Hello, HoTT community.

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I've learned a bit about HoTT in bits of spare time over the = past few years, and have come up with some questions I can't answer on = my own.=C2=A0 It was suggested that I ask them on this list.=C2=A0 I will s= tart with a few small questions, and if anyone in the community here has ti= me to answer them, then I'll continue with others as needed.=C2=A0 Than= k you in advance for any help you can give.

(Where= I'm coming from:=C2=A0 I'm a mathematician; my dissertation was on= intermediate logics, but I haven't focused on logic much for the past = 15 years, instead doing mathematical software and some applied mathematics.= =C2=A0 I have a passion for clear exposition, so as I learn about HoTT, I p= rocess it by writing detailed notes to myself, explaining it as clearly as = I can.=C2=A0 When I can't explain something clearly, I flag it as a que= stion.=C2=A0 I'm bringing those questions here.)

Here are three to start.
1. Very early in the HoTT book, = it talks about the difference between types and sets, and says that HoTT en= courages us to see sets as spaces.=C2=A0 Yet in a set of lecture videos Rob= ert Harper did that I watched on YouTube (which also seem to have disappear= ed, so I cannot link to them here), he said that Extensional Type Theory ta= kes Intuitionistic Type Theory in a different direction than HoTT does, for= malizing the idea that types are sets.=C2=A0 Why does the HoTT book not men= tion this possibility?=C2=A0 Why does ETT not seem to get as much press as = HoTT?
2. When that same text introduces judgmental equality, it claims= that it is a decidable relation.=C2=A0 It does not seem to prove this, and= so I suspected that perhaps the evidence was in Appendix A, where things a= re done more formally (twice, even).=C2=A0 The first of these two formalism= s places some restrictions on how one can introduce new judgmental equaliti= es, which seem sufficient to guarantee its decidability, but at no point is= an algorithm for deciding it given.=C2=A0 Is the algorithm simply to apply= the only applicable rule over and over to reduce each side, and then compa= re for exact syntactic equality?
3. One of my favorite things about Ho= TT as a foundation for mathematics actually comes just from DTT:=C2=A0 Once= you've formalized pi types, you can define all of logic and (lots of) = mathematics.=C2=A0 But then the hierarchy of type universes seem to require= that we understand the natural numbers, which is way more complicated than= just pi types, and thus highly disappointing to have to bring in at a foun= dational level.=C2=A0 Am I right to be disappointed about that or am I miss= ing something?
Thanks in advance for any help you have = time and interest to provide!

Nathan Carter
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