On 8 Apr., 13:51, William Hughes <wpihug...@gmail.com> wrote: > Let D be the set of all finite lines. > > Do you agree with > > If you remove the collection of finite > lines from D you get the empty set. > > Yes or No
It depends on your open answers:
1) Why do you think that the union of FISONs of paths is not a supremum?
2) If 0.111... is not in the list 0.1 0.11 0.111 ... Why do you think so? How can it be distinguished from the finite lines? (Don't answer "by having no last element", because that cannot be checked by the infinite sequence itself but only by its finte definition.) How can it be the infinite union but not the result of infinitely many finite unions?
Given you cannot answer these questions but insist that 0.111... has more indices than every finite line, then my answer is clear: No.
From that what you believe to exist (although it cannot exist without contradictions) you can remove every finite line without reaching the empty set.