Without The Axiom of Choice
Without the axiom of choice, there might exist cardinalities incomparable to (namely, the cardinalities of Dedekind-finite infinite sets). Sets of these cardinalities satisfy the first three characterizations above but not the fourth characterization. Because these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable.
If the axiom of choice holds, the following conditions on a cardinal are equivalent:
- and
- , where and is least initial ordinal greater than
However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriate generalization of "uncountability" when the axiom fails. It may be best to avoid using the word in this case and specify which of these one means.
Read more about this topic: Uncountable Set
Famous quotes containing the words axiom and/or choice:
“It’s an old axiom of mine: marry your enemies and behead your friends.”
—Robert N. Lee. Rowland V. Lee. King Edward IV (Ian Hunter)
“European society has always been divided into classes in a way that American society never has been. A European writer considers himself to be part of an old and honorable tradition—of intellectual activity, of letters—and his choice of a vocation does not cause him any uneasy wonder as to whether or not it will cost him all his friends. But this tradition does not exist in America.”
—James Baldwin (1924–1987)