Finite, Countable and Uncountable Sets
If the axiom of choice holds, the law of trichotomy holds for cardinality. Thus we can make the following definitions:
- Any set X with cardinality less than that of the natural numbers, or | X | < | N |, is said to be a finite set.
- Any set X that has the same cardinality as the set of the natural numbers, or | X | = | N | = ℵ0, is said to be a countably infinite set.
- Any set X with cardinality greater than that of the natural numbers, or | X | > | N |, for example | R | = c > | N |, is said to be uncountable.
Read more about this topic: Cardinality
Famous quotes containing the word sets:
“To the extent to which genius can be conjoined with a merely good human being, Haydn possessed genius. He never exceeds the limits that morality sets for the intellect; he only composes music which has “no past.””
—Friedrich Nietzsche (1844–1900)