Binary Tree and Pairs of Nodes

A path in an infinite binary tree is the representation of a real
number of the interval [0, 1]. (Terminating rational numbers have two
representations each.)

Two paths A and B in the infinite binary tree can be distinguished by
at least one pair of nodes a and b where a belongs to A but not to B,
and b belongs to B but not to A.

Assumption: For every pair of paths we can find a distinguishing pair
of nodes where no node is used to distinguish another pair of path.

Conclusion: The number of paths is not larger than the number of
nodes. The number of nodes is countable. As the number of paths is not
less than the number of reals in the interval [0, 1], the number of
reals in the interval [0, 1] is countable.

Or: The assumption is wrong. There are at least two pairs of paths
that cannot be distinguished by different nodes.

Regards, WM
.

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