In article <d5323000-ffbc-49d9-810f-a91e8652bcbf@z4g2000vbz.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 6 Mrz., 13:18, William Hughes <wpihug...@gmail.com> wrote: > > On Mar 6, 12:48 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > On 6 Mrz., 12:05, William Hughes <wpihug...@gmail.com> wrote: > > > > > > > L_m is a single line if m is a natural number. > > > > > Would you prefer to call L_m infinitely many lines? > > > > > > Nope, I would prefer to call L_m a function > > > > (of time and person). A function may have as > > > > value a "single line of the list" > > > > but calling something that changes a "single line of the > > > > list" is silly.- > > > > > I said always that L_m is a function (of several arguments) and that > > > this function takes as vaules lines of the list. As it takes single > > > lines, I don't see why we should not call them single lines. > > > > Because calling L_m a single line > > is certain to cause miscommunication > > and using language in a way certain > > to cause miscommunication is silly. > > No. L_m is a single line. You have misunderstood as becomes clear from > the following.
If L_n is a single line, where n is a natural number, then there must be a successor line, L_(m+1), since EVERY natural has a successor. At least outside of Wolkenmuekenheim EVERY natural has a successor. > > > > So the statement > > > > "there is no line which contains every > > FIS of d" > > > > becomes in the language of Wokenmuekenheim > > > > "there is no findable line which contains > > every FIS of d" > > > > Similarly, there is no statement about > > the behaviour of "actually infinite" > > sets that does not have an analogue > > in the language of Wolkenmuekenheim. > > > > For example: > > > > in Wolkenmuekenheim you would say > > (about potentially infinite sets) > > > > A subset K of the lines of L > > contains every FIS of d iff > > K has no findable last line. > > No, it is exactly false to require an infinite subset K to contain > every subset of d.
Outside of Wolkenmuekenheim it is exactly true to require an infinite subset of the set of all lines to contain every subset of d.
Every FIS of d is always in one single line. This > line is always the last line. Every other line is not necessary and > not sufficient to contain every FIS of d.
Every FIS of d is a line having a successor line, and for every line in any covering of d a following line is also required in that same covering of d.
So there cannot be a last line, in the ordering by inclusion, in any any set of lines covering all of d.
> Without a last line we can > prove that no line is sufficient and necessary to contain every FIS of > d.
It is only by being without a last line that any set of lines can cover d, at least outside Wolkenmuekenheim. > > This again testifies that you have not understood yet the nature of > potential infinity.
Apparenlty no one can understand it, since WM's arguments about it are too often self-contradictory to show he understands anything. > > > > > to mean the same thing as the > > statement (about "actually infinite sets") > > > > A set of lines K contains > > every FIS of the diagonal > > iff K has infinite cardinality > > That is actual infinity and as such different from the former.
But, at least for actual infiniteness, it is correct, whereas for merely potential infinteness, nothing seems to be correct.
> And, by > the way, the claim is nonsense, since, even in actual infinity, for > every element of K we can prove, that it does not belong to the set of > lines necessary or sufficient to contain every element of d.
However, one can easily prove, at least in standard mathematics, that any infinite set of lines is sufficient and that no finite set of lines is sufficient.
Thus the necessary and sufficient condition on a set of lines or FISs for it to contain d is that set be actually infinite.
at leat everywhere but in wm1.
> > > > This is what I mean when I say > > that "potential infinity" behaves > > like "actual infinity". > > So you have not yet comprehenden the nature of potential infinity.
No one has. > > It is ridiculous to see how many matheologians claim that a set K that > contains every FIS of d although it is clear that none of its FIS can > satisfy this claim.
It is ridiculous to see how WM the WMYTHEOLOGIST fails to see what outside Wolkenmuekenheim is so obvious. > > Everey line of the complete list > > 1 > 1, 2 > 1, 2, 3 > ... > > is not containing the actually infinite set |N. > It is claimed that the union does. > But this list is constructed such that the union is the same as the > sequence.
WRONG!
The sequence, as a set, is {{1}, {1,2}, {1,2,3}, ...}
The union, as a set, is {1,2,3,...}
They are different, at least everywhere outside of Wolkenmuekenheim.
> And the limit |N does not belong to the sequence. > Therefore it does not belong to the union.
Wrong again? The limit of the seqeunce IS the union.
And where is WM's proof that some mapping from the set of all binary sequences to the set of all paths of a CIBT is a linear mapping? WM several times claimed it but STILL cannot seem to prove it. --
