3 ms·
I'm not seeing this yet. Perhaps you could explain it to me in plain terms.
by sharpener 5y ago
I'm not seeing this yet. Perhaps you could explain it to me in plain terms.
- myWindoonn 5y agoPick 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.