3

How to prevent, in a lesson that deals with basic mathematics, that we give two definitions of a metric ? Because there is one, which takes value in $\mathbb{Q}$, to build $\mathbb{R}$, that we do not know already. And one that conveniently takes value in $\mathbb{R}$, for other cases.

fyusuf-a
  • 823
  • If you define $\mathbb R$ as a completion of $\mathbb Q$ as a metric space, then this problem arises. If you define it in any of several other ways (Dedekind cuts, equivalence classes of Cauchy sequences....), there's no such problem. – Michael Hardy Apr 21 '14 at 18:09
  • 1
    What bothers me is that $\mathbb{R}$ is philosophically the completion of $\mathbb{Q}$. – fyusuf-a Apr 21 '14 at 18:21
  • By hindsight I should not have mentioned equivalence classes of Cauchy sequences as an "other" method, since that's what you're doing here. But Dedekind completions won't require you to deal with rational-valued metric and real-valued metric separately. – Michael Hardy Apr 21 '14 at 18:25
  • In principle, you give one definition, then find the completion, then you give a new definition which subsumes the previous one. From that point of view, you don't have two definitions per se, you just realize at some point what the original definition should have been, and retroactively alter it once you have the means (i.e. once you have the concept of $\mathbb{R}$ in hand). That's my point of view anyway – user139388 Apr 21 '14 at 19:00
  • 1
    Personally, I would simply avoid mentioning the general notion of a metric until well after the reals are in hand, but this option may not be available to you. – Dave L. Renfro Apr 21 '14 at 20:40
  • Related: https://math.stackexchange.com/q/1168823/96384 – Torsten Schoeneberg Feb 06 '25 at 16:55

1 Answers1

2

Given the comments below my question, here are some leads :

  1. Define $\mathbb{R}$ as classes of Cauchy sequences (defined as the sequences which verify : $\forall\epsilon\in\mathbb{Q},\epsilon>0\Rightarrow(\exists N\in\mathbb{N},\forall n\geq N,\forall p\geq0,\lvert u_{n+p}-u_{n}\rvert_\mathbb{Q}\leq\epsilon)$, with $\lvert.\rvert_\mathbb{Q}$ being the absolute value in $\mathbb{Q}$). Then there is a strictly increasing ring morphism $i:\mathbb{Q}\rightarrow\mathbb{R}$.

    Define a metric (as usual, with value in the constructed $\mathbb{R}$). Remark that $(x,y)\overset{d}{\rightarrow} \lvert i(x)-i(y)\rvert_\mathbb{R}$ is a metric on $\mathbb{Q}$ (rigourously, $\lvert.\rvert_\mathbb{Q}$ does not have value in $\mathbb{R}$).

    Define the completion of a metric space. The Cauchy sequences defined above are exactly the Cauchy sequences for this metric. And we have $\lvert x-y\rvert_\mathbb{R}=\lim_{n\to\infty}\lvert i(x_n)-i(y_n)\rvert$ with $(x_n)$ and $(y_n)$ being representatives of $x$ and $y$. So actually, $\mathbb{R}$ is the completion of $\mathbb{Q}$.
  2. Use Dedekind cuts.

It is interesting to note that to define a metric with value in $\mathbb{R}$ or with value in $\mathbb{R'}$ which are both ordered field which are archimedian and complete lead to the same concept, as there is a strictly increasing ring isomorphism $R$ between $\mathbb{R}$ and $\mathbb{R'}$. So the metrics with value in $\mathbb{R}$ and those with value in $\mathbb{R'}$ are in bijection : $d\to R\circ d$.
$d$ and $R\circ d$ are topologically equivalent and lead to the same Cauchy sequences, isometries, uniform continuity and Lipschitz functions.

fyusuf-a
  • 823