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
> > > Answers to your three questions. > > > > (1) Does the sequence of decimals > > > 0.1 > > > 0.11 > > > 0.111 > > > ... > > > contain 1/9? > > > No > > > > (2) Does the same sequence understood as FISONs of paths contain the > > > path 1/9? > > > No > > 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.
> > > > > (3) Is the union of these FISONs 1/9? > > > Yes > > > [note for a line to be the union of all lines it is > > necessary and sufficient for it to be a last line. > > There is no last line, so the union of all lines > > is not a line] > > 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.