TITLE: A proper formal definition of the Ordinal numbers?
QUESTION [6 upvotes]: 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?

REPLY [4 votes]: 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 $\lambda$.