[myWindoonn]

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.

[ProfHewitt]

Lawvere's work is elegant! However, Lawvere missed the

crucial importance of Russell's orders on propositions in

blocking the construction of monster propositions using

recursive definitions. Orders on propositions block

construction of I'mFalse, I'mNotSelfapplicable,

I'mUnprovable, and MyTheoremsAreEnumerable.

See the following video for more information:

https://www.youtube.com/watch?v=AJP1VL7shiI

for the following article:

https://papers.ssrn.com/abstract=3603021

[sharpener]

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?

[sharpener]

I'm not seeing this yet. Perhaps you could explain it to me in plain terms.

[myWindoonn]

Pick any Cartesian closed category. Says Lawvere:

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.