More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. For finite sets and sequences it is easy to see that these two notions coincide, since for every number describing a position in a sequence we can construct a set which has exactly the right size, e.g. 3 describes the position of 'c' in the sequence <'a','b','c','d',...>, and we can construct the set {a,b,c} which has 3 elements. However when dealing with infinite sets it is essential to distinguish between the two — the two notions are in fact different for infinite sets. Considering the position aspect leads to ordinal numbers, while the size aspect is generalized by the cardinal numbers described here.
Name:
Anonymous2013-09-01 16:02
Sets and proper classes. These include Von Neumann–Bernays–Gödel set theory, which has the same strength as ZFC for theorems about sets alone, and Morse-Kelley set theory and Tarski–Grothendieck set theory, both of which are stronger than ZFC.