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

 Messages: [ Previous | Next ]
 Virgil Posts: 8,833 Registered: 1/6/11
Re: Finitely definable reals.
Posted: Jan 11, 2013 5:30 PM

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

> 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.

The real line is!
http://en.wikipedia.org/wiki/Real_number
Quote:
The currently standard axiomatic definition is that real numbers form
the unique Archimedean complete totally ordered field (R,+,·,<), up to
isomorphism,[1] whereas popular constructive definitions of real numbers
include declaring them as equivalence classes of Cauchy sequences of
rational numbers, Dedekind cuts, or certain infinite "decimal
representations", together with precise interpretations for the
arithmetic operations and the order relation. These definitions are
equivalent in the realm of classical mathematics.
The reals are uncountable, that is, while both the set of all natural
numbers and the set of all real numbers are infinite sets, there can be
no one-to-one function from the real numbers to the natural numbers
End quote.
>
> But my question aimed at the application of undefined reals in
> mathematics.

Without them one cannot have a complete infinite Archimedean ordered
field such as the real number field.
> >
> > > 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.

But he still showed that the set of real numbers, i.e., the objects
forming the unique Archimedean complete totally ordered field was not a
countable set in the sense that no surjection from |N to that set is
possible.
>
> > 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.

One may know that a real is between 0.1 and 0.2 but still not finitely
definable. In fact one may know a real accurate to any finite number of
decimals places but still have it undefineable any further.

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.

So if you only know its first n digits, that number is one of those
undefineables that WM claims do nt exist..
>
> > 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.

So that every list of finitely defined basal numerals, with base >=4,
is incomplete since its antidiagonal is not listed.

> 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 any assertion that a list of basal numerals is COMPLETE is
falsified by the existence of anti-diagonals which are provably not in
the original list.
>
> > > 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.

To prove countability of a set, one must be able to prove a surjection
from |N to that set, which is, effectively, proving that one can list
all its members.

So that set which cannot be shown to be listable cannot be shown to be
countable.

And when one can show that any attempted listing is incomplete, one has
sown that the set cannot be listed.

At least that is how things work in the mathematics outside of
WMytheology

> In fact *all* list, that have no finite definition are undefined,
> i.e., not existing!

That is irrelevant when showing that any listing is impossible.

> And therefore also their diagonals and anti-
> diagonals are undefined, i.e., not existing.

If a list could exist, so must its anti-diagonal, so the nonexistence of
an antidiagonal proves the nonexistence of any list.

> Therefore there is
> nothing "discernable". It is simply not existing.

Prove of countability implies listability (A surjection from |N to a set
is a list, possibly with repetitions, of the set's members)

Thus disproof of listability proves uncountability.

At least everywhere but in WMytheology
>
> > 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 don't have to know them individually to prove unlistability

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

Prove of countability implies listability (A surjection from |N to a set
is a list, possibly with repetitions, of the set's members)

Thus disproof of listability proves uncountability.

At least everywhere but in WMytheology

The standard definition of "countable":

A set S is COUNTABLE if and only if there is a surjection from the set
of natural numbers, |N, to the set S.
Note that for finite sets, S, this cannot be a bijection.
For countably infinite sets S, it can be but need not be.
For uncountably infinite sets, S , no mapping from |N to S can be
surjective at all.

>
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