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

 Messages: [ Previous | Next ]
 JT Posts: 1,434 Registered: 4/7/12
Re: Finitely definable reals.
Posted: Jan 12, 2013 12:31 AM

On 12 Jan, 06:27, JT <jonas.thornv...@gmail.com> wrote:
> On 11 Jan, 23:30, Virgil <vir...@ligriv.com> wrote:
>
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> > In article

>
> >  WM <mueck...@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.

>
> > --
>
> The real numberline is a joke, the naturals are discrete entities that
> have a continum. There is no 0 in the set of naturals, there is no
> infinity in the set of naturals. There is no 0 in the set of reals
> there is no infinity in the set of reals. They say a real number is a
> value that represents a quantity along a continuous line, but the fact
> you can make the same cut anywhere on the line representing number 3,
> already there tells the truth about natural numbers, they are discrete
> entities counted in and so are the reals 3.14 represent a set of 3
> entities and one fractioned entities. The decimal expansion introduce
> the zeros because of partition of the imaginary fake numberline. That
> is why zeros was introduced in the first place. The idea of zero is
> meaningless there is no number zero between discrete elements, this
> does not mean null isn't a valuable term of description, but it is not
> a mathematical entity. If you line up a series of discrete elements
> and cherrypick where the first positive element within the set start,
> there is no zero entity between -1.+1 just a void depending upon the
> fact that the numberline just isn't there it is created by counting,
> there is no least part of the first natural where it is zero just
> fractions ***as you count them in***

I think my view of numbers very closely relate to Platos idea of
numbers. I have no idea who actually constructed the numberline and
introduced zero, but i think it came with the abacus was us arabic or
hinduic?