Re: The modern mathematical concept of infinity is ...
Virgil
<e5044bec-69e5-48ae...@y13g2000yqn.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 1 Mrz., 22:32, William Hughes <wpihug...@hotmail.com> wrote:
> > On Mar 1, 4:02 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > On 28 Feb., 22:41, William Hughes <wpihug...@hotmail.com> wrote:
> >
> > > > On Feb 28, 4:13 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > > > On 28 Feb., 02:25, William Hughes <wpihug...@hotmail.com> wrote:
> >
> > > > > > You need aleph_0 strokes to distinguish every natural number.
> > > > > > The union of all the lines contains aleph_0 strokes.
> > > > > > Each *element* of this union contain less than aleph_0 strokes.
> >
> > > > > So you can distinguish X numbers by less than X strokes per number?
> >
> > > > Yes, when X is aleph_0.
> >
> > > Done.
> >
> > Good. Let's go on
>
> There is no way to go on.
There are lots of cardinals beyond aleph_0, to those who have not lost
their way as WM has.
> >
> > We start with four statements that you claim as true.
> >
> > Every FISON has a fixed largest element.
> >
> > N does not have a fixed largest element.
> >
> > N is the union of all FISONs.
> >
> > Every finite union of FISONs is a FISON.
> >
> > Please indicate where you find the first mistake.
> >
> > Every FISON has a fixed largest element.
> > N does not have a fixed largest element.
>
> Correct.
> >
> > Therefore: N is not a FISON.
>
> False. N is a FISON. But we can't figure out which.
When WM agrees that
Every FISON has a fixed largest element.
N does not have a fixed largest element.
both are true, as he does above, then it WM who introduces the
self-contradiction by claiming that "N is a FISON."
WM, but nobody else, simultaneously asserts both
N is a FISON.
and
N is not a FISON.
so it is WM's system which is self-contradictory.
> The only way to
> understand that is to use MatheRealism.
Any system in which asserts both both P and not-P, is of little to no
mathematical interest, at least to mathematicians.
WM vies with the red queen.
Then the largest natural
> number and the largest FISON exist, but they can differ relative to
> the observer and they differ with time. If you have spelt out
> (defined) a number, then it exists. If you add one, then the next one
> exists.
That is not a mathematics that even most finitists accept.
Finitists are picky about what has been proved to exist, and is
therefore know to exist, but are silent on what might exist absent
having been proved to exist.
> >
> > N is not a FISON
>
> false
WM's claim of "false" is false everywhere except in his idiot
MathUnrealism.
> > N is the union of all FISONs
>
> correct
> >
> > Therefore: There is a union of FISONs which is not a FISON
>
> false
WM's claim of "false" is false everywhere except in his idiot
MathUnrealism.
> >
> > There is a union of FISONs which is not a FISON
> > Every finite union of FISONs is a FISON.
> >
> > Therefore: There is a union of FISONs which is not finite.
>
> Why aren't you claiming: There is a FISON that is not a FISON (or, to
> veil the contradiction a bit, that is not finite)?
Because there is no FISON which is not a FISON, but every union of an
infinite set of FISONs (which necessarily has no largest member) is
itself not a FISON.
The logical contradictions WM sees are all entirely within WM's own
MathUnrealism, and not in the standard mathematics/set-theory.