Hartogs number

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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.

[edit] 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#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) (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.

[edit] 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.