Thanks for that nice proof of Schroeder-Bernstein.

However, in your proof of Lemma: Any monotone function h: PX –> PX has a fixed point T, I guess the set you consider is the intersection of $latex \{S \in PX: h(S) \subseteq S\}$ instead of $latex \{S \in PX: S \subseteq h(S)\}$ (the intersection of the latter clearly is empty).

Foxx released hisfirst studio disc, Unpredictable, in December 2005. It debuted at number two, selling 598,000 copies in its first week….

(The captcha words for this post were “Mr Scientist,” so I had to write something suitably Baconian.)

Actually, I meant something a little different. I read Slawekk’s comment as dealing with the ease of formal verifiability by computers, but Lamport’s worked up about a different issue (he doesn’t work on computer verification). In his work with asynchronous distributed systems, things can get extremely subtle and counterintuitive. It’s easy to come up with a protocol that sounds like it should work, and even to find a compelling argument for why it should work, when the protocol is inherently defective. If you’re unlucky, the proof will sound good enough that you’re sure formalizing it is merely a matter of writing down a few trivial details. Such “proofs” often get published in this area. The problem is that the objects of study are tremendously complicated and frequently pathological. Plus, the details are crucial: it’s not enough to convince yourself that there exists a protocol that probably works; instead, you want to know whether this particular protocol definitely works.

Most of conceptual mathematics is quite different. There are certainly many subtleties and some pathologies, but there’s a lot of intuition underlying most of mathematics, and mathematicians have spent the last couple of centuries working hard to develop a powerful intuition about the subtleties (quantifying ordering as in uniform continuity, etc.). I don’t know whether this is even possible in Lamport’s area, but it hasn’t been done yet.

So, basically, Lamport honestly believes mainstream mathematicians regularly prove false theorems using techniques that simply cannot be made to work, and he believes this happens on all levels (not just for unimportant theorems in obscure journals nobody reads, but for major, famous theorems). In his view, if we carefully wrote out our work using two-column proofs, we’d find irreparable logical errors in about a third of our proofs. I’m convinced he’s nuts, at least as far as this opinion goes, and he’s convinced that mathematicians are deluding themselves and unwilling to risk trying this project.

Proof: Let T be the intersection of the set $latex \{S \in PX: S \subseteq h(S)\}$. Since h is monotone, $latex h(T) \subseteq h(S) \subseteq S$ for every $latex S$ in this set, and therefore $latex h(T) \subseteq T$. Applying the monotone function h again, $latex h h(T) \subseteq h(T)$. It follows that $latex h(T)$ belongs to this set, and therefore $latex T \subseteq h(T)$. Hence $latex T = h(T)$.

Speaking of Schroeder-Bernstein: most of the proofs I’ve seen are fussy and aesthetically unappealing to me. I prefer a more abstract proof as follows: suppose that f: X –> Y and g: Y –> X are injective. Let PX denote the power set of X. Define a monotone function PX –> PX which sends a subset S of X to not g(not f(S)), where “not” denotes complementation. If this function has a fixed point T [i.e., if g(not f(T)) = not T], then we can define a bijection X –> Y which takes x to f(x) if x belongs to T, and x to g^{-1}(x) if x belongs to not(T).

Lemma: Any monotone function h: PX –> PX has a fixed point T.
Proof: Let T be the intersection of the set {S in PX: h(S)