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The cardinality of the natural numbers is aleph-null (), the next larger cardinality is , then comes and so on. Continuing in this manner, it is possible to define a cardinal number for every ordinal number κ as will be described below.
The concept goes back to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities.
It should be noted that the aleph numbers are unrelated to the ∞ commonly found in algebra and calculus. Alephs measure the sizes of sets. Infinity (∞), however, could roughly be defined as the extreme limit of the real number line. While some alephs are larger than others, ∞ is just ∞.
Aleph-null () is by definition the cardinality of the set of all natural numbers, and is the smallest of all infinite cardinalities. A set has cardinality if and only if it is countably infinite, which is the case if and only if it can be put into a direct one-to-one correspondence (see bijectionIn mathematics, a bijection bijective function or one-to-one correspondence is a function that is both injective ("one-to-one") and surjective ("onto"), and therefore bijections are also called one-to-one and onto . Intuitively, a bijective function creat) with the integers. Such sets include the set of all prime numberIn mathematics, a prime number or prime for short, is a natural number whose only distinct positive divisors are 1 and itself; otherwise it is called a composite number . Hence a prime number has exactly two divisors. The number 1 is neither prime nor coms and the set of all rational numberIn mathematics, a rational number (or informally fraction is a ratio of two integers, usually written as the vulgar fraction a ''b where b is not zero. The set of all rational numbers is denoted by Q or in blackboard bold. Using the set-builder notation is.
Ω is actually pretty useful, if somewhat exotic-sounding. An example application is "closing" with respect to countable operations, e.g. trying to explicitly describe the sigma-algebraIn mathematics, a sigma;-algebra (or sigma;-field X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S''. The concept is important in mathematical anal generated by an arbitrary collection of subsets. This is harder than most explicit descriptions of "generation" in algebra (for example vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (fors, groupAbstract algebra Group theory Group theory is that branch of mathematics concerned with the study of groups. Please refer to the Glossary of group theory for the definitions of terms used throughout group theory. See also list of group theory topics.s, etc.) because in those cases we only have to close with respect to finite operations — sums, products, and the like. The process involves defining, for each countable ordinal, via transfinite induction, a set by "throwing in" all possible countable unions and complements, and taking the union of all that over all of Ω.