- on (called ''addition'' and ''multiplication'', resp.), a Binary Relation ≤ on , satisfying the following properties.
- ) forms a Field . In other words,
- For all ''x'', ''y'', and ''z'' in , ''x'' + (''y'' + ''z'') = (''x'' + ''y'') + ''z'' and ''x'' --- (''y'' --- ''z'') = (''x'' --- ''y'') --- ''z''. ( Associativity of addition and multiplication)
- For all ''x'' and ''y'' in , ''x'' + ''y'' = ''y'' + ''x'' and ''x'' --- ''y'' = ''y'' --- ''x''. ( Commutativity of addition and multiplication)
- For all ''x'', ''y'', and ''z'' in , ''x'' --- (''y'' + ''z'') = (''x'' --- ''y'') + (''x'' --- ''z''). ( Distributivity of multiplication over addition)
- For all ''x'' in , ''x'' + 0 = ''x''. (existence of additive Identity )
- 0 is not equal to 1, and for all ''x'' in , ''x'' --- 1 = ''x''. (existence of multiplicative identity)
- For every ''x'' in , there exists an element −''x'' in , such that ''x'' + (−''x'') = 0. (existence of additive Inverse s)
- For every ''x'' ≠ 0 in , there exists an element ''x''−1 in , such that ''x'' --- ''x''−1 = 1. (existence of multiplicative inverses)
2. (, ≤) forms a Totally Ordered Set . In other words,
- For all ''x'' in , ''x'' ≤ ''x''. ( Reflexivity )
- For all ''x'' and ''y'' in , if ''x'' ≤ ''y'' and ''y'' ≤ ''x'', then ''x'' = ''y''. ( Antisymmetry )
- For all ''x'', ''y'', and ''z'' in , if ''x'' ≤ ''y'' and ''y'' ≤ ''z'', then ''x'' ≤ ''z''. ( Transitivity )
- For all ''x'' and ''y'' in , ''x'' ≤ ''y'' or ''y'' ≤ ''x''. ( Totalness )
- on are compatible with the order ≤. In other words,
- For all ''x'', ''y'' and ''z'' in , if ''x'' ≤ ''y'', then ''x'' + ''z'' ≤ ''y'' + ''z''. (preservation of order under addition)
- For all ''x'' and ''y'' in , if 0 ≤ ''x'' and 0 ≤ ''y'', then 0 ≤ ''x'' --- ''y'' (preservation of order under multiplication)
4. The order ≤ is ''complete'' in the following sense: every non-empty subset of Bounded Above has a Least Upper Bound . In other words,
- If ''A'' is a non-empty subset of , and if ''A'' has an Upper Bound , then ''A'' has an upper bound ''u'', such that for every upper bound ''v'' of ''A'', ''u'' ≤ ''v''.
- ''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 1-1 and Onto .
- ''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'').
The final axiom above is most crucial. Without this axiom, we simply have the axioms which define an Ordered Field , and there are many non-isomorphic models which satisfy these axioms. However, when the completeness axiom is added, it can be shown that any two models must be isomorphic, and so in this sense, there is only one complete ordered field.
We shall not prove that any models of the axioms are isomorphic. Such a proof can be found in any number of modern analysis or set theory textbooks. We will sketch the basic definitions and properties of a number of constructions, however, because each of these is important for both mathematical and historical reasons.
If we have a space where Cauchy Sequence s are meaningful (such as a Metric Space , i.e., a space where distance is defined, or more generally a Uniform Space ), a standard procedure to force all Cauchy sequences to converge is adding new points to the space (a process called Completion ).
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