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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cardinal arithmetic ： ウィキペディア英語版 Cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ''transfinite'' cardinal numbers describe the sizes of infinite sets. Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets. There is a transfinite sequence of cardinal numbers: :$0,\; 1,\; 2,\; 3,\; \backslash ldots,\; n,\; \backslash ldots\; ;\; \backslash aleph\_0,\; \backslash aleph\_1,\; \backslash aleph\_2,\; \backslash ldots,\; \backslash aleph\_,\; \backslash ldots.\backslash$ This sequence starts with the natural numbers including zero (finite cardinals), which are followed by the aleph numbers (infinite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs. Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra, and mathematical analysis. In category theory, the cardinal numbers form a skeleton of the category of sets. == History == The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874–1884. Cardinality can be used to compare an aspect of finite sets; e.g. the sets and 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; e.g. ). Cantor applied his concept of bijection to infinite sets; e.g. the set of natural numbers N = . Thus, all sets having a bijection with N he called denumerable (countably infinite) sets and they all have the same cardinal number. This cardinal number is called $\backslash aleph\_0$, aleph-null. He called the cardinal numbers of these infinite sets transfinite cardinal numbers. Cantor proved that any unbounded subset of N has the same cardinality as N, even though this might appear to run contrary to intuition. He also proved that the set of all ordered pairs of natural numbers is denumerable (which implies that the set of all rational numbers is denumerable), and later proved that the set of all algebraic numbers is also denumerable. Each algebraic number ''z'' may be encoded as a finite sequence of integers which are the coefficients in the polynomial equation of which it is the solution, i.e. the ordered n-tuple (''a''0, ''a''1, ..., ''an''), ''ai'' ∈ Z together with a pair of rationals (''b''0, ''b''1) such that ''z'' is the unique root of the polynomial with coefficients (''a''0, ''a''1, ..., ''an'') that lies in the interval (''b''0, ''b''1). In his 1874 paper, Cantor proved that there exist higher-order cardinal numbers by showing that the set of real numbers has cardinality greater than that of N. His original presentation used a complex argument with nested intervals, but in an 1891 paper he proved the same result using his ingenious but simple diagonal argument. The new cardinal number of the set of real numbers is called the cardinality of the continuum and Cantor used the symbol $\backslash mathfrak$ for it. Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number ($\backslash aleph\_0$, aleph-null) and that for every cardinal number, there is a next-larger cardinal :$(\backslash aleph\_1,\; \backslash aleph\_2,\; \backslash aleph\_3,\; \backslash cdots).\backslash$ His continuum hypothesis is the proposition that $\backslash mathfrak$ is the same as $\backslash aleph\_1$. This hypothesis has been found to be independent of the standard axioms of mathematical set theory; it can neither be proved nor disproved from the standard assumptions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cardinal number」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.