my cache, etc., this is all that I ever see in html mode.

One the other hand, when I view the entry ``ordinal'' in text images mode, I do see an actual definition of ``ordinal''.

Is there a way to fix this discrepancy?
[ reply | up ]
a post for entry "ordinal number" by mathforever on 2004-11-19 09:16:55
I don't really understand the definition:

x = {z \in S | z < x}

as it is written the set on right side DOESN'T contain x at all, so how it could be that it is equal to x?
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