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From: Virgil on 1 Nov 2006 16:08
In article <1162404736.400458.237480(a)i42g2000cwa.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> David Marcus schrieb:
>
> > > I know of a really good logician who told me that he is unable to
> > > translate my proof into ZFC.
> >
> > Hardly surprising. If he is truly a good logician, then he didn't want
> > to tell you that you were talking gibberish.
>
> No, he (and he is not the only one to do so) does understand the
> arguing but does not know how to translate it into ZF language. That is
> all.

Or into any form of purely logical argument. As it requires assumptions
false in ZF or NBG or most set other theories.
> >
> > Fine. Then stop claiming that you can prove ZFC is inconsistent. If you
> > don't like ZFC, that's your concern. But, that is completely different
> > from saying that you have a proof within ZFC of a contradiction.
>
> I have a proof which shows that the real numbers have less elements
> than a countable set.

Since that "proof" requires assumptions which contradict the axioms of
ZFC and other standard set theories, and no axiom syem has been stated
for it nor any formal exposition of it posted, it is neither
mathematically nor logically acceptable.
> >
> > > > OK, let's use our logic (or standard logic or ZFC). The list has
> > > > infinitely many lines and columns, so the number of diagonal elements is
> > > > infinite. Your statement that the number of diagonal elements is finite
> > > > is wrong. You've jumped from "each entry is finite" to "the number of
> > > > columns is finite".
> > >
> > > Please give one letter which requires an actually infinite number omega
> > > of columns.
> > > If you can't, please stop the nonsense talk about an infinite number of
> > > finite numbers.
> >
> > You seem to have skipped right over my statement that I was using
> > standard logic (or ZFC). In standard logic (or ZFC), "the number of
> > columns is finite" is not the same as "each entry in the list has a
> > finite number of columns". It seems simply bizarre that you would mix
> > these up.
>
> It seems that you do not understand what logic is. That branch which
> you call "standard logic" is some kind of theology which is not useful
> to promote clear thinking.

WM is the one indulging in theology, and bad theology at that. Good
theology, if there is such a thing, does not argue illogically, nor
ignore what it does not find easy to answer in the way that WM
habitually does.
>
> The maximum of a set of finite numbers which has no maximum is simply
> not present.

Is it non-existent entirely, or merely somewhere else for the moment?



> It is *not* an infinite number which is larger than any
> finite number, because a maximum must belong to the set. And a supremum
> not belonging to the elements of the set does not yield a diagonal
> digit.

But every number less than a supremum must. Otherwise one would have a
smaller supremum.


So WM's argument turns out to destroy his thesis.
>
> > > > > If the list consists of finite sequences, then the diaogonal is a
> > > > > finite sequence too. Because it cannot be broader than the list
> > >
> > > > In ZFC or in some other system?
> > >
> > > Everywhere.
> >
> > How would you know since you admitted above that your proofs don't work
> > in ZFC? Why make claims that you then immediately contradict?
>
> I know that ZFC claims there are more reals than any countable set has
> elements. Hence my proof contradicts ZFC.

Which demonstrates that WM's system is wrong.
From: Virgil on 1 Nov 2006 16:30
In article <1162405185.816497.74750(a)e3g2000cwe.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:


> > In an infinite binary tree, there are infinitely many paths through each
> > edge and through each node of that tree. So how does WM isolate one of
> > those paths from all the others to be related to only that one edge and
> > no other edges?
>
> In mathematics, there are infinitely many irrational numbers beginning
> with 2,718281828... How do you isolate one of them?

Not by ignoring all the digits at which they agree, as Wm is suggesting.

And why does WM want to ignore all the rationals involved?
>
> Regards, WM
From: Virgil on 1 Nov 2006 16:43
In article <1162405329.073198.286680(a)h48g2000cwc.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> Dik T. Winter schrieb:
>

> > > Irrational numbers have no last digit. Therefore, with a sequence of
> > > digits like the diagonal number is, one can never have a completed
> > > number but only come as close as possible to any number --- or avoid to
> > > do so.
>
> The problem is that:
> For the diagonal number of Cantor's list it is not sufficient to come
> arbitrarily close to a number which is different from any list number
> --- or avoid to do so. .

It is quite enough for the "diagonal" to be constructed so that it
differs from the nth listed number by 1/10^n, even in constructive math.

> >
> > Ah. In that case one or two, depending on the number involved. But
> > the diagonal obviously depends on the actual representative chosen.

Actually, in any base >= 4, the rule for constructing the "diagonal" is
ordinarily chosen in such a way as to be independent of which of the two
representations is chosen where two representations are possible.

For example: let the nth digit of the diagonal be 2 if the nth digit of
the nth number listed is not 2, and be 3 otherwise. This rule is quite
independent of which possible representation is used for any rational in
any fixed base >= 4.
From: Virgil on 1 Nov 2006 16:57
In article <1162405779.057915.322020(a)m7g2000cwm.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> Dik T. Winter schrieb:
>
>
> > > Therefore it is impossible to exchange omega letters in a diagonal.
> >
> > Wrong. For each element of the list a digit is calculated in the diagonal.
> > As there are infinitely many (omega) elements in the list, there are
> > infinitely many (omega) digits in the diagonal.
>
> omega is the supremum, not the maximum. It does not contribute a
> diagonal digit.

But it determines the "number" of diagonal digits, and that "number" is
Card(omega).
>
> An infinite diagonal requires not only an infinite length but also an
> infinite width of the matrix.

Which is easily achieved by making the "width" of the nth listed element
at least as large as n.

A simple solution that seems to be well beyond WM's ability to
comprehend.



> Therefore your absurd infinite number of
> finite lines does not help you. Here we have the same facts as in our
> old problem
> 0.1
> 0.11
> 0.111
> ...
>
> you remember? Without an infinite number in the list there is no
> infinite diagonal defined.
0.222... is an infinite diagonal different from each of WM's "lines"/
0.000... is another.

> >
> > Width and length are equal.
>
> Fine. But the width is finite by definition.

The width of any one line is finite but the set of allowable widths is
not.




>
> This amounts to say that there are infinite natural numbers or that the
> diagonal is longer than any line.
> Impossible.

Only in WM's nutshell. When line n is of length >= n, then an infinite
diagonal is imperative.

> Let us stick to Euclidean geometry. But that is unimportant. I see it
> is impossible to convince you of the existence of reality.

As WM has not even the remotest contact with reality, his inability is
quite to be expected.
From: Virgil on 1 Nov 2006 17:06
In article <1162406116.713481.142930(a)f16g2000cwb.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> Dik T. Winter schrieb:
>
> > In article <1162300936.776151.45540(a)e3g2000cwe.googlegroups.com>
> > mueckenh(a)rz.fh-augsburg.de writes:
> > > Dik T. Winter schrieb:
> > >
> > > > > Attention: 0.111... has only finite initial segments - and nothing
> > > > > more. Only those can be indexed.
> > > >
> > > > You keep stating that, without proof. That it has only finite initial
> > > > segments, I agree. Not the remainder.
> > >
> > > It is you who denies proof. Give an example of a digit which does not
> > > belong to a finite sequence. If you cannot do so, then every digit
> > > belongs to a finite sequence.
> >
> > I do *not* deny that. But there are infinitely many digits, each being
> > part of a finite segment.
>
> That is but a statement without value.

Then Dik must have caught it from WM.

> If you do not deny my assertion,
> then it is not necessary and not useful to assume any infinity, because
> every number we can name belongs to a finite set.

If we only want their names, we do not need sets at all.

If one wishes to be limited to speaking of only some of, but never all
of, the naturals, finite sets may work well enough, but many of us wish
to speak about all of them collectively or of those not in some finite
set, for which we need infinite sets.
> >
> > > The sequence 0.111... consists of every
> > > digit but not of more. Therefore, there is not need and no use of
> > > talking about infinite sequences.
> >
> > How can those infinitely many digits form a finite sequence?
>
> There are not infinitely many digits. That is only your illusion.

Then WM should be able to give us the position number of that last
digit, or else the number of digits.

If he can do neither then he lies.
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Prev: Pi berechnen: Ramanujan oder BBP
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