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In mathematics, there are several ways of defining the real number system as an ordered field. The ''synthetic'' approach gives a list of axioms for the real numbers as a ''complete ordered field''. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic. Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. ==Synthetic approach== The synthetic approach axiomatically defines the real number system as a complete ordered field. Precisely, this means the following. A ''model for the real number system'' consists of a set R, two distinct elements 0 and 1 of R, two binary operations + and × on R (called ''addition'' and ''multiplication'', respectively), and a binary relation ≤ on R, satisfying the following properties. # (R, +, ×) forms a field. In other words, # * For all ''x'', ''y'', and ''z'' in R, ''x'' + (''y'' + ''z'') = (''x'' + ''y'') + ''z'' and ''x'' × (''y'' × ''z'') = (''x'' × ''y'') × ''z''. (associativity of addition and multiplication) # * For all ''x'' and ''y'' in R, ''x'' + ''y'' = ''y'' + ''x'' and ''x'' × ''y'' = ''y'' × ''x''. (commutativity of addition and multiplication) # * For all ''x'', ''y'', and ''z'' in R, ''x'' × (''y'' + ''z'') = (''x'' × ''y'') + (''x'' × ''z''). (distributivity of multiplication over addition) # * For all ''x'' in R, ''x'' + 0 = ''x''. (existence of additive identity) # * 0 is not equal to 1, and for all ''x'' in R, ''x'' × 1 = ''x''. (existence of multiplicative identity) # * For every ''x'' in R, there exists an element −''x'' in R, such that ''x'' + (−''x'') = 0. (existence of additive inverses) # * For every ''x'' ≠ 0 in R, there exists an element ''x''−1 in R, such that ''x'' × ''x''−1 = 1. (existence of multiplicative inverses) # (R, ≤) forms a totally ordered set. In other words, # * For all ''x'' in R, ''x'' ≤ ''x''. (reflexivity) # * For all ''x'' and ''y'' in R, if ''x'' ≤ ''y'' and ''y'' ≤ ''x'', then ''x'' = ''y''. (antisymmetry) # * For all ''x'', ''y'', and ''z'' in R, if ''x'' ≤ ''y'' and ''y'' ≤ ''z'', then ''x'' ≤ ''z''. (transitivity) # * For all ''x'' and ''y'' in R, ''x'' ≤ ''y'' or ''y'' ≤ ''x''. (totalness) # The field operations + and × on R are compatible with the order ≤. In other words, # * For all ''x'', ''y'' and ''z'' in R, if ''x'' ≤ ''y'', then ''x'' + ''z'' ≤ ''y'' + ''z''. (preservation of order under addition) # * For all ''x'' and ''y'' in R, if 0 ≤ ''x'' and 0 ≤ ''y'', then 0 ≤ ''x'' × ''y'' (preservation of order under multiplication) # The order ≤ is ''complete'' in the following sense: every non-empty subset of R bounded above has a least upper bound. In other words, # * If ''A'' is a non-empty subset of R, and if ''A'' has an upper bound, then ''A'' has a least upper bound ''u'', such that for every upper bound ''v'' of ''A'', ''u'' ≤ ''v''. The rational numbers Q satisfy the first three axioms (i.e. Q is totally ordered field) but Q does not satisfy axiom 4. So axiom 4, which requires the order to be Dedekind-complete, is crucial. Axiom 4 implies the Archimedean property. Several models for axioms 1-4 are given below. Any two models for axioms 1-4 are isomorphic, and so up to isomorphism, there is only one complete ordered Archimedean field. When we say that any two models of the above axioms are isomorphic, we mean that for any two models (''R'', 0''R'', 1''R'', +''R'', ×''R'', ≤''R'') and (''S'', 0''S'', 1''S'', +''S'', ×''S'', ≤''S''), there is a bijection ''f'' : ''R'' → ''S'' preserving both the field operations and the order. Explicitly, *''f'' is both injective and surjective. *''f''(0''R'') = 0''S'' and ''f''(1''R'') = 1''S''. *For all ''x'' and ''y'' in ''R'', ''f''(''x'' +''R'' ''y'') = ''f''(''x'') +''S'' ''f''(''y'') and ''f''(''x'' ×''R'' ''y'') = ''f''(''x'') ×''S'' ''f''(''y''). *For all ''x'' and ''y'' in ''R'', ''x'' ≤''R'' ''y'' if and only if ''f''(''x'') ≤''S'' ''f''(''y''). 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「Construction of the real numbers」の詳細全文を読む スポンサード リンク
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