Search results
Results from the WOW.Com Content Network
Cardinality can be used to compare an aspect of finite sets. For example, the sets {1,2,3} and {4,5,6} are not equal, but have the same cardinality, namely three. This is established by the existence of a bijection (i.e., a one-to-one correspondence) between the two sets, such as the correspondence {1→4, 2→5, 3→6}.
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.
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 } .
The initial definition of a cardinal number is an equivalence class of sets, where two sets are equivalent if there is a bijection between them. The difficulty is that almost every equivalence class of this relation is a proper class , and so the equivalence classes themselves cannot be directly manipulated in set theories, such as Zermelo ...
is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the cardinality (or the cardinal number) of the set. A set that is not a finite set is called an infinite set. For example, the set of all positive integers is infinite:
For instance, in our example the number 1 is paired with the subset {4, 5}, which does not contain the number 1. Call these numbers non-selfish. Likewise, 3 and 4 are non-selfish. Using this idea, let us build a special set of natural numbers. This set will provide the contradiction we seek. Let be the set of all non-selfish natural numbers.
The definition of a finite set is given independently of natural numbers: [3] Definition: A set is finite if and only if any non empty family of its subsets has a minimal element for the inclusion order. Definition: a cardinal n is a natural number if and only if there exists a finite set of which the cardinal is n. 0 = Card (∅)
Notably, ℵ ω is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory not to be equal to the cardinality of the set of all real numbers 2 ℵ 0: For any natural number n ≥ 1, we can consistently assume that 2 ℵ 0 = ℵ n, and moreover it is possible to assume that 2 ℵ 0 is as least as large ...