While reading up on ordinals I came across the following lines on Wikipedia:
Essentially, an ordinal is intended to be defined as an isomorphism class of well-ordered sets: that is, as an equivalence class for the equivalence relation of "being order-isomorphic". There is a technical difficulty involved, however, in the fact that the equivalence class is too large to be a set in the usual Zermelo–Fraenkel (ZF) formalization of set theory. But this is not a serious difficulty.
The original definition of ordinal numbers, found for example in the Principia Mathematica, defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned in ZF and related systems of axiomatic set theory because these equivalence classes are too large to form a set.
These lines had no citation so I was unable to follow them back to a source, nor could I find any information regarding sets 'too large' for ZF. So what did the author mean by this?
I have not formally studied set theory so have minimal knowledge of ZF(C) and ordinals, so I apologise if this question makes no sense/has been answered already, I couldn't find any information on it.