In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> Do you think it is not a contradiction, to have the statements: > 1) 0.111... has more 1's than any finite sequence of 1's. > 2) But if we remove all finite sequences of 1's, then nothing remains.
In proper English (1) should read "the infinite sequence represented by 0.111... has more 1's in it than in any finite sequence of 1's."
And if WM wishes to prevail, he WM must explain how he intends to remove all finite sequences of 1's without removing all 1's in the process. > >
> > you have not shown that we could remove all finite lines > > without changing the union in any way. > > You reverse the facts.
WH may have reversed SOME statements, but he has not reversed any FACTS.
The fact is that one cannot remove every set containing a natural from a family of sets some of which contain that natural of without removing that natural from the union of set of remaining sets.
(Under the standard definition of union, the union of a set is the set containing all members of members of the original set. Thus the union of an empty set is, outside of Wolkenmuekenheim, again an empty set.
So after removing all lines from a set of lines, as WM would do, the the set of lines remaining unremoved is empty, and its union ie equally empty, and contains no naturals at all.
At least in the sane world outside of Wolkenmuekenheim.
> > WH: Yes, given any set of lines K, every element of K has the property > that it can be removed without changing the union of all lines. Yes, > the set of lines that has this property is the complete set K.
Quite true, any one line, and every the compliment in the set of lines of any infinite set of lines can be removed without eliminating a single natural from the union of those lines left.
But WM has not proved, and cannot prove, at least outside of Wolkenmuekenheim, that one can remove all but finitely many lines and still have all naturals, or even infinitely many naturals, in the union of the finite set of lines remaining. > > WH: We both agree that you have not shown that we can do something > which leaves no lines and does not change the union.
Which is correct,everywhere outside Wolkenmuekenheim, regardless of WM's endless whining about it. >
> Of course in set theory we can construct the set of all finite lines > which are subject to my proof. Of course we can subtract ithis set > from the list that contains all natural numbers.
WM, as usual, is confusing sets with members in impossible ways, at least unless he is using something like the von Neumann model for the naturals in which every line/FISON is also a natural.
> Of course that does > what my proof shows
We have yet to see anything by WM that qualifies as a proof outside his Wolkenmuekenheim, and we automatically reject things that are only valid inside Wolkenmuekenheim.
> Otherwise my proof would be wrong
Which it is everywhere outsie Wolkenmuekenheim. > > Do you wish to withdraw your approval?
Of your claim of proof for you anti-theorem?
We have never granted any such approval.
And won't until your arguments are valid outside Wolkenmuekenheim. --