What is the definition of greater than or less than in terms of real numbers?

  1. A thought just struck me today after watching a lecture on the construction of the rational numbers. What is the definition of 'greater than (>)' and 'less than (<)' in the real number system. The only way I can think to describe it is to make some reference to the Euclidean distance between the number and zero and see which distance is greater or smaller but of course that is just using the term in the definition itself. I also thought about the number of steps it takes to construct the number from ZFC set theory. Again though, you have to have some concept of greater than or less than to determine which took more steps.
    Any help would be appreciated
  2. jcsd
  3. Think in terms of a set of greater cardinality having all sets of lesser cardinality as proper subsets.
  4. disregardthat

    disregardthat 1,814
    Science Advisor

    One popular construction of the real numbers is to start with the natural numbers N with their natural ordering defined by 0 < n for all natural numbers n.

    Then you continue to construct the integers Z = N x N / ~ where ~ is the equivalence relation such that (a,b) ~ (c,d) if a+c = d+b, where n = [(n,0)], and -n = [(0,n)] for positive n. We induce an ordering on Z by [(a,b)] < [(c,d)] if a+d > c+b. Note that this gives the natural order on Z we are used to.

    Then we construct Q = Z x (N-{0}) / ~, where ~ is defined by (a,b) ~ (c,d) if ad = bc, and [(a,b)] < [(c,d)] if ad < bc, where < is the order of Z.

    Finally, we construct R by looking at the cauchy-sequences of Q^N (i.e. sequences or rational numbers that are cauchy). A sequence (q_n) of rational numbers is cauchy if for every rational number e, there is a natural number N such that |q_n-q_m| < e for all n,m >= N. Let this set of cauchy-sequences be C.

    We define R = C /~ where (q_n) = (p_n) if the sequence (q_n -p_n) converge to 0. The order of R induced is defined as [(q_n)] < [(p_n)] if there is a rational number e such that there exists a natural number N such that p_n-q_n >= e for all n >= N. It will require a proof of that this in fact is a well-defined ordering, but when you do that it will be the natural ordering of R we are used to.

    From this definition of R we can prove all the known axioms of R we need, most importantly the least-upper-bound property.
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