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

 Messages: [ Previous | Next ]
 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: Finitely definable reals.
Posted: Jan 11, 2013 7:52 AM

On 11 Jan., 12:36, Zuhair <zaljo...@gmail.com> wrote:
> On Jan 11, 12:49 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>

> > Please answer one question: What shall undefinable reals be good for?
>
> Explaining continuity of space? Possibly?

Something that is underfined and hence unexplained should be able to
explain something?
Further space is not continuous.

But my question aimed at the application of undefined reals in
mathematics.
>
> > They cannot spring off Cantor's argument.
>
> They do of course, they are a consequence of his arguments.

No, you misunderstand again. Cantor's opinion was (and did not change
until he died) that undefined items are nonsense. And ofcourse he was
absolutely right.

> Cantor proved that the
>

> > *definable* reals (those which are definitely different from all reals
> > of his list) cannot be put in bijection with |N.

>
> You mean * discernible* reals, there is a difference between
> discernible reals and finitely definable reals, two reals might be
> discernible (i.e. differ at some finite position of their decimal
> expansions) and yet each one of them might be non finitely definable!

Nonsense. If a real number is not finitely definable, then it has no
positions. If you know, say, only the digits of the first three finite
positions, then you have not an undefined real but you have an
interval with two rationals as limits, in decimal you have the
interval between 0.abc000... and 0.abc999...

You cannot define a real number by increasing step by step the number
of known digits. You would never arrive at a point. All you do is
shrinking the interval. In order to define a real number you need a
finite definition that describes all nested intervals.

> YES Cantor proved that his Diagonal real is * discernible* from all
> the other members of the list, AS FAR AS THAT LIST IS COUTNABLE, but
> that doesn't make out of it *finitely definable*; for it to be
> finitely definable it must UNIQUELY satisfy some finite predicate and
> proving it discernible doesn't by itself make out of it finitely
> definable. Cantor's arguments tells us that we do have MORE
> discernible reals than finitely definable ones. We do have UNCOUNTABLY
> many discernible reals but we have only COUNTABLY many finitely
> definable reals.

There are two cases:
1) If a Cantor list is finitely defined, then you know the entry in
every line and you know every digit of the diagonal.
2) If a Cantor list is undefined and has only, as usual, the first
three lines and then an "and so on", then you do neither know the
following entries nor the digits of the diagonal. Nothing is
"discernible" then except the theorem that two decimals which differ
at some place are not identical. But that is not a deep recognition.

> But we know that they> are countable. Undefinable reals are not elements of mathematics and
> > of Cantor-lists. They cannot help to make the defined diagonals belong
> > to an uncountable set.

>
> No some of Non finitely definable reals ARE members of Cantor-lists.
> Actually for some lists the diagonala is provabley (by Cantor's
> arguments) non finitely definable!

Actually some *lists* are not finitely definable (not only the
diagonals), and therefore these lists are undefinable.
In fact *all* list, that have no finite definition are undefined,
i.e., not existing! And therefore also their diagonals and anti-
diagonals are undefined, i.e., not existing. Therefore there is
nothing "discernable". It is simply not existing.

> But of course all elements on
> Cantor's list and the diagonal (or antidiagonal) all are definitely
> discernible (i.e. differ from each OTHER real at some finite position
> of their decimal expansions).

But as you don't know the "each" and "other" you don't know anything.
>
> You are confusing * discern-ability* with *finite definability*

No. You are confusing intervals and numbers and defined lists and
undefined "lists", i.e., not existing "lists".

Regards, WM