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Topic: Finitely definable reals.
Replies: 52   Last Post: Jan 18, 2013 2:37 PM

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 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: Finitely definable reals.
Posted: Jan 11, 2013 5:01 PM
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On 11 Jan., 22:47, Virgil <vir...@ligriv.com> wrote:
> In article
> <9d94157d-444e-478e-86a0-61b57259f...@f4g2000yqh.googlegroups.com>,
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:

> > On 11 Jan., 10:16, Zuhair <zaljo...@gmail.com> wrote:
> > > Lets say that a real r is finitely definable iff there is a predicate
> > > P that is describable by a Finitary formula that is uniquely satisfied
> > > by r.

>
> > Let's say a real r is finitely definable if every mathematician can
> > understand the definition.

>
> That would make reals language-dependent!

Of course. If there is no language, nobody can talk about numbers.
>
> It may well be that a real number does not have any complete decimal
> expansion, but has other definition on which people can agree

Numbers without complete decimal expansion cannot result from Cantor's
list. So they are irrelevant for the present discussion.

> But even if undefined, not necessarily non-existent.

Matheology.
>
>

> > > and many known first order languages are of that sort and
> > > they are proven to be consistent and even supportive of a proof
> > > system.

>
> > But they can only be understood by very patient listeners.
>
> Which WM is not.

That was irony. Infinite languages can only be understand by people
who lack any understanding.

> > In any case this kind of nonsense has nothing to do with Cantor who
> > held the opinion that infinite words are nonsense.

>
> But Cantor also managed to prove that there are more than countably many
> infinite binary strings possible

Cantor managed to prove that there are more than countably many finite
binary strings possible. Remember, the part behind a_nn of a_n is not
relevant for his proof.

> and that the standard definition of the
> standard complete ordered real number field requires the existence of
> more than countably many real numbers.

for people with predominantly mud in their heads.

> > And
> > uncountable sets which are in bijection with |N, namely the set of all
> > distinguishable real numbers.

>
> Being in bijection with |N is a special case of being countable.

I see.

Regards, WM

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