Hartogs number

In mathematics, specifically in axiomatic set theory, a Hartogs number is a particular kind of cardinal number. It was shown by Friedrich Hartogs in 1915, from ZF alone (that is, without using the axiom of choice), that there is a least wellordered cardinal greater than a given wellordered cardinal.

To define the Hartogs number of a set it is not in fact necessary that the set be wellorderable: If X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X cannot be wellordered, then we can no longer say that this α is the least wellordered cardinal greater than the cardinality of X, but it remains the least wellordered cardinal not less than or equal to the cardinality of X.

Proof

Given some basic theorems of set theory, the proof is simple. Let $\alpha =\{\beta \in {\textrm {Ord}}|\exists i:\beta \hookrightarrow X\}$. First, we verify that α is a set.

1. X × X is a set, as can be seen in axiom of power set#Consequences.
2. The power set of X × X is a set, by the axiom of power set.
3. The "set" W of all reflexive wellorderings of subsets of X is a definable subset of the preceding set, so is a set by the axiom schema of separation
4. The "set" of all order types of wellorderings in W is a set by the axiom schema of replacement, as

   (Domain(w) , w) $\cong$ (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.

References

• Hartogs, Friedrich (1915). "Über das Problem der Wohlordnung". Mathematische Annalen. 76: 438–443. doi:10.1007/BF01458215.
• Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2.