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 well-ordered cardinal greater than a given well-ordered cardinal.

To define the Hartogs number of a set it is not in fact necessary that the set be well-orderable: 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 well-ordered, then we can no longer say that this α is the least well-ordered cardinal greater than the cardinality of X, but it remains the least well-ordered cardinal not less than or equal to the cardinality of X. The map taking X to α is sometimes called Hartogs' function.

Proof

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

1. X × X is a set, as can be seen in axiom of power set.
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) $\cong$ (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, Fritz (1915). "Über das Problem der Wohlordnung". Mathematische Annalen (in German). 76 (4): 438–443. doi:10.1007/BF01458215. JFM 45.0125.01.
• Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2.
• Charles Morgan. "Axiomatic set theory" (PDF). Course Notes. University of Bristol. Retrieved 2010-04-10.