Re: Euclidean Geometry in Schools




On 26/06/2005 16:36, Timothy Murphy wrote:

> Jean-Claude Arbaut wrote:
>
>>>> Construction is so messy. Let's instead teach axioms.
>>>> The reals are a complete Archimedean ordered field.
>>>
>>> What exactly does "complete" mean here?
>>
>> Every Cauchy series converges.
>
> So you have a metric on the field?
> How do you define a metric without the reals?

Very funny. That's where you see a good contruction of R is very
helpful ;-)

1st, it can be defined by that property, dunno the exact construction
in that case, but you probably only need a metric on Q, which you have, with
(x,y) -> |x-y|. Ok, not a *metric* by usual definition (which uses R), but
it's enough.

2nd, it can be defined by Dedekind cuts, and you will find the property as a
consequence.

The only *needed* axioms are those of Q (with almost no changes to apply to
R), and "every majorized subset of R has a sup". Once your construction has
given these properties, you can call them axioms of R, and reinvent real
analysis if you wish. That's the way I learnt analysis in 1st year after
high school.

.