Homotopy Type Theory excluded middle > history (Rev #1)


Excluded middle says that there is a term of the sum of a proposition PP and the type of functions PP \to \emptyset.

Ptypep: a:P b:Pa= Pbq:P+P\frac{P\ \mathrm{type} \quad p:\prod_{a:P} \prod_{b:P} a =_P b}{q:P + P \to \emptyset}

In universes

Excluded middle is said to hold in a universe 𝒰\mathcal{U} if the universe comes with a dependent function

p: P:𝒰( a:𝒯 𝒰(P) b:𝒯 𝒰(P)a= 𝒯 𝒰(P)b)(𝒯 𝒰(P)+𝒯 𝒰(P))p:\prod_{P:\mathcal{U}} \left(\prod_{a:\mathcal{T}_\mathcal{U}(P)} \prod_{b:\mathcal{T}_\mathcal{U}(P)} a =_{\mathcal{T}_\mathcal{U}(P)} b\right) \to \left(\mathcal{T}_\mathcal{U}(P) + \mathcal{T}_\mathcal{U}(P) \to \emptyset\right)

See also

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