On 9 Apr., 20:15, William Hughes <wpihug...@gmail.com> wrote:
> > > You said that the Binary Tree, when constructed from its FISONs, does > > not contain their suprema. > > nope.
So it contains the suprema? Uncountably many? > > > > > You said that the sequence > > 0.1 > > 0.11 > > 0.111 > > ... > > does not contain 1/9. > > yep > > > But this list contains an infinite sequence of > > finite unions. > > Nope. The list only contains lines.
Every line is the union of itself and all its predecessors. Unless you 'd like to consider indices, consider the sequences of FISONs
1 1, 2 1, 2, 3 ...
There it is clear.
> No line is an infinite > sequence of finite unions.
But if the list contains infinitely many (more than any finite number of) FISONs, then it contains infinitely many (more than any finite number of) unions, doesn't it?
> So now we have: > > D is the collection of all finite > lines. > If you remove the collection of all finite > lines from D
i.e., if you remove, according to induction, with FIS n also FIS n+1
> nothing is left > If you remove any one line (and all its predecessors) > every natural number is left. > > This is exactly what you keep saying > is a contradiction.-
Not at all! A contradiction would only exist, if the list would be or contain an infinite set. A contradiction is your claim:
0.1 0.11 0.111 ...
does not contain 1/9, infinitely many appended 1's are not sufficient to yield the decimal fraction of 1/9, but *there is* a decimal fraction of 1/9. And this is accomplished by the union of all lines (if not understandable use the FISONs above and |N instead of 1/9).
And a contradiction to one of these claims will be your statement about the infinite paths in the Binary Tree, whether you claim the supremum being present (infinite union of FISONs yields |N) or not (infinite sequence of FISONs does not contain |N).
This is the contradiction: In my list the infinite sequence is the infinite union. According to set theory both yield different results with respect to the existence of the limit.