Virgil
Posts:
4,482
Registered:
1/6/11
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Re: Mathematics in brief
Posted:
Dec 8, 2012 6:09 PM
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In article
<cf33770c-c642-4baf-9f2d-fb593524c2a6@10g2000yqo.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 8 Dez., 10:41, Zuhair <zaljo...@gmail.com> wrote:
>
>
> I will give two answers, the first one depends on the finity of the
> universe, the second one does not.
>
> > Mathematics is "discourse
>
> and discourse needs the tools available to us. They all are finite.
> >
> > How those forms are known to us? the answer is through their
> > exemplification as part of the discourse of consistent theories about
> > form.
>
> Consistent means that discourse about all elements is possible.
> We know that all finite words belong to a countable set. We know that
> no infinite word can be mentioned without having a finite name.
>
> Therefore this silly argument is really silly:
> "I can decide for a real number x whether a real number y deviates in
> its decimal (or any other) expansion from that of x."
> The complete infinite expansion of x is never known, not even in an
> infinite universe, but only the finite formula allowing expansion to
> any required level.
It only takes finitely many decimal places to determine that the
decimals for two real number differ.
>
> So the argument is silly that there are uncountably many x because x
> has an infinite expansion. No x is known without a finite formula,
> name, word, ...
The definition of a set being uncountable was that there cannot be any
surjection from the naturals. The set of all reals provably satisfies
this requirement. Does you argument prove such a surjection does exist?
If not, it is irrelevant.
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