User blog comment:Alemagno12/Huge Ordinal Analysis I: Z 2/@comment-1605058-20180203205324

Your definition of a well-ordering is wrong. There are a couple of ways to define a well-ordering which are equivalent over Z2 and yours is neither of them.

First of all, an order is always a relation between pairs of elements. This can be either a formula with two free variables like CatIsFluffy mentions, or a set of numbers each of which encoding a pair.

In the context of proof theory, we only care about orders which are recursive (so using the former representation, we only look at recursive formulas, and for the latter we only care about computable sets of numbers), and such an order, which I'll denote by \(\prec\) to not confuse with <, is a well-ordering if essentially what you wrote holds for all recursive \(\varphi\).

Therefore a well-ordering is not "a series of formulas". A well-ordering is a relation < for which what you say is true for all recursive formulas \(\varphi\). Note that you do not have the freedom of choice of \(\varphi\).