Here's a list of errors we've found in the Fall 2006 Quantum Gravity Seminar notes. If you discover any more errors, please let me know, and we'll eventually try to correct them, or at least add them to this list.
John wrote:Some of you were wondering about which order to evaluate the expression in exercise 1.1. Now that I look at it, I see there's no ambiguity: Selinger packs this expression with enough parentheses to make it unambiguous. Just do stuff inside parentheses first!Also, something of the form λ f.λx.t is unambiguous, since λf.λx makes no sense; so it's λf.(λx.t). As for λz.z + 1, this is ambiguous if you can add a function and a number, since that is what (λz.z) + 1 does; but Selinger probably meant λz.(z + 1).
While I'm here, I'll make a few comments based on Derek's notes.
While you're contrasting classical logic with quantum logic here, there is a (different) contrast of classical with constructive logic (also called "intuitionistic" logic). I see the lambda calculus a lot in the metamathematics of constructive mathematics (which also involves a lot of Cartesian closed categories). So while the lambda calculus is classical as opposed to quantum, it's quite happily constructive as well.
Interestingly, the "easy proof" of existence of noncomputable functions is nonconstructive, while the "better proof" is constructive. Indeed, in the Russian (Markov's) school of constructive mathematics, it is accepted as a fact (axiomatic) that every total function from the set of natural numbers to itself is recursive/computable! (Thus, there is a set --the set of Turing machines that never halt-- with an injection to a countable set --the set of Turing machines-- and a surjection to an uncountable set --the set of total functions. This is consistent with, say, Aczel's constructive set theory CZF.) However, all constructivists (well, except finitists) must agree that the partial function given on the top of page 3 is uncomputable. (Using classical logic, you could turn this into a total function, and many classical mathematicians would do so, but it's not necessary.)
Also (this is less philosophical, more along the lines of errata), I've never seen the term "α-reduction", only "α-equivalence"; this is because there is no preferred direction (unlike with β-reduction, which simpifies (λx.fx)a to fa, for example, but would rather not expand fa to (λx.fx)a). Even "η-reduction" is viewed with some suspicion, because the correct direction seems to a matter of dispute. I see that Selinger's notes give a direction to it, but it's not the direction that you might expect!
Luckily, in the free cartesian closed category on one object X equipped with an isomorphism α : X ≅ hom(X,X), we have X ≠ hom(X,X). (I should say even more about this, but I won't now.)