In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 23 Mrz., 15:20, William Hughes <wpihug...@gmail.com> wrote: > > On Mar 23, 3:13 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > On 23 Mrz., 15:01, William Hughes <wpihug...@gmail.com> wrote: > > > > > > On Mar 23, 2:43 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > > > On 23 Mrz., 10:31, William Hughes <wpihug...@gmail.com> wrote: > > > > > > We both agree that you have not shown that we can > > > > > > do something which leaves no lines and does not > > > > > > change the union. > > > > > > > No, of course we do not. > > > > WH: this does not mean that one can do something > > WH: that does not leave any of the lines of K > > WH: and does not change the union of all lines. > > > > WM: That is clear > > > > > > WH: this does not mean that one can do something > > > > > Of course we cannot really do infinite things. This is only an > > > abbreviation. > > > > > I say that there is no finite line that changes the union. > > > > Correct > > > > > So the union would be the same if there was no finite line. > > > > Nope, does not follow. > > It follows in ordinary logic. The negation of "no finite line changes > the union"
But changing finite lines may well change unions of sets of finite lines. The distinction is on whether the set of lines in question has a maximal member when ordered by inclusion.
If it does then the removal of the maximal member changes the union, so that WM's claim is not true in general, but can only be true for the special case in which the set of sets does not have a maximal member by inclusion.
And or a set sets of of naturals like WM's sets of lines not to have a maximal member requires that for every line in it there is a longer line in it which is a superset of that previous line.
Thus it follows that theses things that WM claims for sets of lines can only hold when such sets have n maximum by inclusion member.
Outside of Wolkenmuekenheim, such sets of sets are called infinite. Inside Wolkenmuekenheim, they cannot get visas to enter.
> is "at least one finite line changes the union". But this > is excluded by my proof.
WM's poofs convince no one. > > It seems you wish to refrain from two-valued logig.
it is WM who refrains from two-valued logic.
It is only in places like Wolkenmuekenheim that tertuim non datur is rejected and more than two logical values are allowed.
In the standard logic that WM rejects, TERTIUM NON DATUR still holds!
In Wolkenmuekenheim, there seem to be an unlimited list of logical values, practically a spectrum of them. --