- Hartogs number
In
mathematics , specifically inaxiomatic set theory , a Hartogs number is a particular kind ofcardinal number . It was shown byFriedrich Hartogs in 1915, from ZF alone (that is, without using theaxiom of choice ), that there is a leastwellordered 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 . First, we verify that α is a set.
#"X" × "X" is a set, as can be seen inaxiom of power set#Consequences .
# Thepower set of "X" × "X" is a set, by theaxiom of power set .
# The "set" "W" of all reflexive wellorderings of subsets of "X" is a definable subset of the preceding set, so is a set by theaxiom schema of separation
# The "set" of allorder type s of wellorderings in "W" is a set by theaxiom 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".References
*
*
Wikimedia Foundation. 2010.