Weaker Forms
There are several weaker statements that are not equivalent to the axiom of choice, but are closely related. One example is the axiom of dependent choice (DC). A still weaker example is the axiom of countable choice (ACω or CC), which states that a choice function exists for any countable set of nonempty sets. These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.
Other choice axioms weaker than axiom of choice include the Boolean prime ideal theorem and the axiom of uniformization. The former is equivalent in ZF to the existence of an ultrafilter containing each given filter, proved by Tarski in 1930.
Read more about this topic: Axiom Of Choice
Famous quotes containing the words weaker and/or forms:
“By weaker accents, what’s your praise
When Philomel her voice doth raise?”
—Sir Henry Wotton (1568–1639)
“Two forms move among the dead, high sleep
Who by his highness quiets them, high peace
Upon whose shoulders even the heavens rest,
Two brothers. And a third form, she that says
Good-by in the darkness, speaking quietly there,
To those that cannot say good-by themselves.”
—Wallace Stevens (1879–1955)