Re: The real numbers, and general comments

From: GEIvey (George.Ivey_at_gallaudet.edu)
Date: 09/20/04 

Date: Mon, 20 Sep 2004 13:27:41 +0000 (UTC)

On 15 Sep 2004, Andrew Usher wrote:
>mattgrime@o2.co.uk (matt grime) wrote in message news:<f9a5a56b.0409140942.13d974b4@posting.google.com>...
>
>> > > > It gives the rule to construct all the other terms, therefore it's a
>> > > > finite definition.
>> > > >
>> > > And it is a way of defining the real numbers contradicting your original
>> > > claim, I think. I don't have the original post anymore and can't be
>> > > bothered to search google.
>> >
>> > I defined N, not R. Pay atterntion or don't bother replying.
>>
>> Erm, from the format of your reply you seemed to be agreeing that my
>> definitions were "finite". Why don't you pay attention? Or define your
>> terms properly.
>
>I said a definition was finite if it gives the rule to make all of the
>set. Since no rule is possible giving all Dedekind cuts, or all Cauchy
>sequences, R does not have a finite definition.
>
>Andrew Usher

  You wrote {1, 2, 3,...} and then ASSERTED that that was a "rule" giving all natural numbers. There is no "rule" if you don't already know what the natural numbers are so that couldn't be used as a definition. I don't understand what you mean by a "finite definition".

  Since several people have already given the Dedekind cut definition of real numbers, here's another, equivalent definition:
  Start with the set of all non-decreasing, bounded sequences of rational numbers. We say that two such sequences {a_n} and {b_n} are equivalent if and only if the sequence {a_n- b_n} converges to 0. The real numbers are the equivalence classes defined by that equivalence relation.

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