T h e i n t r i n s i c t o p o l o g y o f a M a r t i n - L o f u n i v e r s e Martin Escardo, University of Birmingham, UK. February 2012, updated 17 Feb 2012, 06 April 2012. This is a proof in intensional Martin-Lof type theory, extended with the propositional axiom of extensionality as a postulate, written in Agda notation. The K-rule or UIP axiom are not used, except in a few instances where they can be proved. The proof type-checks in Agda 2.3.0. A b s t r a c t. We show that a Martin-Lof universe `a la Russell is topologically indiscrete in a precise sense defined below. As a corollary, we derive Rice's Theorem for the universe: it has no non-trivial, decidable, extensional properties. I n t r o d u c t i o n This universe indiscreteness theorem may be surprising, because types like the Cantor space of infinite binary sequences are far from indiscrete in the sense considered here, as they have plenty of decidable properties. The Cantor space also fails to be discrete, because it doesn't have decidable equality, and this fact shows up in the proof of Rice's Theorem. We need to postulate the axiom of extensionality, but nothing else (the univalence axiom would give a slightly sharper result). In particular, Brouwerian continuity axioms are not postulated, even though this is about topology in constructive mathematics. We show that the universe Set, in Agda notation, is indiscrete, in the sense that every sequence of types converges to any desired type. Convergence is defined using ℕ∞, the generic convergent sequence, constructed in the module GenericConvergentSequence, but briefly introduced below. For the sake of motivation, let's define convergence for sequences of elements of types first. We say that a sequence x : ℕ → X in a type X converges to a limit x∞ : X if one can construct a "limiting sequence" x' : ℕ∞ → X such that x n = x'(under n) x∞ = x' ∞ where under : ℕ → ℕ∞ (standing for "underline") is the embedding of ℕ into ℕ∞. It is easy to see that every function of any two types now becomes automatically continuous in the sense that it preserves limits, without considering any model or any continuity axiom within type theory. The collection of convergent sequences defined above constitutes the intrinsic topology of the type X. This is motivated as follows. There is an interpretation of type theory (Johnstone's topological topos) in which types are spaces and all functions are continuous. In this interpretation, ℕ is the discrete space of natural numbers and the space ℕ∞ is the one-point compactification of ℕ. Moreover, in this interpretation, convergence defined in the above sense coincides with topological convergence. Using a general construction by Streicher, assuming a Grothendieck universe in set theory, one can build a space in the topological topos that is the interpretation of the universe. Voevodsky asked what the topology of this interpretation of the Martin-Lof universe is. I don't know the answer, but it follows from what we prove here that the quotient by type isomorphism is the indiscrete topology. (Moreover, I conjecture that the Grothendieck universe with the indiscrete topology can be given the structure needed to interpret a Martin-Lof universe `a la Russell. But this may be a bit too audacious.) A space is indiscrete if the only open sets are the empty set and the whole space. It is an easy exercise, if one knows basic topology, to show that this is equivalent to saying that every sequence converges to any point. The appropriate notion of equality for elements of the universe Set of types is isomorphism. Hence we reformulate the above definition for limits of sequences of types as follows. We say that a sequence of types X : ℕ → Set converges to a limit X∞ : Set if one can find a "limiting sequence" X' : ℕ∞ → Set such that X n ≅ X'(under n) X∞ ≅ X' ∞ If one assumes the univalence axiom, one can replace the isomorphisms by equalities to get an equivalent notion. But notice that in the topological topos isomorphism is not the same thing as equality. In this Agda module we show that every sequence of types converges to any type whatsoever. This explains, in particular, why there can't be non-trivial extensional functions Set → ₂, where ₂ is the discrete type of binary numbers. Such functions are the (continuous characteristic functions of) clopen sets of the universe, and indiscreteness shows that there can be only two of them, so to speak. This is Rice's Theorem for the universe Set. (NB. The auxiliary modules develop much more material than we need (and many silly things on the way - apologies).) \begin{code} {-# OPTIONS --without-K #-} module TheTopologyOfTheUniverseOld where open import Cantor open import CurryHoward open import Equality open import Extensionality open import GenericConvergentSequence open import IsomorphismOld open import Naturals hiding (_+_) hiding (_≣_) open import SetsAndFunctions open import Two open import HSets \end{code} The following is the crucial construction. It attaches the singleton type 𝟙 as the limit to any given sequence of types. The idea is that if u ≡ n then a function of type u ≡ n → X n simply picks an element of X n, so that (u ≡ n → X n) ≅ X n, and otherwise such a function has empty graph so that (u ≡ n → X n) ≅ 1. In particular, if u ≡ ∞, then the latter holds for every n, so that X' ∞ ≅ (n : ℕ) → 1 ≅ 1 \begin{code} attach-𝟙 : (ℕ → Set) → (ℕ∞ → Set) attach-𝟙 X u = (n : ℕ) → u ≣ n → X n \end{code} Notice that this is isomorphic to (s : Σ \(n : ℕ) → u ≣ n) → X(π₀ s) by (un)currying. We first show that the constructed limiting sequence extends the given sequence: \begin{code} _[_] : (ℕ∞ → Set) → (ℕ → Set) X [ i ] = X(under i) attach-𝟙-lemma : ∀(X : ℕ → Set) → ∀ i → attach-𝟙 X [ i ] ≅ X i attach-𝟙-lemma X i = (g i , f i , funext(S i) , funext(R i)) where f : (i : ℕ) → X i → attach-𝟙 X [ i ] f i x n r = subst {ℕ} {X} (under-mono r) x g : (i : ℕ) → attach-𝟙 X [ i ] → X i g i h = h i refl R-lemma : ∀ i → ∀(h : attach-𝟙 X [ i ]) → ∀ n → ∀(r : under i ≡ under n) → subst (under-mono {i} {n} r) (h i refl) ≡ h n r R-lemma i h n r = trans (lemma q h) (cong (h n) (ℕ∞-hset (cong under q) r)) where q : i ≡ n q = under-mono r A : ∀ i n → i ≡ n → Prp A i n q = ∀(h : attach-𝟙 X [ i ]) → subst q (h i refl) ≡ h n (cong under q) lemma : ∀{i n} → ∀(q : i ≡ n) → A i n q lemma = J A (λ i h → cong (h i) refl) R : ∀ i h → f i (g i h) ≡ h R i h = funext(λ n → funext(R-lemma i h n)) S : ∀ i x → g i (f i x) ≡ x S i x = cong (λ r → subst {ℕ} {X} r x) lemma where lemma : under-mono refl ≡ refl lemma = ℕ-hset (under-mono refl) refl \end{code} And then we show that the added limit point is what we claimed: \begin{code} attach-𝟙-lemma∞ : ∀(X : ℕ → Set) → attach-𝟙 X ∞ ≅ 𝟙 attach-𝟙-lemma∞ X = only-one-iso the-only-element lemma₁ where the-only-element : attach-𝟙 X ∞ the-only-element n r = unique-from-∅(∞-is-not-ℕ n r) lemma₀ : ∀(f g : attach-𝟙 X ∞) → ∀ n r → f n r ≡ g n r lemma₀ f g n r = unique-from-∅(∞-is-not-ℕ n r) lemma₁ : ∀(f g : attach-𝟙 X ∞) → f ≡ g lemma₁ f g = funext(λ n → funext(lemma₀ f g n)) \end{code} The following follows the above pattern: We give a series of constructions of limiting sequences, each one followed by two lemmas, one saying what the limiting sequence is and the other saying what its limit (value at ∞) is. We begin with a particular case of the above, for the sake of clarity, and because it is used twice: the constant sequence ∅, where ∅ is the empty type, converges to the singleton type 𝟙: \begin{code} constant-∅-converging-to-𝟙 : ℕ∞ → Set constant-∅-converging-to-𝟙 = attach-𝟙(λ i → ∅) constant-∅-converging-to-𝟙-lemma : ∀(i : ℕ) → constant-∅-converging-to-𝟙 [ i ] ≅ ∅ constant-∅-converging-to-𝟙-lemma = attach-𝟙-lemma(λ i → ∅) constant-∅-converging-to-𝟙-lemma∞ : constant-∅-converging-to-𝟙 ∞ ≅ 𝟙 constant-∅-converging-to-𝟙-lemma∞ = attach-𝟙-lemma∞(λ i → ∅) \end{code} The constant sequence ∅ converges to any desired type Y, by multiplying the previous sequence pointwise by Y: \begin{code} constant-∅-converging-to-anything : Set → (ℕ∞ → Set) constant-∅-converging-to-anything Y u = (constant-∅-converging-to-𝟙 u) × Y constant-∅-converging-to-anything-lemma : ∀(Y : Set) → ∀(i : ℕ) → constant-∅-converging-to-anything Y [ i ] ≅ ∅ constant-∅-converging-to-anything-lemma Y i = ≅-trans (lemma[X≅X'→X×Y≅X'×Y] (constant-∅-converging-to-𝟙-lemma i)) lemma[∅×Y≅∅] constant-∅-converging-to-anything-lemma∞ : ∀(Y : Set) → constant-∅-converging-to-anything Y ∞ ≅ Y constant-∅-converging-to-anything-lemma∞ Y = ≅-trans (lemma[X≅X'→X×Y≅X'×Y] constant-∅-converging-to-𝟙-lemma∞) lemma[𝟙×Y≅Y] \end{code} The constant sequence 𝟙 converges to ∅, by applying the "negation" function X ↦ (X → ∅) to the constant sequence ∅ that converges to 𝟙. \begin{code} constant-𝟙-converging-to-∅ : ℕ∞ → Set constant-𝟙-converging-to-∅ u = (constant-∅-converging-to-𝟙 u) → ∅ constant-𝟙-converging-to-∅-lemma : ∀(i : ℕ) → constant-𝟙-converging-to-∅ [ i ] ≅ 𝟙 constant-𝟙-converging-to-∅-lemma i = ≅-trans (lemma[X≅X'→[X→Y]≅[X'→Y]] (constant-∅-converging-to-𝟙-lemma i)) lemma[[∅→∅]≅𝟙] constant-𝟙-converging-to-∅-lemma∞ : constant-𝟙-converging-to-∅ ∞ ≅ ∅ constant-𝟙-converging-to-∅-lemma∞ = ≅-trans (lemma[X≅X'→[X→Y]≅[X'→Y]] constant-∅-converging-to-𝟙-lemma∞) lemma[[𝟙→∅]≅∅] \end{code} By multiplying the previous limiting sequence pointwise with the sequence that converges to one, we get that any sequence converges to ∅: \begin{code} attach-∅ : (ℕ → Set) → (ℕ∞ → Set) attach-∅ X u = (attach-𝟙 X u) × (constant-𝟙-converging-to-∅ u) attach-∅-lemma : ∀(X : ℕ → Set) → ∀(i : ℕ) → attach-∅ X [ i ] ≅ X i attach-∅-lemma X i = ≅-trans (lemma[X≅X'→Y≅Y'→[X×Y]≅[X'×Y']] (attach-𝟙-lemma X i) (constant-𝟙-converging-to-∅-lemma i)) lemma[Y×𝟙≅Y] attach-∅-lemma∞ : ∀(X : ℕ → Set) → attach-∅ X ∞ ≅ ∅ attach-∅-lemma∞ X = ≅-trans (lemma[X≅X'→Y≅Y'→[X×Y]≅[X'×Y']] (attach-𝟙-lemma∞ X) (constant-𝟙-converging-to-∅-lemma∞)) lemma[Y×∅≅∅] \end{code} Finally, by adding pointwise a sequence that converges to ∅ to the constant sequence ∅ converging to anything, we get any sequence X to converge to any type Y we please: \begin{code} attach : (ℕ → Set) → Set → (ℕ∞ → Set) attach X Y u = (attach-∅ X u) + (constant-∅-converging-to-anything Y u) attach-lemma : ∀(X : ℕ → Set) → ∀(Y : Set) → ∀ i → attach X Y [ i ] ≅ X i attach-lemma X Y i = ≅-trans (lemma[X≅X'→Y≅Y'→[X+Y]≅[X'+Y']] (attach-∅-lemma X i) (constant-∅-converging-to-anything-lemma Y i)) lemma[Y+∅≅Y] attach-lemma∞ : ∀(X : ℕ → Set) → ∀(Y : Set) → attach X Y ∞ ≅ Y attach-lemma∞ X Y = ≅-trans (lemma[X≅X'→Y≅Y'→[X+Y]≅[X'+Y']] (attach-∅-lemma∞ X) (constant-∅-converging-to-anything-lemma∞ Y)) lemma[∅+Y≅Y] \end{code} We pack the previous three lemmas together to get the Universe Indiscreteness Theorem, which says that any type Y : Set can be attached as a limit of any given sequence X : ℕ → Set: \begin{code} Universe-Indiscreteness-Theorem : ∀(X : ℕ → Set) → ∀(Y : Set) → (∀ i → attach X Y [ i ] ≅ X i) ∧ attach X Y ∞ ≅ Y Universe-Indiscreteness-Theorem X Y = ∧-intro (attach-lemma X Y) (attach-lemma∞ X Y) \end{code} As a corollary of The Universe Indiscreteness Theorem, we get Rice's Theorem for the universe, which can be found in the module RicesTheoremForTheUniverse.