Aristotelian logic is nothing more (nor less) than a path through a binary tree. Change the structure of all the paths and all the “conclusions” follow that change.
Arguments in mathematics can take one of two forms: proof by contradiction or by “showing” (a la Wittgenstein) via construction.
I have shown (the “Real deal” and “Real deal, too” posts) via construction that the diagonal “proof” by contradiction is only the result of ignoring part of the picture…by an arbitrary truncation of the full range of structures for binary trees.
It is by “showing” alternate constructions of binary trees that we can see the shortfall inherent in the diagonal “proof”.
Cantor’s construction of the diagonal arbitrarily leaves out a vital degree of freedom that exists. It cannot be got rid of arbitrarily….that changes nothing of the reality of its existence.
There exist non-trivial automorphisms of “R”. They are there to be seen in the DD and anti-DD trees. No proof by contradiction can deny that fact. Any such “proof” assumes a certain structure of “R” that it claims to prove and is thereby simply circular. Poincaré was correct. It is an arbitrary restriction of a real existing degree of freedom inherent in the set of binary strings.
21st Century Paradox 