Re: Cantor Confusion


Virgil schrieb:


The union of all such restricted infinite binary trees will be a subtree
of the compete infinite binary tree.

So far, I hope WM and I agree.

Of course.

The issue between us is whether that union will be a proper subtree of
the complete infinite binary tree or will be the whole tree.

This question can easily be decided by finding a node which is in the
whole tree but not in the union. Should that be impossible, then a
property of the subtree is that it is not a proper subset but the
complete tree.

Since every path in every one of the restricted infinite binary trees in
the union is eventually constant, every path in the union of all those
trees will also be eventually constant.

That would be correct for a finite union of finite trees. What is
correct for the finite case need not be correct for the infinite case.

So that, for example, the union will not contain any path which
alternates between branching left and branching right.

That would be correct for a finite union of finite trees. What is
correct for the finite case need not be correct for the infinite case.

Further, it will not contain any path which has infinitely many
branchings in both directions.

That would be correct for a finite union of finite trees. What is
correct for the finite case need not be correct for the infinite case.

Regards, WM

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