Hartogs Number - Proof

Proof

Given some basic theorems of set theory, the proof is simple. Let . First, we verify that α is a set.

1. X × X is a set, as can be seen in axiom of power set.
2. The power set of X × X is a set, by the axiom of power set.
3. The class W of all reflexive well-orderings of subsets of X is a definable subclass of the preceding set, so it is a set by the axiom schema of separation.
4. The class of all order types of well-orderings in W is a set by the axiom schema of replacement, as

   (Domain(w), w) (β, ≤)

   can be described by a simple formula.

But this last set is exactly α.

Now because a transitive set of ordinals is again an ordinal, α is an ordinal. Furthermore, if there were an injection from α into X, then we would get the contradiction that α ∈ α. It is claimed that α is the least such ordinal with no injection into X. Given β < α, β ∈ α so there is an injection from β into X.