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A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4. Aleph-null, the smallest infinite cardinal. In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set.
As is standard in set theory, we denote by the least infinite ordinal, which has cardinality ; it may be identified with the set of natural numbers.. A number of cardinal characteristics naturally arise as cardinal invariants for ideals which are closely connected with the structure of the reals, such as the ideal of Lebesgue null sets and the ideal of meagre sets.
The continuum hypothesis says that =, i.e. is the smallest cardinal number bigger than , i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis is independent of ZFC , a standard axiomatization of set theory; that is, it is impossible to prove the continuum ...
In the other direction, the binary expansions of numbers in the half-open interval [,), viewed as sets of positions where the expansion is one, almost give a one-to-one mapping from subsets of a countable set (the set of positions in the expansions) to real numbers, but it fails to be one-to-one for numbers with terminating binary expansions ...
Existence of a cardinal number κ of a given type implies the existence of cardinals of most of the types listed above that type, and for most listed cardinal descriptions φ of lesser consistency strength, V κ satisfies "there is an unbounded class of cardinals satisfying φ".
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that κ {\displaystyle \kappa } is a regular cardinal if and only if every unbounded subset C ⊆ κ {\displaystyle C\subseteq \kappa } has cardinality κ {\displaystyle \kappa } .
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Cardinal functions are widely used in topology as a tool for describing various topological properties. [2] [3] Below are some examples.(Note: some authors, arguing that "there are no finite cardinal numbers in general topology", [4] prefer to define the cardinal functions listed below so that they never taken on finite cardinal numbers as values; this requires modifying some of the ...