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 is an axiom in political science that unless a people are educated and enlightened it is idle to expect the continuance of civil liberty or the capacity for self-government.”
—Texas Declaration of Independence (March 2, 1836)
“But real action is in silent moments. The epochs of our life are not in the visible facts of our choice of a calling, our marriage, our acquisition of an office, and the like, but in a silent thought by the way-side as we walk; in a thought which revises our entire manner of life, and says,—”Thus hast thou done, but it were better thus.””
—Ralph Waldo Emerson (1803–1882)