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Equivalence of All Constructions of R

 Equivalence of All Constructions of R  Having made the effort to abstract the properties of the integers, the rationals and the real numbers, we are now in a position to reap their fruits: we can show that all constructions of R are isomorphic, that is are in a one-toone correspondence which preserves their algebraic properties (addition and multiplication and their associated properties), their order properties, and their least upper bound properties. That is if R and R 0 are two constructions,  then for all a, b ∈ R and a 0 , b0 ∈ R 0 with a ↔ a 0 and b ↔ b 0 we have a + b ↔ a 0 + b 0 , ab ↔ a 0 b 0 and a ≤ b ⇐⇒ a 0 ≤0 b 0 ,  while for all S ⊆ R and S 0 ⊆ R 0 with S ↔ S 0 we have sup S ↔ sup S 0 . Formally, Theorem 4.1 Every complete totally ordered field is both isomorphic and order-isomorphic to R, so in this sense all constructions of R are equivalent.  Proof : In what follows, suppose R is any construction of the reals satisfying the axioms given in Const...
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Construction of the real numbers

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  Construction of the real numbers [ edit ] Main article:  Construction of the real numbers The theorems of real analysis rely on the properties of the  real number  system, which must be established. The real number system consists of an  uncountable set  ( � ), together with two  binary operations  denoted  +  and  ⋅ , and an  order  denoted  < . The operations make the real numbers a  field , and, along with the order, an  ordered field . The real number system is the unique  complete  ordered field , in the sense that any other complete ordered field is  isomorphic  to it. Intuitively, completeness means that there are no 'gaps' in the real numbers. This property distinguishes the real numbers from other ordered fields (e.g., the rational numbers  � ) and is critical to the proof of several key properties of functions of the real numbers. The completeness of the reals is often...
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Explicit constructions of models

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  Explicit constructions of models [ edit ] 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. The first three, due to  Georg Cantor / Charles Méray ,  Richard Dedekind / Joseph Bertrand  and  Karl Weierstrass  all occurred within a few years of each other. Each has advantages and disadvantages. A major motivation in all three cases was the instruction of mathematics students. Construction from Cauchy sequences [ edit ] A standard procedure to force all  Cauchy sequences  in a  metric space  to converge is adding new points to the metric space in a process called  completion . �  is defined as the completion of  Q  with respect to the metric | x - y |, ...
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