[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.