To anyone else, I recommend starting at the 25 minute mark; that’s where Voevodsky starts discussing his own ideas. (Before that it’s just a review of very basic formal logic.)

The Q&A period begins at the 45 minute mark. It’s worth listening to the first question and Voevodsky’s reply. The questioner posed another possibility besides the three choices on slide 5: “We know we can’t prove the consistency of mathematics, but we can always hope.”

To reiterate, although I have a bemused view of Voevodsky’s philosophical claims (in this lecture), I don’t want to cast shade on the Univalent Foundations Project. Motivations are one thing, math another!

]]>And to get back to your dialogue with JB, my point in recalling Voevodsky’s conference was the ‘nebulousness’ you were discussing with JB could very well resonate with Voevodsky’s position in the talk and his proposal to overcome the shakiness of the foundations. ]]>

- 1. Univalent foundations naturally include ”axiomatization” of the categorical and higher categorical thinking.
- 2. Univalent foundations can be conveniently formalized using the class of languages called dependent type systems.
- 3. Univalent foundations are based on direct axiomatization of the ”world” of homotopy types instead of the world of sets.
- 4. Univalent foundations can be used both for constructive and for non-constructive mathematics

All these motivations strike me as more appealing than as a defense against some as yet undiscovered inconsistency.

As for the third bullet on slide 5: “Admit that the sensation of knowing in this case is an illusion and that the first order arithmetic is

inconsistent.” I’m sort-of OK with the first half (depending on what you mean by “illusion”), but the inconsistency of PA just doesn’t follow at all.

Slides 17 and 18 (on constructing “reliable” proofs) made me think of George Kreisel’s critique of Hilbert’s program:

…the finitist reduction … consisted [for Hilbert] in clearing the fair name of mathematics which had been sullied by the paradoxes. [Hilbert also said] that the paradoxes simply have nothing to do with the theory of sets of numbers: it is hard to see why this remark, if true, has not cleared the name at least of analysis unless one believes that stained reputations can only be cleared with a great deal of ceremony.

We haven’t found an inconsistency in ZFC, even fortified with large cardinal axioms, for a good long while! If the issue is, “how can we ever trust set theory (or even arithmetic) again?” after Gödel’s theorem—well, like the advice columnists always say, you rebuild trust one day at a time.

IMO, Voevodsky is pulling a fast one on slide 5. The trick is the slipperiness of “know”. We “know” that PA is consistent only in the sense that (most of us) believe it pretty strongly, just like we believe we’re not really lobsters dreaming we’re humans. Voevodsky gives the main reason for this belief (the consistency of PA, not the thing about lobsters) in the first bullet on the next slide: the “formulas as subsets” interpretation of the induction schema. Well of course that *does* lead to a proof of consistency of sorts, namely the usual proof of Con(PA) in ZF. Claiming that this sort of “knowledge” means that “second incompleteness theorem is false as stated” is a non-sequitur.

The third bullet on slide 5 likewise contains a non-sequitur. So I guess I’d plump for the second bullet, except that I’d amend “‘transcendental’, provably unprovable knowledge” to “I’m really strongly convinced of this, but if an inconsistency in PA turns up tomorrow, I won’t kill myself.”

]]>(The list is ever-renewing—it’s *always* 57 items long, no matter how much I learn. Kind of like the bowl in Greek mythology. Now if *everyone else* would just stop discovering new stuff…)

“Types” are objects. The “untyped” λ-calculus should be called the “one-typed” λ-calculus, because it has just one type. While it’s bizarrely fun (and insanely dangerous) to have all data be of the same type when you’re doing computation, it’s sadly limiting and a bit mystifying from the category-theoretic point of view to focus on categories with just one object.

I recommend Lambek and Scott’s book for more on all this, or my lecture notes.

]]>