My definition for the ordinal number is "von-Neumann ordinal". I thought this is the only definition for the ordinal, but i found some other definitions in wikipedia.
What is the usual definition of "ordinal number"?
If the usual definition is NOT von-Neumann ordinal, then could you please give me a reference so that i can study the standard definition?
On
Definition: An ordinal number is the order type of a well-ordered set, and the order type of a well-ordering is defined as the set of all well-orderings order-isomorphic to that well-ordering.
For a good introduction, see https://www.encyclopediaofmath.org/index.php/Ordinal_number and references therein.
On
Abstractly, an "ordinal number" is an order type of well-orderings -- that is, intuitively whichever "thing" (order-)isomorphic well-ordered collections have in common is their ordinal number.
The Von Neumann ordinals are a popular way of choosing one concrete set-theoretical object to represent each (set-sized) ordinal number. This identification is so convenient and well-behaved that it is common (and unproblematic) to say that the Von Neumann ordinals are the ordinal numbers.
Context is important in mathematics.
In some context, "ordinal number" represents an equivalence class of a well-ordered set when considering the equivalence relation to be order isomorphism. Sometimes, the context demands this to be a set, so we find a way to "trim the equivalence class" into a set, or even pick one canonical representative -- as in the case of the von Neumann ordinals. And at other times, ordinals can be just any equivalence class of an ordered set, not just a well-ordered one.
Part of growing up, mathematically, is being able to understand and infer the context on your own, from reading a few sentences. So it is often the case where we don't really say what exactly we mean, because if we would say exactly what we mean, no one would understand what we are say because of all the details.
When I say "ordinal" I mean the von Neumann ordinals.