Cardinality and infinity
Cardinality is the size of a set, measured by trying to match its members one to one with the members of another set. Two sets have the same cardinality when such a matching exists, which is the only definition of size that still works when both sets are infinite. Cantor showed that the whole numbers and the fractions can be matched, so there are no more fractions than whole numbers, but that the real numbers cannot be matched with the whole numbers. Infinity therefore comes in more than one size, and there is no largest one.
Why it matters
It is the clearest case in mathematics of a careful definition overturning an intuition that everyone had, and it opened the whole subject of set theory.
Also written: cardinal number, countable, uncountable
Where it comes up
Read about Cardinality and infinity on Wikipedia
Sources
- Cardinality Wikipedia
- Cantor's diagonal argument WikipediaThe one-page proof that the real numbers are uncountable. Work through it line by line.
- Infinity Stanford Encyclopedia of Philosophy
