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Maybe I'm just being a bit stupid with where I'm searching here, but I can't actually find a proper definition of these things wherever I look, books or online. The definition I find most is that it's the order type of a well-ordered set, which wouldn't be so bad if the definition of order type seemed to exist anywhere beyond simply saying two sets have the same order type when they're order isomorphic. That's great but it doesn't really say what an order type actually is, I can't help but think there must surely be a better definition for either ordinals or order type?

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Have you seen, "Von Neumann definition of ordinals"? – Rudy the Reindeer Nov 8 '11 at 11:19
Wow, I'm giving up on Wolfram and stuff like that, really should have seen that coming, I feel pretty stupid for that now. God only knows how I missed that section. Thanks mate – AaronM Nov 8 '11 at 11:33
No problem! And don't feel stupid. – Rudy the Reindeer Nov 8 '11 at 11:35
I think that I have defined ordinals on this very website more than once. – Asaf Karagila Nov 8 '11 at 11:45
Isomorphism classes of well-orderings. – Jon Beardsley Nov 8 '11 at 22:56

The naive definition of an ordinal number is that it is an "order type" of a well ordered set, where "order type" is a kind of abstract property so that two well ordered sets have the same order type if and only if they are order isomorphic. The problem is how to represent an "order type" as a particular mathematical object. You can get a similar idea if you think of the "height type" of a person as an abstract property which is shared by all people who are the same height. We can then represent each "height type" by a real number, replacing the abstract object with a mathematical one.

One approach to representing ordinal numbers is to define an ordinal number to be an equivalence class of well ordered sets under the equivalence relation nduced by order isomorphism. This works in some settings, but in ZFC set theory it has the problem that these equivalence classes will not be sets, except for the ordinal 0 (only the empty set can be well-ordered with this order type).

The usual approach in ZFC is to pick a single representative from each equivalence class and declare these representatives to be "ordinal numbers". In particular, the standard definition by von Neumann says that an ordinal number is defined to be a transitive set which is well-ordered by the set membership relation. It can be proved in ZFC that every well-ordered set is order isomorphic to exactly one of these, so they serve as an adequate class of representatives.

The von Neumann definition has several elegant features which make it particularly nice to work with. One is that the well ordering on an ordinal number is just the set membership relation. Another is that each ordinal is, in fact, the set of all smaller ordinals, so that $\lambda < \kappa$ and $\lambda \in \kappa$ mean the same thing for ordinals $\kappa$ and $\lambda$.

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First we show that given $(a,R), (b,S)$ two well ordered sets then either $a$ isomorphic to an initial segment of $b$ or vice versa. We can further show that the embedding is unique. This is done by sending the minimal element of $a$ to the minimal element of $b$ and so on. We will either exhaust $a$, or have $b$ embed into it (not necessarily into a proper segment, as this may end up an isomorphism).

If the sets are isomorphic then this isomorphism is unique. This means that there is a linear ordering in the order types of well orders. In contrast consider $\mathbb Q$ and $\mathbb Q\setminus\{q\mid 0<q<1\}$ can be embedded into one another, but are not isomorphic.

Ordinals are representatives of the equivalence classes. The von Neumann ordinals are sets well ordered by $\in$ which are also transitive, i.e. every element is a subset.

From the bottom up we can define:

  • $0 =\varnothing$,
  • $\alpha+1=\alpha\cup\{\alpha\}$, and
  • if we have defined $\beta$ for all $\beta<\alpha$ then $\alpha=\bigcup_{\beta<\alpha}\beta$.

Exercise: Show the above definition gives transitive sets which are well ordered by $\in$, and if a transitive set is well ordered by $\in$ then it has the above form.

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How do we know if we've defined β for all β<α? – Chan Bae May 9 at 9:29
That is the induction hypothesis. You assume that you have defined $\beta$ for all smaller ordinals, and from this assumption you define $\alpha$. – Asaf Karagila May 9 at 9:30

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