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How to build the real number system?

thanks.

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I vote to close as 'not a real question' because nobody there is no answer that isn't just a wikipedia link. –  anon Sep 5 '10 at 18:19
    
    
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closed as not constructive by Old John, sdcvvc, Sasha, Michael Greinecker, William Sep 1 '12 at 4:37

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6 Answers

up vote 2 down vote accepted

Let's start by defining the natural numbers (a version including 0) by Peano's Axioms (sometimes called the Peano Postulates):

  1. 0 is a number.
  2. If n is a number, the successor of n is a number.
  3. 0 is not the successor of a number.
  4. Two numbers of which the successors are equal are themselves equal.
  5. (induction axiom.) If a set S of numbers contains 0 and also the successor of every number in S, then every number is in S.

Now, the integers are the natural numbers and their additive inverses and the rational numbers are numbers that can be expressed as a ratio of an integer to a non-zero integer. The real numbers are the set of numbers that are limits of Cauchy sequences of rational numbers (which is more or less a convenient way of saying that the sequence will converge in the reals).

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Alternatively, you can look at Cauchy sequences: http://en.wikipedia.org/wiki/Construction_of_the_real_numbers (and other approaches than can be found there).

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Have you learnt of dedikind cuts. If not please look at this: http://en.wikipedia.org/wiki/Dedekind_cut

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You have a lot of possibilities, which dependes on what you want to achieve. See, for example wikipedia, which also reports the following quote:

Few mathematical structures have undergone as many revisions or have been presented in as many guises as the real numbers. Every generation reexamines the reals in the light of its values and mathematical objectives.

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This is a great article that constructs the real numbers as classes of rational Cauchy sequences:

http://math.mit.edu/~tkemp/18.024/R.pdf

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The construction of $\mathbb{R}$ depends on your theory from which you want to start. For example it is reasonable for analysis to start with the axiom "There is a complete ordered field", show that this field is actually unique up to unique isomorphism and call it $\mathbb{R}$. Then you may define the other number systems, too. Or we start from $\mathbb{N}$ and use Grothendieck constructions and completions to get $\mathbb{R}$ (see the other answers).

Anyway, we use $ZFC$ to make all these constructions work. There we have no chance to define(!) the natural numbers by means of the Peano axioms (because we have to prove the Peano axioms from $ZFC$ for $\omega$, which may be defined as the smallest inductive set). So take care of Isaac's first sentence. Also, if we start with a postulated complete ordered field, this actually follows from $ZFC$, but for the basics of analysis, this does not play any role and may be postponed to the next algebra or logic course.

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