A categorical explanation of why Churchs Thesis holds in the - PowerPoint PPT Presentation

CT2018 - University of Azores A categorical explanation of why Church’s Thesis holds in the Effective Topos Fabio Pasquali University of Padova j.w.w. M. E. Maietti and G. Rosolini

Arithmetic doctrines ◮ P : C op → Heyt ◮ C has finite products ◮ for f : X → Y the map P ( f ): P ( Y ) → P ( X ) has (natural) a left and a right adjoint E A f : P ( X ) → P ( Y ) f : P ( X ) → P ( Y ) ◮ C is weakly cartesian closed (wcc) o s � N � N ◮ C has a parametrized nno (pnno) 1 ◮ P satisfies the induction principle on N

Examples

Examples Subobjects C is Sub C : C op → Heyt ◮ elementary topos ◮ nno

Examples Subobjects C is Sub C : C op → Heyt ◮ elementary topos ◮ nno Weak subobjects C is ◮ lex Ψ C : C op → Heyt ◮ finite co-products A �→ ( C / A ) po ◮ weakly lcc ◮ pnno

Internal language A in C f : X → A α ∈ P ( A ) P ( f )( α ) ∈ P ( X ) a : A x : X | f ( x ): A a : A | α ( a ) x : X | α ( f ( x ))

Internal language A in C f : X → A α ∈ P ( A ) P ( f )( α ) ∈ P ( X ) a : A x : X | f ( x ): A a : A | α ( a ) x : X | α ( f ( x )) φ 1 ∧ ... ∧ φ n ≤ ψ in P ( A 1 × ... × A k ) becomes a 1 : A 1 , ..., a k : A k | φ 1 ( a 1 , ..., a k ) , ..., φ n ( a 1 , ..., a k ) ⊢ ψ ( a 1 , ..., a k )

Internal language A in C f : X → A α ∈ P ( A ) P ( f )( α ) ∈ P ( X ) a : A x : X | f ( x ): A a : A | α ( a ) x : X | α ( f ( x )) φ 1 ∧ ... ∧ φ n ≤ ψ in P ( A 1 × ... × A k ) becomes a 1 : A 1 , ..., a k : A k | φ 1 ( a 1 , ..., a k ) , ..., φ n ( a 1 , ..., a k ) ⊢ ψ ( a 1 , ..., a k ) α = ⊤ A becomes a : A ⊢ P α ( a )

The equality predicate E � id X , id X � ( ⊤ X ) ∈ P ( X × X )

The equality predicate E � id X , id X � ( ⊤ X ) ∈ P ( X × X ) abbreviated by = X becomes x : X , x ′ : X | x = X x ′

The equality predicate E � id X , id X � ( ⊤ X ) ∈ P ( X × X ) abbreviated by = X becomes x : X , x ′ : X | x = X x ′ P has comprehensive diagonals if for all f , g : A → X f = g iff a : A ⊢ P f ( a ) = X g ( a )

Formal Church’s Thesis P is arithmetic. N N is a weak exp.

Formal Church’s Thesis P is arithmetic. N N is a weak exp. Formal Church’s Thesis (CT) ⊢ P ∀ x : N ∃ y : N R ( x , y ) → ∃ e : N ∀ x : N ∃ y : N [T( e , x , y ) = N 1 ∧ R ( x , U( y ))]

Formal Church’s Thesis P is arithmetic. N N is a weak exp. Formal Church’s Thesis (CT) ⊢ P ∀ x : N ∃ y : N R ( x , y ) → ∃ e : N ∀ x : N ∃ y : N [T( e , x , y ) = N 1 ∧ R ( x , U( y ))] Formal Type-Theoretic Church’s Thesis (TCT) ⊢ P ∀ f : N N ∃ e : N ∀ x : N ∃ y : N [T( e , x , y ) = N 1 ∧ U( y ) = N ev ( x , f )]

Formal Church’s Thesis P is arithmetic. N N is a weak exp. Formal Church’s Thesis (CT) ⊢ P ∀ x : N ∃ y : N R ( x , y ) → ∃ e : N ∀ x : N ∃ y : N [T( e , x , y ) = N 1 ∧ R ( x , U( y ))] Formal Type-Theoretic Church’s Thesis (TCT) ⊢ P ∀ f : N N ∃ e : N ∀ x : N ∃ y : N [T( e , x , y ) = N 1 ∧ U( y ) = N ev ( x , f )] Rule of choice (RC) if a : A ⊢ P ∃ b : B R ( a , b ), there is f : A → B s.t. a : A ⊢ P R ( a , f ( a ))

Formal Church’s Thesis P is arithmetic. N N is a weak exp. Formal Church’s Thesis (CT) ⊢ P ∀ x : N ∃ y : N R ( x , y ) → ∃ e : N ∀ x : N ∃ y : N [T( e , x , y ) = N 1 ∧ R ( x , U( y ))] Formal Type-Theoretic Church’s Thesis (TCT) ⊢ P ∀ f : N N ∃ e : N ∀ x : N ∃ y : N [T( e , x , y ) = N 1 ∧ U( y ) = N ev ( x , f )] Rule of choice (RC) if a : A ⊢ P ∃ b : B R ( a , b ), there is f : A → B s.t. a : A ⊢ P R ( a , f ( a )) (TCT) + (RC) + full weak comprehension ⇒ (CT)

� � � � � Weak comprehension C op P Heyt id C E · { |−| } · ⊣ C op Ψ C

� � � � � Weak comprehension C op P Heyt id C E · { |−| } · ⊣ C op Ψ C E Weak comprehension is full iff { |−| } = id P .

� � � � � Weak comprehension C op P Heyt id C E · { |−| } · ⊣ C op Ψ C E Weak comprehension is full iff { |−| } = id P . E Theorem: { |−| } = id Ψ C iff P satisfies (RC) [Maietti, Pasquali, Rosolini. Tbilisi Mathematical Journal. 2017]

Elementary quotient completion [Maietti, Rosolini. Elementary quotient completion. 2013] [Maietti, Rosolini. Unifying exact completions. 2015]

Elementary quotient completion C op P � Heyt [Maietti, Rosolini. Elementary quotient completion. 2013] [Maietti, Rosolini. Unifying exact completions. 2015]

� � � � � Elementary quotient completion C op � � P ∇ op Heyt · P � Q op P P [Maietti, Rosolini. Elementary quotient completion. 2013] [Maietti, Rosolini. Unifying exact completions. 2015]

� � � � � Elementary quotient completion C op � � P ∇ op Heyt · P � Q op P P P has effective quotients. � � P is the free such on P . [Maietti, Rosolini. Elementary quotient completion. 2013] [Maietti, Rosolini. Unifying exact completions. 2015]

� � � � � � � � Elementary quotient completion C op � � P ∇ op Heyt · P � Q op P P P has effective quotients. � � P is the free such on P . Q op Ψ C � Ψ C � Heyt · Sub C ex/lex C op ex / lex [Maietti, Rosolini. Elementary quotient completion. 2013] [Maietti, Rosolini. Unifying exact completions. 2015]

Elementary quotient completion and full comprehension P : C op → Heyt has full weak comprehension and C is lex

� Elementary quotient completion and full comprehension P : C op → Heyt has full weak comprehension and C is lex � P I eqc P

� � Elementary quotient completion and full comprehension P : C op → Heyt has full weak comprehension and C is lex � Sub C ex/lex P I eqc I eqc Ψ C P

� � � Elementary quotient completion and full comprehension P : C op → Heyt has full weak comprehension and C is lex � Sub C ex/lex P I eqc I eqc � Ψ C P ⊥

� � � � Elementary quotient completion and full comprehension P : C op → Heyt has full weak comprehension and C is lex � � Sub C ex/lex P I eqc I eqc � Ψ C P ⊥

� � � � � � Elementary quotient completion and full comprehension P : C op → Heyt has full weak comprehension and C is lex � � Sub C ex/lex Q P C ex/lex P I eqc I eqc ∇ P ∇ Ψ P C � Ψ C P ⊥

� � � � � � � Elementary quotient completion and full comprehension P : C op → Heyt has full weak comprehension and C is lex L � � Sub C ex/lex Q P � C ex/lex P ⊥ R I eqc I eqc ∇ P ∇ Ψ P C � Ψ C P ⊥

� � � � � � � Elementary quotient completion and full comprehension P : C op → Heyt has full weak comprehension and C is lex L � � Sub C ex/lex Q P � C ex/lex P ⊥ R I eqc I eqc ∇ P ∇ Ψ P C � Ψ C P ⊥ R is full and faithful L preserves finite products

Elementary quotient completion and (TCT), (CT)

� � � Elementary quotient completion and (TCT), (CT) P C op ∇ P ( N ) is a pnno in Q P Heyt � � ∇ op � P P Q op P

� � � Elementary quotient completion and (TCT), (CT) P C op ∇ P ( N ) is a pnno in Q P Heyt � � ∇ op � P P Q op P Theorem: ◮ P satisfies (TCT) if and only if � P satisfies (TCT) ◮ P satisfies (CT) if and only if � P satisfies (CT)

� Full comprehension and (TCT), (CT) E P � Ψ C ⊥ E (RC) iff { |−| } = id Ψ C { |−| }

� Full comprehension and (TCT), (CT) E P � Ψ C ⊥ E (RC) iff { |−| } = id Ψ C { |−| } α ≤ β in P ( A ) iff { | α | } ≤ { | β | } in Ψ C ( A )

� Full comprehension and (TCT), (CT) E P � Ψ C ⊥ E (RC) iff { |−| } = id Ψ C { |−| } α ≤ β in P ( A ) iff { | α | } ≤ { | β | } in Ψ C ( A ) { | = A | } = = A { | P ( f )( α ) | } = Ψ C ( f ) { | α | } { | α ∧ β | } = { | α | } ∧ { | β | } { | α → β | } = { | α | } → { | β | } A { | f φ | } = Π f { | φ | }

� Full comprehension and (TCT), (CT) E P � Ψ C ⊥ E (RC) iff { |−| } = id Ψ C { |−| } α ≤ β in P ( A ) iff { | α | } ≤ { | β | } in Ψ C ( A ) { | = A | } = = A { | P ( f )( α ) | } = Ψ C ( f ) { | α | } { | α ∧ β | } = { | α | } ∧ { | β | } { | α → β | } = { | α | } → { | β | } A { | f φ | } = Π f { | φ | } E { | α ∨ β | } = { |−| } [ { | α | } ∨ { | β | } ] E E { | f φ | } = { |−| } [Σ f { | φ | } ]

Full comprehension and (TCT), (CT) R ∈ P ( A × B ) R has a Skolem arrow for B if there is f : A → B s.t. x : A | ∃ y : B R ( x , y ) ⊢ R ( x , f ( x ))

Full comprehension and (TCT), (CT) R ∈ P ( A × B ) R has a Skolem arrow for B if there is f : A → B s.t. x : A | ∃ y : B R ( x , y ) ⊢ R ( x , f ( x )) Theorem: if R has a Skolem arrow for B E E { | π φ | } = { |−| } [Σ π { | φ | } ] = Σ π { | φ | } where π : A × B → A , i.e. { |∃ y : B φ ( x , y ) | } = Σ y : B { | φ | } ( x , y )

Full comprehension and (TCT), (CT) (TCT) ∀ f : N N ∃ e : N ∀ x : N ∃ y : N [T( e , x , y ) = N 1 ∧ U( y ) = N ev ( x , f )]