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There are two ways to define the "cardinality of a set": The cardinality of a set A is defined as its equivalence class under equinumerosity. A representative set is designated for each equivalence class. The most common choice is the initial ordinal in that class. This is usually taken as the definition of cardinal number in axiomatic set theory.
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.
An attempt to define the cardinality of a set as the equivalence class of all sets equinumerous to it is problematic in Zermelo–Fraenkel set theory, the standard form of axiomatic set theory, because the equivalence class of any non-empty set would be too large to be a set: it would be a proper class.
Formally, assuming the axiom of choice, the cardinality of a set X is the least ordinal α such that there is a bijection between X and α. This definition is known as the von Neumann cardinal assignment. If the axiom of choice is not assumed we need to do something different.
The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers.For a well-orderable set U, we define its cardinal number to be the smallest ordinal number equinumerous to U, using the von Neumann definition of an ordinal number.
Besides the cardinality, which describes the size of a set, ordered sets also form a subject of set theory. The axiom of choice guarantees that every set can be well-ordered, which means that a total order can be imposed on its elements such that every nonempty subset has a first element with respect to that order.
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 ...
This page includes a list of large cardinal properties in the mathematical field of set theory.It is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given property.