Date: Jan 27, 2013 3:59 PM
Author: Virgil
Subject: Re: ZFC and God

In article 
WM <> wrote:

> On 27 Jan., 15:49, "Jesse F. Hughes" <> wrote:

> > >> That is, for each i in N, the i'th digit of 0.777... is defined and is
> > >> 7.

> >
> > > And do you have problems to find this confirmed as possible in the
> > > complete set of terminating decimals? Any digit or index missing?

> >
> > I've no idea what you mean when you ask whether I can "find this
> > confirmed as possible".  But, for each i in N, the i'th digit of
> > 0.777... is defined and equals 7.  Is there anything more I need to
> > know in order to claim that it is a non-terminating decimal?

> You need to know whether this n is an element of a finite initial
> segment of {1, 2, 3, ..., n, n+1, n+2, ..., n^n}.

Since Jesse did not mention any "n", what "this n" are you talking
about, WM? And, secondly, it seems reasonably obvious that every n in |N
is a member of {1,2,3,...,n^n}, though why that fact is of any
importance here is totally obscure.
> > I didn't ask whether it was a finite definition.  I asked whether it
> > was terminating.  And it is not sufficient to note that each t_i is a
> > terminating decimal to conclude that d defined by
> >
> >   d(j) = 7 if j > k or t_j(j) != 7
> >   d(j) = 6 if j <= k and t_j(j) = 7.
> >
> > is also terminating.  This simply does not follow.

> Here you are simply wrong.

Not outside of Wolkenmuekenheim he isn't.

> Prove by induction: If every d(j) is
> element of a finite initial segment, then all d(i) with i < j are
> elements of one and the same finite initial segment. If you are unable
> to prove this, try to find a counter example and fail. (I would be
> very surprised if you would succeed. But I calmly await your answer.)

The answer is that it is just as irrelevant here as your n in
{1,...,n^n} was above.
> >
> > This is analogous to the fact that limits of sequences of rational
> > numbers may be irrational.

> Please refrain from handwaving analogies.

Why should he refrain from doing what you do all the time?

> > We begin with a list of terminating
> > decimals, but it doesn't follow that the anti-diagonal is also
> > terminating.  YOU HAVE TO PROVE THAT.

> On the contrary, you have to prove that it is possible to express the
> anti-diagonal or any irrational number by a sequence of digits.

It is already known that every convergent sequence of rationals
converges to a real, and every infinite decimal effectively defines, and
is defined by, just such a convergent sequence of rationals, and thus
defines a real.

Though one notes that the result need not be irrational.

For example, 0.777... = 7/9.

> You
> are so accustomed to that nonsense that you think the contrary must be
> proved.

Wm is hardly in a position to criticize others for being "accustomed to
> No, you have to prove that the 2500 years old proof of Hippasos is
> wrong, saying that it is *not* possible to express sqrt(2) by a
> fraction. You have to prove that not and "never" can be egalized by
> "actually infinite".

Actually, Jesse does not have to prove anything, because none of WM's
claims have been proved by arguments valid outside his own little
Wolkenmuekenheim, and many of them have been proved false outside that
> Regards, WM