Date: Feb 9, 2013 3:59 PM
Subject: Re: Matheology � 222 Back to the roots
WM <email@example.com> wrote:
> On 8 Feb., 23:52, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <64c6e6d9-d039-48bf-9cd6-7c614cee3...@j4g2000vby.googlegroups.com>,
> > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 8 Feb., 23:26, William Hughes <wpihug...@gmail.com> wrote:
> > > > More WM logic
> > > > L is a potentially infinite
> > > > list and d is the potentially infinite
> > > > anti-diagonal
> > > > From
> > > > i. For every natural number n, d
> > > > is not the nth line of L
> > > correct.
> > > > ii. i. implies that there is no
> > > > natural number m such that
> > > > d is the mth line of L
> > > No such m can be fixed.
> > It is "fixed" in the sense of not existing at all!
> > > > iii. d may or may not be a line of L
> > > There is no part of d(potential) that is surpassing every line of a
> > > suitable list.
> > If every member of the list has a last digit but d does not,
> That is one side of the medal, but it is not the only side.
> It is exactly as if you would prove that the even numnbers are larger
> than the odd numbers, by showing that for every off number there is a
> larger even number. Of course the latter is right, but it does not
> prove the claim.
It does prove that there is an even as large as any given odd, which is
more to the point.
> > then for
> > every member of the list there will be a first FIS of d surpassing it,
> and for every FIS of d there will be a first line of the remaining
> list surpassing it.
But no finite cap on the length of d unless there is finite cap on the
lengths of the set of FISs, which there is not.
And in standard set theories not finite means actually infinite.
> > and following it, a lot more of them following that first one..
> and following this first line there a lot more with the same surplus.
But no finite cap on either, thus an actual infinity of both.
> > At least outside the idiotic constraints of WMytheology.
> There are no constraints. Is every FIS of d surpassed by a line of the
> list or is there a first FIS that is not surpassed? In mathematics the
> defender of such a position should be able to either prove it or to
> show an example.
But equally, every line of the list is surpassed by a FIS, thus both the
set of lines in the list AND the set of FISs of the diagonal must be
not-finite. Each is a strictly increasing in lengths sequence without a
maximum so is clearly NOT FINITE.
And NOT FINITE means INFINITE everywhere outside WMytheology.
> You have already agreed hat d is not actually infinite
When or where do you allege that I have done anything so foolish?
In ZF, and elsewhere outside Wolkenmuekenheim , there is provably a set
having a first element and for each element another greater than it.
In S+ZF, for example, each member of such a set is a proper subset of
each of its successors.
Such sets are provably not finite. Which everywhere outside of
Wolkenmuekenheim is also called infinite.