The topology of the set of all types

• 30 March 2012
• Martin Escardo
• Computation, Constructive math, Guest post

It is well known that, both in constructive mathematics and in programming languages, types are secretly topological spaces and functions are secretly continuous. I have previously exploited this in the posts Seemingly impossible functional programs and A Haskell monad for infinite search in finite time, using the language Haskell. In languages based on Martin-Löf type theory such as Agda, there is a set of all types. This can be used to define functions $\mathbb{N} \to \mathrm{Set}$ that map numbers to types, functions $\mathrm{Set} \to \mathrm{Set}$ that map types to types, and so on.

Because $\mathrm{Set}$ itself is a type, a large type of small types, it must have a secret topology. What is it? There are a number of ways of approaching topology. The most popular one is via open sets. For some spaces, one can instead use convergent sequences, and this approach is more convenient in our situation. It turns out that the topology of the universe $\mathrm{Set}$ is indiscrete: every sequence of types converges to any type! I apply this to deduce that $\mathrm{Set}$ satisfies the conclusion of Rice's Theorem: it has no non-trivial, extensional, decidable property.

To see how this works, check:

• A short paper with the proofs in mathematical vernacular, and further discussion of the intuitions, motivations and consequences.
• Literate proofs in Agda of the universe indiscreteness theorem and Rice's Theorem for the universe.
• Agda proofs of related facts.

The Agda pages can be navigated be clicking at any (defined) symbol or word, in particular by clicking at the imported module names.

Comments

alaska

29 November 2012 at 04:50

Thanks for taking the time to discuss this, I feel strongly about it and love learning more on this topic. If possible, as you gain expertise, would you mind updating your blog with more information? It is extremely helpful for me...thanks

Circles, Contractible Spaces, and the Axiom of Choice | infinite cardinals

12 May 2014 at 21:38

[…] Martín Escardó points out, “types are secretly topological spaces and functions are secretly continuous.” This […]