Date: Apr 4, 2013 3:34 PM
Author: Virgil
Subject: Re: Matheology � 224

In article 
WM <> wrote:

> On 4 Apr., 16:01, William Hughes <> wrote:
> > On Apr 2, 10:39 pm, WM <> wrote:
> >

> > > On 1 Apr., 15:30, William Hughes <> wrote:
> > <snip>
> > > > You can only have no lines left if you
> > > > remove an infinite thing.

> >
> > > Piffle.
> >
> > So what non infinite thing do you remove?

> I remove all finite things, namely all finite lines that are subject
> to induction.

When WM removew all finite things from a set of infinitely many
all-finite things , he remove infinitely many things and end up with an
empty thing. Much like his head!
> > Sure you can remove any one finite line,
> and its predecessor.

But WM wants to rewmove ALL lines.
> > but that does not leave no lines.
> In actual infinity it leaves no lines.

Removing any one line an all its predecessors doe not remove any of that
lines successors, of which there are many more than of those you have

> You can apply Cantor's argument to digonalize every line at a finite
> position n.

The anti-diagoanal asstandardly defined, and as required for the Cantor
argument, deals with every line at a different position than any other
line. That WM is trying to change things so that they will no longer
work as Cantor intended does not weaken Cantor's original argument.

> Why don't you object in this case, that that does not leave no lines,
> i.e., that the diagonalization does not apply to the complete list
> including all lines?

Because, as originally contructed, it still does!
> > If you remove "every finite line"
> > your are removing an infinite thing
> > "an infinite collection of finite things"

> If an infinite collection of infinite things exists actually, i.e., IF
> it is not only simple nonsense, to talk about an actually infinite set
> of finite numbers, then I can remove this infinite thing because it
> consists of only all finite things for which induction is valid.

Sow us a formal induction argument for your alleged induction claim, if
you expect us to swallow it
> > is an infinite thing. "every finite line"
> > is "an infinite collection of finite
> > things".

> This is, by God, a very special trick to be successful. Induction
> holds for all natural numbers, but if they form an actually infinite
> thing, then induction does not hold for all of them.

Outside of Wolkenmuekenheim, standard induction only holds if the set of
naturals IS infinite. For finite sets, one can use non-standard or
finite induction, which is not at all the same thing.

> Cantor's
> argument, however, holds for this infinite thing, because set theory
> is defined this way.

> Zermelo made the first mistake, when taking Dedekind's potential
> infinity as actual infinity.

There is, as yet, no reason to suppose that to be a mistake.

And actual infinitenes does allow solutions to Zeno's paradoxes, which
in Wolkenmuekenheim still prove motion to be impossible.

> Set theorists now make the second
> mistake, when claiming that Cantor's diagonalization does not apply to
> line after line but to all lines simultaneously.

A rule which applies uniformly to all lines WM would not allow to apply
to some lines? Which lines does WM claim it will not apply to?

> And you make the
> third mistake, when claiming that induction does not apply to the set |
> N generated by this erroneous actual infinity, but only to a set that
> in priciple is not existing in set theory, namely a potential non-
> exhaustable set. Remember: The inductive set in set theory is actual
> and finished. Why can't we use it completely?

WE do use it completely, it is only WM who wants to limit everyone to
only a finite part of it.