Number of elements

For finite sets, cardinality recovers the usual concept of size as "number of elements." However, it is more often difficult to ascribe "sizes" to infinite sets. A system of cardinal numbers can be developed to extend the role of natural numbers in answering "how many". Most commonly, the Aleph numbers⁠${\displaystyle \aleph _{0},\aleph _{1},\aleph _{2},...\aleph _{\omega },\aleph _{\omega +1}...}$⁠ are used, since their definition naturally extends the process of counting, and it can be shown that every infinite set has cardinality equivalent to some Aleph.