On 8 Apr., 13:18, William Hughes <wpihug...@gmail.com> wrote: > On Apr 8, 12:56 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > 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? > > I said the "collection of all lines" (this is the same > as the "collection of all finite lines"). The collection > of all finite lines is D and it is not possible > to remove D without leaving the empty set.
That is not in accordance with your statement: "Because the fact that for every finite union P is true, says nothing about whether P is true for an infinite union."
> Do you agree > > There is no contradiction in saying that you can > remove any one line (plus its predecessors) without > changing the union but you cannot > remove the collection of all lines without changing > the union. > > Yes or No
That depends on your statement quoted above and your answer to my third question: (3) Is the union of these FISONs 1/9? You said yes.
Obviously we can remove from an actually infinite line 1/9 = 0.111... all FIS of indices.
Remember the definition of actually infinite for that case: The number of indices in 0.111... is larger than every finite number of indices and therefore cannot be reached by a sequence that contains only finite numbers of indices. 0.1 0.11 0.111 ... Therefore it cannot be removed by removing all finite lines.
In order to proceed, you have to assume a difference between an infinite sequence of finite unions and an infinite union. But that will be difficult to explain.