I just noticed something funky. Let $X$ denote an $I$-indexed family of sets. There is a projection $$\pi_X: \bigsqcup_{i:I} X_i \rightarrow I.$$ It isn’t necessarily surjective, of course, because one or more of the $X_i$ may be empty. Anyway, I noticed that the set $\prod_{i:I} X_i$ can be identified with the set of sections of […]

The standard way of using ZFC to encode the rest of mathematics is sometimes criticized because it introduces unnecessary, strange properties such as, for example $1\in 2$ if we encode integers by ordinals. From a constructivist type theorist’s perspective, uples or functions or complex numbers should have nothing to do with the membership relation, we […]

My question is related to the formal presentation of type theory as stated in the context of Homotopy Type Theory. Every formalization is grounded on typing judgements like $$ a: A $$ where mostly it is said that $a$ is an object that can be built due to several formation or term-forming rules and $A$ […]

What are some good online/free resources (tutorials, guides, exercises, and the like) for learning Lambda Calculus? Specifically, I am interested in the following areas: Untyped lambda calculus Simply-typed lambda calculus Other typed lambda calculi Church’s Theory of Types (I’m not sure where this fits in). (As I understand, this should provide a solid basis for […]

As an exercise in HoTT basics, I am trying to construct a term that has the type $Id_{Nat}(S(O),O)\to\bot$. If this were a Coq proof, I’d be done after a single inversion on the premise, as the impossible identity would leave zero cases to consider. I guess I could do something similar here, but I’m not […]

I like to distinguish between sets and subsets. We imagine that sets are floating free in the universe, and that the elements of a set are constructed according to some kind of recursive rules. Like the elements of $\mathbb{N}$ can be constructed using the rules: $$\frac{}{0 \in \mathbb{N}} \;\; \frac{n \in \mathbb{N}}{n+1 \in \mathbb{N}}$$ On […]

This is similar to What does it take to divide by $2$? about $(A\sqcup A\cong B\sqcup B)\Rightarrow A\cong B$ which is valid in $\textsf{ZFC}$ by using cardinalities and also in $\textsf{ZF}$ by some combinatorial argument, but where it remained unclear whether constructive arguments suffice. This questions concerns products instead of disjoint union, so: Question: Does […]

I’ve picked up the Homotopy Type Theory book for leisure. I’m comfortable with strongly typed languages and familiar with dependently typed languages and I enjoy topology, so I thought that the HoTT book was a good opportunity to learn some of the math underlying the type systems (and, by HoTT’s reputation, a new way of […]

I’ve been trying to find a proof that the pullback functors in a locally cartesian closed category have right adjoints (used to model the notion of indexed product inside a category (rather than indexed by a set), or, equivalently, dependent products in models of dependent type theories). I found a proof in Awodey’s book, but […]

I am becoming increasingly convinced that Wildberger’s views are, if a little bizarre, at least not hopelessly inconsistent. When I was reading the comments in the video following (MF17), somebody said something that shocked me a bit, because I was unable to give a rebuttal that I found satisfactory: The reason I consider [the axiom […]

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