In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 8 Apr., 12:18, William Hughes <wpihug...@gmail.com> wrote: > > On Apr 8, 12:10 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > > > > > > > > > On 8 Apr., 10:31, William Hughes <wpihug...@gmail.com> wrote: > > > > > > On Apr 8, 9:48 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > > > On 7 Apr., 23:25, William Hughes <wpihug...@gmail.com> wrote: > > > > > > Note that any line plus its predecessors is a finite set. > > > > > > So, there is no contradiction in saying that you can > > > > > > remove any one line (plus its predecessors) but you cannot > > > > > > remove the collection of all lines > > > > > > Is there a contradiction, Yes or No > > > > Please answer the question > > You can remove a finite set of lines without removing the infinite > set. That is obvious. But why should it not be possible to remove all > finite lines?
If there were only finitely many of them then it would indeed be possible, but that is not the case.
> That set is well defined.
But is not a finite set.
> And then something remains, if > you are right in saying that an infinite union is more than an > infinite sequence of finite unions.
The limit of a a strictly increasing by inclusion sequence of sets can be "more" than any term of that sequence just like the limit, if it exists, of a strictly increasing set of rationals must be larger than any term in the sequence.
If one has an infinite sequence of sets each of which is a proper superset of all its predecessors, the limit set, if it is to exist at all, it cannot be any one of the terms of that sequence.
At least outside of Wolkenmuekenheim. > > > > > A side-question: Why then can the anti-diagonal of a list of finite > > > lines be an infinite line? > > > > The number of elements in the anti-diagonal of a list of finite > > lines is the supremum of the line lengths, not the maximum. > > Why do you think that the union of FISONs of paths is not a supremum?
Why do you think he thinks that? > > > > The infinite union is already built into the construction principle of > > > the sequence: Every line l_n is the union of all lines l_1 to l_n. Why > > > do you think there would be a difference when the infinite union is > > > explicitly mentioned? > > > > Because the fact that for every finite union P > > is true, says nothing about whether P is true > > for an infinite union.- > > Why do you think that an infinite sequence of finite unions is not an > infinite union?
Because an infinite sequence is not just a set and an infinite union is just a set.