Re: abundance of irrationals!)

In article <1115325968.421135.253130@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> Virgil wrote:
>
>
> > > It is not a matter of understanding, but a matter of contradiction.
> >
> > WE have a number of axiom systems, such as the Zermelo-Frankel
> system,
> > in which infinite sets are REQUIRED by the axioms.
>
> The axiom requires n+1 for given n. That means An Em with n < m.
> It does not require Em An n < m.

How is that related to the axiom of infinity?

The axiom of infinity says that:
there exists a set S which (1) contains the empty set as a member, {},
and (2) whenever S contians x as a member, S also contains
(x union {x}) as a member.

It may be shown , by using the mapping x --> (x union {x}), that any
such set meets Cantor's definition of an infinite set.
>
>
> > > 0. a b c ...
> > > 1 2 3
> > >
> > > The digit-positions of my listed numbers are enumerated by natural
> > > numbers. Can you find a number n e N which does not enumerate a
> > > digit-position?


For all but a FINTITE few of WM's listed numbers, n does not enumerate a
digit position in the nth listed number.


> >
> > I can conceive of "infinitely many" natural numbers, each marking the
>
> > last digit of one of your listed numbers, and none of your listed
> > numbers having more than finitely many non-zero digits.
>
> Don't blather! Tell me one (1) digit of any real number r e (0,1) which
> cannot be found in my list. Then claim it incomplete.
The digit 2 does not appear in any of your numbers, but does appear in
many real numbers.
>
> I can ask for that number n e N which is sufficient to have all digits
> of Sqrt(2) determined. How many digits are necessary to distinguish the
> number 1/Sqrt(2) from each other real number? Is it not in my list
> because no one can answer that question. But it is given here:
> 0.7071... ???
>
> > > Or there are infinitely many natural numbers, then there are
> infinitely
> > > many digit positions.


> >
> > There are infinitely many digit positions possible, but in WM's list,
>
> > each listed number uses only finitely many of them.
>
> The great many comes at later positions.

Too late!
> >
> > If this were not so, WM would be able to point to one which needs
> > infinitely many decimal places.
>
> No. Nobody can point to such a number in binary or decimal
> representation.

Because in WM's list they do not exist.

> But in my list, I avoid the mistake of interminging
> partial sums and limits. All partial sums are written down separately.
> I can tell you the position of every partial sum. You believe that such
> a series icorporates its limit by definition.



> Therefore, these limits
> are in my list.

According to the definition of a limit of a sqeuence, a sequence can,
have only one limit, but now WM claims that his sequence has infinitely
many.

Now as an infinite SET of values, WM's set can be dense in the open
inteval (0,1) and thus have all of (0,1) as CLUSTER POINTS, but not as
LIMITS, and not as members of the set itself.

There are important distinctions between cluster points of a set and a
limit of a sequence. And WM does not seen to be aware of them.

Wow, a list containing all reals of (0,1), well ordered
> and shown countable.
>
> >
> > Claims of existence, such as WM keeps making, require proofs. The
> most
> > direct proof of existence is to produce one of the things alleged to
> > exist.
>
> Every serious matematician will consider a number existing, if I can
> tell him the position of any digit he may ask for.
>

On the contrary, mathematicians will require that you give the digits
of ALL postitions, not the position of one digit. WM again gets thing
backwards.
>
> > Does WM dispute this and claim the there is an nth number in his list
>
> > requiring more than n decimal places? Again, let him produce such an
> n
> > and its corresponding number.
>
> Of course, these numbers are written down somewhat later.


This rule applies to ALL your listings, so there is nothing "later" that
all of them.
>
> > >
> > > Further I understand that Cantor exchanged the quantifiers, to
> obtain,
> > > from what today is called axiom of infinityor induction or
> > > Archimedean, grossly
> > > An Em : n < m,
> > > his
> > > Em An : n < m
> > > with m called aleph_0.
> > > A typical quantifier mismatch.
> >

> > That is entirely WM's mishmash, and none of Cantor's.
> > Cantor NEVER claims that aleph_0 is a member of N, that falsehood is
> > entirely WM's.
>
> I did never claim that above m called aleph_0 is a member of N. It is a
> whole number, acording to Cantor. However, the axiom gives only An Em :
> n < m.

While "A n in nN : E m in N : n < m" is true, it is not the axiom of
infinity.

> > > And all of you who accused me to do so, all of you don't realize
> that.
> >
> > We realize that what ever you think you are talking about is riddled
> > with internal contradictions, but what we are talking about has been
> > examined by experts for decades without finding any internal
> > contradictins.
>
> The fault is to believe that experts had examined that. Why should they
> examine such an "obvious" fact?

They examined the parallel postulate of Eucidean geometry and found it
unneccesary. A comparable exertion has been expended on Cantor with no
discovery of internal contradictions.

To claim that someone who cannot evn keep his quantifications straight
has done what the likes of Goedel could not do is unwarranted arrogance.
.

Relevant Pages

  • Re: abundance of irrationals!)
    ... >> It is not a matter of understanding, but a matter of contradiction. ... > in which infinite sets are REQUIRED by the axioms. ... The axiom requires n+1 for given n. ... > last digit of one of your listed numbers, ...
    (sci.math)
  • Re: An uncountable countable set
    ... index digit number n of K and so cover K up to digit number n. ... of index positions is infinite. ... The number of digit positions is irrelevant. ... My basic assumption is the existence of this axiom. ...
    (sci.math)
  • Re: Cantor Confusion
    ... For each element of the list a digit is calculated in the diagonal. ... As there are infinitely many (omega) elements in the list, ... infinite width of the matrix. ... If your axiom contradicts this, ...
    (sci.math)
  • Re: An uncountable countable set
    ... A finite string can never cover an ... index digit number n of K and so cover K up to digit number n. ... of index positions is infinite. ... My basic assumption is the existence of this axiom. ...
    (sci.math)
  • Re: The Modified Halting Problem, Take ??? .
    ... Turing tapes" will not be a set or even a proper class in NAFL, ... each tape itself is an infinite entity. ... They're usually described as potentially infinite or finitely unbounded. ... and there is no last digit of Pi ever computed which one would think ...
    (sci.logic)