In article <firstname.lastname@example.org>, 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. --