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:
“The writer who neglects punctuation, or mispunctuates, is liable to be misunderstood.... For the want of merely a comma, it often occurs that an axiom appears a paradox, or that a sarcasm is converted into a sermonoid.”
—Edgar Allan Poe (1809–1845)
“... given a choice between hearing my daughter say “I’m pregnant” or “I used a condom,” most mothers would get up in the middle of the night and buy them herself.”
—Joycelyn Elders (b. 1933)