Re: Cantor and the binary tree



Virgil wrote:

> > If you prefer "is", you may use it. That does not matter. It is
> > obviously impossible that the set of paths is uncountable when the set
> > of nodes is countable, because every pair of paths springs off from one
> > node, while one path leads to that node. Try to find an error n the
> > arguing, not in the result.
>
> For finite paths

There are no finite paths in my tree. Terminating rationals are
completed by strings of zeros. For that sake enough zeros are in the
tree.
>
> The set of these unbounded paths can mapped bijectively to P(N), whereas
> the set of nodes can be mapped bijectively to N.

Consistency of set theory is questioned, hence I do not accept Cantor's
proof as an argument.

line number n
0 0.
1 0 1
2 0 1 0 1
... ..................


1) Each real number of (0,1) is given by a path stretching over
infinitely many nodes (bits).
2) All nodes (bits) of the tree belong to a countable set.
3) A node can only exist within a path.
4) Any node increases the number of paths by 1 from 1 coming in, to 2
going out. 2 - 1 = 1.
5) Any node increases the number of nodes by 1.

Please point out which of these simple steps is (are) wrong.

Regards, WM

.

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