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Re: Finitely definable reals.
Posted:
Jan 11, 2013 9:44 AM
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On Jan 11, 3:52 pm, 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.
>
> 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
I'm not confusing anything, I gave Exact definitions and I wrote them
formally, You are the one who is speaking in the air without any
formalization to support your claims and always giving your terms
different meanings and consequently being trapped into utter confusion
by using them.
I defined "finitely definable" real as
r is a finitely definable real iff r is a real & Exist phi. for all y.
phi(y) <-> y=r
where phi(y) is any finitary formula.
I defined r to be discernible real iff r is a real & for every real
r'. r' =/= r -> Exist n. d_n of r' =/= d_n of r.
It is easy to prove that ALL reals are discernible.
Now the question is: Is all reals finitely definable?
or Equivalently: is it the case that for each real r if r is
discernible then r must be finitely definable.
or Equivalently: Can there exist a real r that is discernible but not
finitely definable.
YOUR answer to that question is in the negative, i.e. you think that
there cannot exist a real that is discernible and not finitely
definable. That's your stance.
While Cantor's arguments PROVES the existence of at least one real
that is discernible but not finitely definable.
Now YOU don't have any proof of your assertion, i.e. you asserted that
EVERY real is finitely definable. But you didn't give any PROOF of
that assertion. Your method to prove your assertion is to challenge
others to present two reals each of which is not finitely definable
and yet discernible. You say what are the first two or three digits of
some undefinable reals that are discernible. Those are silly questions
of course, and they don't establish any proof of your assertion. And
it is very easy to give you examples of those really.
Take the list of ALL finitely definable reals. You hold that this list
is countable, i.e. there is a function f such that f is bijective from
N to the set of all finitely definable reals.
Ok it is easy to prove that there can be infinitely many such
bijective functions.
Take f1 and f2 to be some of those bijective functions, where f1 sends
1 to the real 0.110000.... and sends 2 to the real 0.0011111... and we
don't need to know anything of the other entries since you asked to
show the first two digits of some undefinable real. Now the
antidiagonal on f1 would have the first two digits being 0.01. Now
take f2 to send 1 to 0.001111... and sends 2 to 0.1010000, now the
first two digits of the decimal expansion of the antidiagonal on f2
would be 0.11
and clearly those two are discernible from the first digits actually.
And of course both anti-diagonals are PROVABLY non definable! And
knowing the first two digits of their decimal expansions didn't render
them definable!
Now you need to prove that for the definitions given to terms
"finitely definable" and terms "discernible" as depicted above, then
it is the case that: EVERY discernible real is finitely definable.
Which you didn't give. While Cantor's (whether he was conscious of it
or not) arguments leads to the proof of the existence of at least one
real that is not finitely definable, and this is pretty much accepted
as standard mathematics, among most LEADING mathematicians of the last
century up till today. Actually the standard is that MOST of reals are
not finitely definable! And yet ALL of them are discernible! i.e. each
one of them do differ from all other reals at some finite position of
its decimal expansion.
Your ideas are against what is accepted as Standard by leading
mathematicians and philosophers of the fields concerned, and
unfortunately they are all derived by misguided delusional account of
what others had presented and you are trapped into your own
misinterpretations and false inferences. Its a pity really for a
mathematician (if you are one) to succumb into such shattered
disorganized thought.
Anyhow I don't think that further discussion with you would be
fruitful. Actually from earlier posts I came to realize that, but I
wanted to present thoughts here for those who can fathom them, and
unfortunately you are not among those.
Best Regards
Zuhair
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