Virgil
Posts:
4,482
Registered:
1/6/11
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Re: TRANSFINITE would be very weak if there was an alternate theory
Posted:
Dec 4, 2011 4:59 PM
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In article
<746512c6-8590-4f8e-ada9-76da30c1c0f9@t16g2000vba.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 4 Dez., 18:56, Daryl McCullough <stevendaryl3...@yahoo.com> wrote:
> > On Sunday, December 4, 2011 12:29:31 PM UTC-5, WM wrote:
> > > On 4 Dez., 17:06, Daryl McCullough <stevend...@yahoo.com> wrote:
> > > > On Sunday, December 4, 2011 10:10:50 AM UTC-5, WM wrote:
> > > > > The first hint was given by the problem, originally not at all
> > > > > considered by Cantor, that the number 0.1 can be in the list in form
> > > > > of 0.0999....
> >
> > > > That's complete nonsense. If you use diagonalization on
> > > > a list of REPRESENTATIONS for reals, then you end up producing
> > > > a REPRESENTATION that is not on the list. Because a real number
> > > > can have more than one decimal representation, you have to do
> > > > a little extra work to show that there is a real such that none
> > > > of its representations is in the list, but that isn't difficult.
> >
> > > Nevertheless it has to be done. Cantor, in his original argument did
> > > not provide for that possibility.
> >
> > That's not true. Cantor's original proof was not about reals, but
> > about FUNCTIONS from the naturals into a finite alphabet (which we
> > can take to have just two elements, {0,1})
>
> or, as in his original work, w and m. Then it becomes clear
> that .wmmm... cannot be equal to .mwww...
Mirabile dictu, WM right for once!
>
>
> > > The assumption that there is a complete set |N implies that
> > > there is a complete set of FISONs too, each defined by its
> > > last element. But for every FISON and, hence, for every set
> > > of FISONs (without limit) finite mathematics is valid
> > > (as is for every term of every sequence unless a limit is
> > > calculated): Every such set of FISONs contains all natural
> > > numbers that belong to that set (= are in the union of that set of FISONs)
> > > in one and the same FISON.
> >
> > That's true for every FINITE set of FISONs.
>
> That is true, in finite mathematics, for every FISON, hence for every
> set of FISONs, because every FISON is *by definition* (part of) a
> finite set.
It is also, by definition, part of an infinite set, namely |N!
> An actually infinite set of FISONs is as likely to be
> observed or to be meaningfully defined as a set of 10^3 two-digit-
> numbers.
What is or is not observable in WOLKENMUEKENHEIM is of no consequence in
ZFC or other standard set theories.
>
>
> > If you want to assume
> > it is true for EVERY FISON, then that's an additional assumption.
> > Let's call it WMs Axiom.
>
> No, it is true for every finite set in simple finite mathematics.
But ZFC is not "simple finite mathematics" so what you claim is
irrelevant in ZFC.
>
> > So what you're saying is that you have an axiom that contradicts
> > Cantor's theorem. Okay, fine. Why should anyone care about your
> > axiom? Why should anyone believe that it is true?
>
> Perhaps one should care about mathematics.
Those who care find your claims ridiculous, you logic flawed, and your
arrogance idiotic.
> And if not, if one thinks
> to know better, then one should show at least two FISONs that are
> required to contain together more than each of them. Otherwise one is
> disproved by mathematics.
Claimed but never proven, and claims made without proof may be rejected
without proof.
>
> > For any finite set of FISONs, the union of all them is equal to
> > the largest element. If you want to claim that this is true
> > for INFINITE collections of FISONs, then you need an axiom to
> > assert this. But mathematics without that axiom is perfectly
> > consistent, and is more interesting.-
>
> Finite mathematics is valid for FISONs - for every FISON - with no
> regard to the frequencey of appearance. Only if a set contains more
> than every FISON, then, may be, finite mathematics is no longer vaild.
Since for every fison, there is a natural not in it, no set of fisons
whose union is also a fison can ever cover every natural. And this is
trivially true in any sane set theory.
And the union of any finite set of fisons is a fison.
And the union of any infinite set of fisons is |N.
> But as I remarked above: In order to have a set of FISONs that
> contains more natural numbers than every FISON of the set, you need to
> have at least one infinite FISON in that set.
WM may need to, but no one else does, as long as they have a set of
infinitely many fisons.
WMs confusion of an infinitely large fison with a set of infinitely many
fisons seems to be just another expression of the kink in his thinker
causing his quantifier dyslexia and other logical problems.
> And even if that is not
> a contradiction in set theory with its finished infinities and empty
> multitudes, it is a contradiction in every finite mathematics.
Only in WM's eyes.
--
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