> The set of finite things has larger cardinality m than any finite > number n: > For all n exists m : m > n
True, if m is indeed, a cardinality, and not a finite number .
> But there is no cardinality m that is larger than every finite number > n > Exists m for all n : m > n > is wrong.
http://en.wikipedia.org/wiki/Axiom_of_infinity The axiom of infinity, one of the basic axioms of set theory, says precisely that m exists . If you reject it,not only are we left unable to talk about real numbers ,banning them as invalid, we are left unable to talk about your binary tree .
Because it contains ALL Finite initial segments , the binary tree should not exist . It's wrong .
We've been using 'magic'/infinity in our mathematics up until now . An important point to make is IT WORKS . The same way we've been using 'i' in our mathematics .Modern physics is based on 'i' . As long as it will CONTINUE TO WORK , we will CONTINUE TO USE IT .
Opinions on infinity are varied, but the paradoxes you keep rambling about DON'T EXIST FOR ANYONE ELSE .Nor do you have any results to show. Unless you manage to understand that, you'll make no further progress .
"Wanting to reform the world without discovering one's true self is like trying to cover the world with leather to avoid the pain of walking on stones and thorns. It is much simpler to wear shoes." - Ramana Maharshi
"Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties."