Re: Repaired: (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: 01/27/05
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Date: Thu, 27 Jan 2005 00:51:40 -0500
In <ct06h4$9f$1@gander.coarse.univie.ac.at>, on 01/26/2005
at 03:45 AM, piotr5@unet.univie.ac.at (Piotr Sawuk) said:
>....and y is in R\Q. However, this strategy does pre-suppose too much
>about the nature of R and Q,
What do you mean by R and Q? If you're talking about systems described
by the standard axioms, then the statements are simple consequences of
the axioms. If by R you mean any of the standard constructions from Q,
then the statements are simple consequences of the construction,
except that instead of referring to Q itself you need to refer to its
image under the natural imbedding.
>....which I fail to believe is equivalent to anything in ZFC.
It's a simple consequence of the construction of Q from Z, and it
doesn't even require ZFC; ZF is more than adequate.
>No, C definitely does take a set as input, s does output a set when
>given a natural number,
That may have been your intent, but it's not what you wrote. If you
want anybody to understand what you're trying to say then you need to
keep your notation explicit and consistent. You've used the names of
functions, without arguments, in contexts where that clearly made no
sense.
>True, I did bind the variable z and after ending the sentence I did
>continue using this variable as if the sentence never ended. Maybe I
>should start using XML for my mathematical expressions to emphasize
>more clearly when a bound variable does get out of range.
Try using different letters for different variables.
>i.e. the set S containing only irrational numbers (under the
>condition that there exists 2 irrational numbers with no rational
>number inbetween) such that there exists no pair (A,B) of open sets
>such that intersection of A and B is empty and A intersected with S
>and B intersected with S are both non-empty.
Google for "connected".
>It's got something to do with a topology
>of which R is said to not have that kind of topology,
Huh? R is connected. Q and R\Q are not.
>Did I write that?
You wrote several things that were inconsistent with each other, which
leaves everyone else guessing which ones are typos. That's why it is
important to write everything in formal language and then proofread
it.
>Well, I meant g(x):=intersection of all elements of {G(x,i)|i
>element of C(s)} with C being defined as a function which takes a
>set of sets, and returns a set containing one element for each of
>those sets inside of the set given as an argument, and s would be
>the set {G(z',z) with z'<z both in Q, z>x a free variable,
That's an example of what I was talking about; the value of s
dependens on x, so you're actually defining a function s(x). You need
to remove any dependencies that aren't indicated.
-- Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel> Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org
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