As a side note, I think the B in BHK is more honorific for his position in philosophy of mathematics.

A minor point: has structures in even higher dimensions, which might not fit into the three-dimensional type theory.

I’m not expert on this topic, but I like to think of free choice sequences in terms of processes. It’s a stream whose generator is inaccessible to you, coming from an independent process. You’re required to reason in such a way that you impose no assumptions about how that process generates its outputs. I find it fascinating that Brouwer used this as his conception of the continuum. This contrasts with “recursive analysis” which represents reals as Cauchy sequences given by a Turing machine index. It’s not very intuitionistic in that it is self-consciously computable, whereas the beauty of intuitionism is that you just “do math” in such a way that a computable interpretation is always available.

Update: I misinterpreted the Constable and Bickford paper. Their result is conditional on the Fan “Theorem”, and Th. Coquand pointed out to me that FT is not provable in ETT, contrary to what I said above. Apologies to all. (Perhaps it should be called the Fan Principle or somesuch when used in contexts in which it is not a theorem.)

It’s worth noting that Brouwer was motivated to invent intuitionism out of his distaste for the idea that the continuum is a set (i.e., the natural numbers come out of the “first act” of intuitionism, and the continuum comes from the “second act”). In HTT this has a very cute formalization: the interval is a higher inductive type with two points and a declared path between them, and so is trivially not an h-set!

But it’s unclear to me whether this is sufficient to capture his thinking.

1. What is the status of the fan theorem? I don’t know enough about HTT to guess what properties like compactness look like in this setting.

2. He also posited free choice sequences as part of intuitionistic analysis, and they are weird. They look like streams, but they aren’t, since (a) there is no rule to produce them, but (b) streams necessarily have such a rule (since they are all constructed from an unfold).