如何在Agda中构造一个可能非空的Set

Question

I know that (A \\/ ~A) is not provable in general. How does one go about constructing an example of a set A where (A \\/ ~A) is not provable, is this possible? And if it is possible, is it possible without quantifiers?

Answer 1

I know that (A / ~A) is not provable in general. How does one go about constructing an example of a set A where (A / ~A) is not provable,

You have already given an example: A \\/ ~A itself.

open import Level
open import Data.Empty
open import Relation.Nullary
open import Data.Sum

lem-for : ∀ {α} -> Set α -> Set α
lem-for A = A ⊎ ¬ A

lem : ∀ {α} -> Set (suc α)
lem = ∀ {A} -> lem-for A

lem-lem : ∀ {α} -> Set (suc α)
lem-lem = lem-for lem

lem says "for all A A is either true or false". lem-lem says "the law of excluded middle is either true or false". But we know that constructively lem is not true and, since Agda is not anti-classical, lem is not false either.

Other classical logic axioms (taken from the Software Foundations book) are

Definition peirce := ∀P Q: Prop,
  ((P→Q)→P)→P.
Definition classic := ∀P:Prop,
  ~~P → P.
Definition de_morgan_not_and_not := ∀P Q:Prop,
  ~(~P ∧ ¬Q) → P∨Q.
Definition implies_to_or := ∀P Q:Prop,
  (P→Q) → (¬P∨Q).

These all + lem are equivalent.

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Here is a fancier example:

open import Relation.Binary.PropositionalEquality
open import Data.Bool.Base
open import Data.Fin

eq : Set₁
eq = Fin 2 ≡ Bool

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"Extensional" predicates are of the same sort. The simplest is function extensionality, but we can also say

open import Coinduction
open import Data.Nat.Base
open import Data.Stream

zeros : Stream ℕ
zeros = 0 ∷ ♯ zeros

eq₂ : Set
eq₂ = zeros ≡ 0 ∷ ♯ zeros