Date: Nov 21, 2012 2:13 PM
Subject: Re: Matheology § 154: Consistency Proof!

On 21 Nov., 19:23, William Hughes <> wrote:
> On Nov 21, 1:57 pm, WM <> wrote:

> > On 21 Nov., 18:43, William Hughes <> wrote:
> > > On Nov 21, 1:20 pm, WM <> wrote:
> > > Nope.  The limit of the set of digits to the left of the decimal
> > > point is not a set of digits to the right of the decimal.

> > Of course it is not, but it does not prohibit that there are digits on
> > the right.

> > > If we change the limit to the set of digits to the left or right of
> > > the decimal point we still get {}.  {} is not a real number
> > > and does not have a reciprocal.

> > We cannot conclude from set theory that the digits on the right of the
> > decimal point vanish.

> Yes we can.

No. Set theory does not destruct the set of all natural numbers. Set
theory only shows that the set can be enumerated by the set of all
even numbers, .i.e., all inidces disappear left and gather right.

> > > > Every infinite sequence of real numbers either has no limit or has a
> > > > limit in the real numbers or the improper limit oo. In any case there
> > > > are never two or more limits!

> > > Piffle. You really know nothing about limits do you.
> > In my book on analysis I write: a sequence may have many accumulation
> > points, If there is only one accumulation point, we call it the limit
> > of the sequence. (But I did not invent that definition.)

> Anyone who is writing a book on analysis should understand that
> accumulation point depends on the topology used.

If the book concerns real analysis only, then the topology is clear.
By the way, this topology is explained in the book.

> Sure, one usually use the "standard" topology
> derived from the standard metric, and one may even
> use language suggesting that this is the only possible
> topology, but you still have to know there are
> other possibilities.

Of course, how shouldn't I. Perhaps I will be going to write another
book on other topologies in the future. Why not? That is not an
esoteric lore. But my sequence is a sequence of real numbers and it
has one and only one real number (or infinity) as its (improper)
limit. And just for this very case set theory shows that the limit is
less than 1. No, it is not attempted to calculate any other limit. We
consider only the one and only possible limit of the real sequence in
the real numbers.

Therefore all your struggling is in vain. Perhaps a sect of
matheologians will survive for a while and die out only very slowly
(their last words being: There is no contradiction!) but every sober
mind outside of that sect will recognize that it is impossible to have
a house and at the same place to find no bricks (or other material).

Regards, WM