Diagonal arguments and cartesian closed categories with author commentary

Lawvere uses category theory to formulate his result, but there´s an exposition not using category theory by Yanofsky:

A universal approach to self-referential paradoxes, incompleteness and fixed points

Both papers are freely available.

]]>It’s too bad you are not aware of current research in the history of mathematics. You should educate yourself:

Ryskamp, John Henry, “Paradox, Natural Mathematics, Relativity and Twentieth-Century Ideas” (May 19, 2007). Available at SSRN: http://ssrn.com/abstract=897085

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Also, you can use latex code in WordPress blogs now (including the comments) just by enclosing the relevant stuff with dollar signs (as usual) as well as the word “latex” after the first dollar sign.

And thanks for pointing out the typo – I’ll fix it now.

]]>Actually, I view the fundamental result in Cantor’s proof as something even more general, a sort of fixed-point result: Given f : A -> (A ->B) and g : B -> B, there is some D : A -> B such that D is very simply definable from f and g, and satisfies the property that the image under D of the preimage under f of D contains only fixed points of g. Furthermore, this can be done uniformly, in the sense that there is a fixed definition of D from f and g which works no matter what f and g (and, for that matter, A and B) are. That fixed definition, of course, is the one of diagonalization, giving D a = g(f a a).

Cantor’s Theorem in its conventional form then follows from taking B to be the type of truth values and g the operation of negation; since g has no fixed points, it will follow that D cannot be in the range of f. Indefinability of satisfaction follows as above. But also, rather than focusing on such impossibility results, we can instead consider cases where f and g are such that f’s range must contain everything definable from f and g in an appropriately simple way, and then what the theorem gives us are fixed-point existence results; for example, versions of Kleene’s recursion theorem can be viewed through this lens as simple instances of Cantor’s result, as can the existence of fixed point combinators in the lambda calculus (the one that falls right out of Cantor’s proof would be the plain-vanilla Y combinator).

]]>Anyway, that’s quite interesting to see that something like Cantor’s Theorem is the basic result here. Although, if you’re talking about a surjection onto *definable* predicates, then there’s something like a truth or satisfaction notion right there in the notion of definability. So it’s not totally clear to me which one I would put as more basic. Cantor’s Theorem was understood decades before Gödel’s theorems, but it’s also true that the liar paradox was known millenia before that.

Of course, this is exactly the same, except for perhaps the perspective of the approach, as the argument above, but it is the perspective which impresses me; that the indefinability of satisfaction is pretty much the same fact, in some sense, as Cantor’s Theorem, despite the latter being presumably so much better known than the former.

I then think of the indefinability of truth under standard encodings of sentences as then something of a technical curiosity which happens to arise as a consequence of this more fundamental discovery (of Cantor’s Theorem in its full strength). This would be similar to, say, knowledge of the fact that the decision problem for first-order logic is undecidable in the presence of a single binary relation or of two unary functions or whatever the various minimality results are, I don’t even know; it’s an interesting, in a way, that the machinery needed for the fundamental result can be reduced, through various tricks, to such things, but the real crux of the phenomenon, the location of all the important action and where to go to find insight, would be in the pre-reduced version. (The incompleteness of PA is a similar thing; it’s interesting that the machinery for encoding sequences, and through them computable functions, can be reduced to natural number addition and multiplication, but this is also basically a parlor trick)

Of course, since discussions of TRUTH are discussions of a big, important, capital-letter word in a way which discussions of satisfaction or surjections into power sets are not, it’s to be expected that people will naturally get caught up in viewing the phenomenon from that end. But if one really wants to wrap one’s mind around it, I think thinking about the phenomenon as fundamentally a phenomenon about truth is misguided.

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