Re: (sketch of a) Proof that the set of Real Numbers doesn't exist

From: Shmuel (Seymour J.) Metz (spamtrap_at_library.lspace.org.invalid)
Date: 02/20/05 

Date: Sat, 19 Feb 2005 19:27:17 -0500

In <cv1c7q$cb$1@gander.coarse.univie.ac.at>, on 02/17/2005
   at 05:45 PM, piotr5@unet.univie.ac.at (Piotr Sawuk) said:

>exactly! I have seen several definitions of "open set" in R, and
>since the representation of R does differ, so does this definition.

No. You have seen several models of R, but they are canonically
isomorphic and homeomorphic. You have not seen multiple topologies on
R; the definitions of open sets on each are equivalent.

>I choose to define "open set" as F(x) for some element x and some
>Cauchy-filter F.

If you choose a nonstandard topology then it is your obligation to
explicitly state that whenever you refer to topological notions.

>a theorem who's proof in ZFC I have not seen yet (for the definition
>of R through Cauchy-filters).

In order to legitimately refer to "the definition of R through
Cauchy-filters: you must prove that the objects, functions and
relations satisfy the axioms for R, i.e., that you have constructed a
complete Archimedean field. Once you have done so, then the fact that
each non-empty open set contains both rational and irrational numbers
is a simple theorem and the proof does not need to go back to your
construction.

>What I wish is an algorithm for creating a bunch of models for R.

I don't see the utility.

>what Cauchy-filters do offer me is a description of the properties a
>model of R should fulfill,

No. The properties that a model of R should[1] fulfil are those of a
complete Archimedean field, nothing more.

>Cauchy-sequences and Dedekind-cuts are just 2
>models, each useful for some limited amount of applications.

The whole point of those models is to show that a complete Archimedean
field exists. For that purpose, the simpler the better.

>I would wish for a more general model

Any model is general.

>Cauchy-filters on the other hand do not need to be used when working
>with R, you only need to prove some properties related to them before
>you can introduce your own favourite model of R to work with.

No, you don't need to prove any properties of Cauchy filters in order
to introduce a model that doesn't use Cauchy filters, and there is no
point to dragging them in.

>yes, "{q1(n)|n in N} is {q>x in Q}" equals "(forall n in N: q1(n) in
>{q>x in Q}) and {q>x in Q}\rng(q1)=empty". I decided to write the
>shorter expression. was that the wrong decision?

Writing "the shorter expression" is only legitimate if it expresses
the same thing. Writing a shorter expression that doesn't express what
you meant is always the wrong decision.

>> You need to indicate all of the functional
>> dependencies. Please try to fix collect all of your definitions, fill
>> in the holes, carefully proofread them and only then post them.

You still haven't done that.

>x is a fixed variable

There is no such thing. You need to make explicit what quantifiers are
involved and what their scopes are. You can do that in English, but it
takes more care than you have been willing to invest in your articles.

>I3_x=intersection(I3_x_n),I3_x_n=(x,q1(n)), q1_x(n)>x and {q1_x(n)}
>is {q>x in Q}.

Did you mean

I3_x=intersection(I3_x_n),I3_x_n=(x,q1_x(n)), q1_x(n)>x and {q1_x(n)}
in {q>x in Q}

? Why specify ">x" twice?

>I1_x=I5_x=intersection(F(x,y)),y>x in R\Q

Scattering definitions over multiple articles makes it difficult to
follow what you're trying to say. Fix the typos, *PROOFREAD* and post
it all in one place.

[1] Actually, must fulfill, because otherwise it isn't a model of R. 

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Shmuel (Seymour J.) Metz, SysProg and JOAT  <http://patriot.net/~shmuel>
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