In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 7 Apr., 23:25, William Hughes <wpihug...@gmail.com> wrote: > > D is the set of all finite lines, so it is not more than > > all finite lines. If you remove the collection of all > > finite lines then nothing remains. > > Consider the lines > 0.1 > 0.11 > 0.111 > ... > and so on. > > If you have the union 0.111..., then you can remove all finite lines > without removin 1/9.
WM may claim to be able to remove everything from a set and not have the result being the empty set, but it is WMs claims that are empty. > > > > You have agreed: If you remove any finite > > set E from D a non empty set (a different one for every E) > > remains. > > > > You have agreed: If you remove any finite set E from D > > what remains contains every natural number. > > Of course. > > > > 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 > > See above. There you can remove the collection of all finite lines > without removing the limit since it is not a finite line.
When one removes the collection of all finite lines from the collection of all finite lines, everywhere but in Wolkenmuekenheim, one ends with the empty set.
The "limit", being here the union of a set of lines, cannot exist for a set having no lines, even though WM claims it can.
> This is the > same if |N is an actually infinite set.
With actual infiniteness, one does not get non-empty unions of empty sets.