Martin Escardo, December 2012, based on earlier work, circa 2010. Searchable ordinals via squashed sums (without using the Cantor space). We can define plenty of searchable sets by transfinitely iterating squashed sums. These are countable sums with an added limit point at infinity (see the module SquashedSum). \begin{code} {-# OPTIONS --without-K #-} module SearchableOrdinals where open import SetsAndFunctions open import AlternativeCoproduct open import Naturals open import SquashedSum open import Searchable \end{code} We use ordinal encodings that are slightly different from those considered in the module "Ordinals" (Church & Brouwer): \begin{code} data SO : Set where One : SO Add : SO → SO → SO Mul : SO → SO → SO SumPlusOne : (ℕ → SO) → SO \end{code} The above are searchable ordinals codes. (The empty ordinal is excluded because it is not searchable. It is merely exhaustible or omniscient (see the module Searchable for a partial discussion of this). The reason why including the empty ordinal causes insurmountable problems is discussed in research papers.) The decoding function (or semantic interpretation, or evaluation function) is this: \begin{code} ordinal : SO → Set ordinal One = 𝟙 ordinal (Add α β) = ordinal α +' ordinal β ordinal (Mul α β) = ordinal α × ordinal β ordinal (SumPlusOne α) = Σ¹ \(i : ℕ) → ordinal(α i) \end{code} All sets in the image of the function ordinal are searchable: \begin{code} searchable-ordinals : ∀(α : SO) → searchable(ordinal α) searchable-ordinals One = one-searchable searchable-ordinals (Add α β) = binary-sums-preserve-searchability(searchable-ordinals α)(searchable-ordinals β) searchable-ordinals (Mul α β) = binary-Tychonoff(searchable-ordinals α)(searchable-ordinals β) searchable-ordinals (SumPlusOne α) = squashed-sum-searchable (λ i → searchable-ordinals(α i)) \end{code} Classically, the squashed sum is the ordinal sum plus 1. Brouwer ordinal codes can be mapped to searchable ordinal codes, so that the meaning is not necessarily preserved, but so that it is bigger or equal. \begin{code} open import Ordinals brouwer-to-searchable-code : B → SO brouwer-to-searchable-code Z = One brouwer-to-searchable-code (S α) = Add One (brouwer-to-searchable-code α) brouwer-to-searchable-code (L α) = SumPlusOne(λ i → brouwer-to-searchable-code(α i)) \end{code} Relatively "small" example: a type which amounts to the ordinal ε₀ in set theory: \begin{code} ε₀-ordinal : Set ε₀-ordinal = ordinal(brouwer-to-searchable-code B-ε₀) searchable-ε₀-ordinal : searchable ε₀-ordinal searchable-ε₀-ordinal = searchable-ordinals(brouwer-to-searchable-code B-ε₀) \end{code} We can go much higher using the work of many people, including Hancock and Setzer. To do: prove that these searchable types are really ordinals in the sense of the paper "Infinite sets that satisfy the principle of omniscience in all varieties of constructive mathematics". That is: they are linearly ordered (in a suitable constructive sense), and every decidable inhabited subset as a least element (found using the selection function that exists by searchability). This is proved in that paper for subsets of the Cantor space. This file constructs the same ordinals but without having them inside the Cantor space, but the proof (omitted here for the moment) is essentially the same.