How to construct ordinals, sets and classes without circular definitions?

Question

Gödel's constructible universe is constructed using ordinals. Ordinals are constructed as equivalence classes of well-ordered sets. Classes are constructed as collections of sets.

What is the proper order to construct them?

Answer 1

In the usual way the foundations are set up, sets are axiomatized using the ZFC axioms. Note that this is not really a construction or definition of sets: we don't actually say exactly what sets are, but rather simply posit that whatever they are, they satsify a certain list of axioms. These axioms say, for instance, that there is a set with no elements, and that for any set, there is another set whose elements are exactly the subsets of the first set. In this sense, then, we say that we can "construct" the empty set or the power set of any set using the axioms.

Within the framework of ZFC, we can define pretty much any kind of mathematical structure of interest as sets. Ordinals are defined as sets with certain properties (note that they are normally defined not as equivalence classes of well-ordered sets, but as certain nice representatives of each equivalence class, since the ZFC axioms actually prove that the equivalence classes do not exist as sets). For each ordinal $\alpha$, the stage $L_\alpha$ of the constructible universe can similarly be defined as a set. Classes do not actually exist at all, but are instead just a convenient way of talking about predicates on sets: given a property $\varphi$ we can write down, then we can talk about sets which satisfy $\varphi$, and informally refer to the collection of such sets as a "class" (even though this collection may not actually exist as a set). If we are using ZFC as our foundation for mathematics, then talking about classes is always just a shorthand for talking about specific predicates.

There are variants on this setup. For instance, instead of ZFC, you can use NBG as your axioms, which are axioms for both sets and classes simultaneously (so classes actually exist). Using NBG you could define ordinals as equivalence classes of well-ordered sets (since those equivalence classes do actually exist as classes), but this is still highly inconvenient and it is better to define them as sets like in ZFC.

You can find far more details in any introductory text on axiomatic set theory.