cardinals and ordinals, problem with proper class A while ago I wanted to educate myself about cardinals and ordinals and stubled across this pdf (enjoyed it a great deal!). Visit www.math.ksu.edu/~nagy/real-an/ap-c-ord.pdf to see the pdf! 
It was only when I was nearly finished, that I realised that there was a problem with the fundamental definitions.
Regarding ordinals, specifically, order-isomorphisms between well-ordered-sets were defined and an ordinal was said to be an equivalence class with respect to the corresponding equivalence relation.
What makes me uncomfortable now is that this is an "equivalence relation" on a proper class, and while I "understand" what's going on, the fact remains that the notion of a proper class is merely informal in ZFC.
A visit to wikipedia taught me that there is another (more set-theoretic) way to define ordinals using reflexive sets.


*

*Is there a way to "rescue" the results of the above mentioned pdf without having to deal with proper classes?

*Can someone suggest a - preferably freely available - source to learn about the construction of ordinals using the reflexive sets, that covers the contents in the above mentioned pdf?
Thanks a lot!
 A: You are right to be skeptical about defining equivalence relations on proper classes. While there are various ways you can approach the problem rigorously within ZF(C), it's not entirely straightforward. While ordinals have enough structure that this isn't a huge problem, for cardinals, you have to be careful.
Now, given the set-theoretic definition of ordinals (i.e. a set whose elements are well-ordered by $\in$), one can show that any well-ordered set is order isomorphic to a unique ordinal, so this does capture the notion of the equivalence classes. It dodges the worry with proper classes by explicitly defining representatives of the equivalence classes and then saying that any well-ordered set is in one of the equivalence classes so represented. (By the way, we aren't defining a set of ordinals; we're defining a property that makes a particular set an ordinal. This is because the collection of all ordinals is a proper class). 
For cardinals, if you use the axiom of choice, you can well-order any set, so you can use a particular ordinal as the representative of a particular equivalence class of sets (under bijection). This ordinal is defined to be the least ordinal in this equivalence class: well-ordering of the proper class of ordinals shows that even though this equivalence class is proper, we have a unique least ordinal in it. 
However, without the axiom of choice, it's difficult to make the definition of cardinals as equivalence classes under the bijection relation rigorous. You use the axiom of regularity and apply something called "Scott's Trick" - see here - which relies on the Von Neumann cumulative hierarchy of sets.
