Hi, Lawvere pummelled your position into the ground a while ago: http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf Your critique involves repeatedly crossing the boundary between the inside and outside of the system in question; Lawvere works entirely inside the system, and shows that the paradoxes of self-reference arise from our interpretations. https://arxiv.org/abs/math/0305282v1 explains with many examples.
Hi downvoters: Use your words when somebody is wrong. Your downvotes aren't helpful here for finding the truth.
Lawvere's paper appears to suggest that if I refute one diagonal argument then I refute them all. Would you consider that to be an accurate description?
If there exists t : Y -> Y such that t;y != y for all y : 1 -> Y then for no A does there exist a surjection A -> (A -> Y).
(He actually says something much stronger.) Note that the first half of this is saying "if there exists t such that t has no fixed points..."
Let our category be Set, the category of sets and functions; it is well-known to be Cartesian closed. Let A be the set of natural numbers and let Y be the Booleans. Then Lawvere is saying that there is no surjection N -> (N -> 2), and thus definitely no bijection, because there is a function 2 -> 2 with no fixed points: the negation function which swaps true and false has no fixed point.
It does not get much plainer without actually reading Lawvere and/or Yanofsky directly, sorry. I hope that this helps explain how inescapable this sort of theorem is.
Hi downvoters: Use your words when somebody is wrong. Your downvotes aren't helpful here for finding the truth.