Lecture 4: Models
Teacher: Bruno Barras
The soundness statement involves 3 formalisms: domain, codomain, and the formalism in which the proof of soundness has been done (usually: primitive recursive arithmetic)
Soundness is used to prove:
 Consistency of $A$ (w.r.t $Γ$): $Γ A\not⊢ ⊥$
 Independence of $A$ (w.r.t. $Γ$): $Γ \not⊢ A$
Different kinds of models:
 Settheoretical: for everyday mathematics, to convince set theorists that our theory is consistent (and see type theory settheoretically)
 Groupoid model (HofmanStreicher, 95): types interpreted by groupoids ⟹ from that: Homotopy Type Theory (effective to check theorem of homotopy theory)
 Strong normalisation models (realisability): all the typable terms are strongly normalising (idea: interpret types and proofs by programs that strongly normalise)
Set theoretical model

Dependent product, $λ$abstractions interpreted by their counterparts in set theory (set of functions depending on their argument and one function respectively)

Application correspond to function application

$Prop$ interpreted as $𝒫(∅)$ ($≅ \lbrace 0, 1 \rbrace$ classically, but contains intermediate sets as well in intuitionistic logic)
 Problem: $A ⇒ A$ is function between booleans if $A$ is a proposition, but is a boolean as well: everything is collapsed to the empty set. Peter Aczel function encoding: instead of couples $(x, f(x))$ ⟶ $(x,y) \quad y ∈ f(x)$, so a function $x ⟼ ∅$ is interpreted as $∅$

$Type_i$ interpreted as $U_i$, in a Grothendieck universe

Inductive types interpreted by least fixed points (only strictly positive occurrences allowed)
Incomplete model: because all of these are not provable:
 consistency of Classical logic
 Choice axioms
 Functional extensionality
 Propositional extensionality
But it also means that we can add them to type theory without breaking consistency
Realizability
What enable us to show that intuitionistic type theory is constructive: we can extract programs/objects satisfying the property from proofs.
Double negation translation:
\[⊢_{\text{classic}} A ⟺ ¬¬A \qquad\text{ and } \qquad ⊢_{\text{classic}} A \text{ implies } ⊢_{\text{intuit.}} ¬¬A\]Normalizing an existential/disjunction proof ⟹ extraction: you get a witness
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