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

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 Virgil Posts: 8,833 Registered: 1/6/11
Re: Finitely definable reals.
Posted: Jan 11, 2013 8:14 PM

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 11 Jan., 22:47, Virgil <vir...@ligriv.com> wrote:
> > In article
> >
> >  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.

So that for someone who does not understand German, "Ein" is not a
number?

I was under the impression that the value of a nmber was independent of
the language that it was expressed in.
> >
> > 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.

Not at all!

None of the real numbers in such a listing of reals need be known beyond
a finite number of places in order to show that there is a number
missing from that list.
>
> > But even if undefined, not necessarily non-existent.
>
> Matheology.

Everything else is WMytheology.
> >
> >

> > > > 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.

Enough of infinite languages can be understood to prove WM wrong.

Note that there are very few people who know ALL of even finite
languages like English so one may well know enough of an infinite
language to get on with without knowing all of it.
>
> > > 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 and thus more than countably many reals.

>
> 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.

Quite so, but that in no way weakens 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.

For people who wish to work with a complete Archimedean ordered Field
like the standard real field.

One wonders what sort of a "real field" WM works with, since it cannot
be the standard real field.
>
> > > 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.

The number of things you do NOT see appears to be uncountable.
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