Role of Axiom of Choice
The cardinality of any infinite ordinal number is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its initial ordinal. Any set whose cardinality is an aleph is equinumerous with an ordinal and is thus well-orderable.
Each finite set is well-orderable, but does not have an aleph as its cardinality.
The assumption that the cardinality of each infinite set is an aleph number is equivalent over ZF to the existence of a well-ordering of every set, which in turn is equivalent to the axiom of choice. ZFC set theory, which includes the axiom of choice, implies that every infinite set has an aleph number as its cardinality (i.e. is equinumerous with its initial ordinal), and thus the initial ordinals of the aleph numbers serve as a class of representatives for all possible infinite cardinal numbers.
When cardinality is studied in ZF without the axiom of choice, it is no longer possible to prove that each infinite set has some aleph number as its cardinality; the sets whose cardinality is an aleph number are exactly the infinite sets that can be well-ordered. The method of Scott's trick is sometimes used as an alternative way to construct representatives for cardinal numbers in the setting of ZF.
Read more about this topic: Aleph Number
Famous quotes containing the words role of, role, axiom and/or choice:
“A famous theatrical actress
Played best in the role of malefactress.
Yet her home-life was pure
Except, to be sure,
A scandal or two just for practice.”
—Anonymous.
“Our role is to support anything positive in black life and destroy anything negative that touches it. You have no other reason for being. I don’t understand art for art’s sake. Art is the guts of the people.”
—Elma Lewis (b. 1921)
““You are bothered, I suppose, by the idea that you can’t possibly believe in miracles and mysteries, and therefore can’t make a good wife for Hazard. You might just as well make yourself unhappy by doubting whether you would make a good wife to me because you can’t believe the first axiom in Euclid. There is no science which does not begin by requiring you to believe the incredible.””
—Henry Brooks Adams (1838–1918)
“It was a real treat when he’d read me Daisy Miller out loud. But we’d reached the point in our relationship when, in a straight choice between him and Henry James, I’d have taken Henry James any day even if Henry James were dead and not much of a one for the girls when living, either.”
—Angela Carter (1940–1992)