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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

f : P(X) → P(Y )

A

f : P(X) → P(Y ) ◮ C is weakly cartesian closed (wcc) ◮ C has a parametrized nno (pnno) 1

• N

s

N

◮ P satisfies the induction principle on N

Examples

Examples

Subobjects

C is

◮ elementary topos ◮ nno

SubC:C op → Heyt

Examples

Subobjects

C is

◮ elementary topos ◮ nno

SubC:C op → Heyt Weak subobjects

C is

◮ lex ◮ finite co-products ◮ weakly lcc ◮ pnno

ΨC:C op → Heyt A → (C/A)po

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)) φ1 ∧ ... ∧ φn ≤ ψ in P(A1 × ... × Ak) becomes a1: A1, ..., ak: Ak | φ1(a1, ..., ak), ..., φn(a1, ..., ak) ⊢ ψ(a1, ..., ak)

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)) φ1 ∧ ... ∧ φn ≤ ψ in P(A1 × ... × Ak) becomes a1: A1, ..., ak: Ak | φ1(a1, ..., ak), ..., φn(a1, ..., ak) ⊢ ψ(a1, ..., ak) α = ⊤A becomes a: A ⊢P α(a)

The equality predicate

E

idX ,idX (⊤X) ∈ P(X × X)

The equality predicate

E

idX ,idX (⊤X) ∈ P(X × X)

abbreviated by =X becomes x: X, x′: X | x =X x′

The equality predicate

E

idX ,idX (⊤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. NN is a weak exp.

Formal Church’s Thesis

P is arithmetic. NN is a weak exp. Formal Church’s Thesis (CT) ⊢P ∀x:N∃y:NR(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. NN is a weak exp. Formal Church’s Thesis (CT) ⊢P ∀x:N∃y:NR(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 :NN∃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. NN is a weak exp. Formal Church’s Thesis (CT) ⊢P ∀x:N∃y:NR(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 :NN∃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:BR(a, b), there is f : A → B s.t. a: A ⊢P R(a, f (a))

Formal Church’s Thesis

P is arithmetic. NN is a weak exp. Formal Church’s Thesis (CT) ⊢P ∀x:N∃y:NR(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 :NN∃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:BR(a, b), there is f : A → B s.t. a: A ⊢P R(a, f (a)) (TCT) + (RC) + full weak comprehension ⇒ (CT)

Weak comprehension

Cop

P

• idC
• Heyt

Cop

ΨC

• E

·

• · {

|−| }

• ⊣

Weak comprehension

Cop

P

• idC
• Heyt

Cop

ΨC

• E

·

• · {

|−| }

• ⊣

Weak comprehension is full iff E { |−| } = idP.

Weak comprehension

Cop

P

• idC
• Heyt

Cop

ΨC

• E

·

• · {

|−| }

• ⊣

Weak comprehension is full iff E { |−| } = idP. Theorem: { |−| } E = idΨC 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

P

• Heyt

Q op

P

• P
• ·
• [Maietti, Rosolini. Elementary quotient completion. 2013]

[Maietti, Rosolini. Unifying exact completions. 2015]

Elementary quotient completion

C op

P

• ∇op

P

• Heyt

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

P

• Heyt

Q op

P

• P
• ·
• P has effective quotients.

P is the free such on P. Qop

ΨC

• ΨC

Heyt

C op

ex/lex

• SubCex/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

P

Ieqc

Elementary quotient completion and full comprehension

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

• P

SubCex/lex P

Ieqc

• ΨC

Ieqc

Elementary quotient completion and full comprehension

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

• P

SubCex/lex P

Ieqc

• ⊥

ΨC

• Ieqc

Elementary quotient completion and full comprehension

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

• P

SubCex/lex

• P

Ieqc

• ⊥

ΨC

• Ieqc

Elementary quotient completion and full comprehension

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

• P

SubCex/lex

• P

Ieqc

• ⊥

ΨC

• Ieqc
• QP

Cex/lex C

∇P

• ∇ΨP

Elementary quotient completion and full comprehension

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

• P

SubCex/lex

• P

Ieqc

• ⊥

ΨC

• Ieqc
• QP

R ⊥

Cex/lex

L

• C

∇P

• ∇ΨP

Elementary quotient completion and full comprehension

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

• P

SubCex/lex

• P

Ieqc

• ⊥

ΨC

• Ieqc
• QP

R ⊥

Cex/lex

L

• C

∇P

• ∇ΨP
• R is full and faithful

L preserves finite products

Elementary quotient completion and (TCT), (CT)

Elementary quotient completion and (TCT), (CT)

∇P(N) is a pnno in QP

C op

P

• ∇op

P

• Heyt

Q op

P

• P

Elementary quotient completion and (TCT), (CT)

∇P(N) is a pnno in QP

C op

P

• ∇op

P

• Heyt

Q op

P

• 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)

P

⊥ { |−| }

ΨC

E

• (RC) iff

{ |−| } E = idΨC

Full comprehension and (TCT), (CT)

P

⊥ { |−| }

ΨC

E

• (RC) iff

{ |−| } E = idΨC α ≤ β in P(A) iff { |α| } ≤ { |β| } in ΨC(A)

Full comprehension and (TCT), (CT)

P

⊥ { |−| }

ΨC

E

• (RC) iff

{ |−| } E = idΨC α ≤ β in P(A) iff { |α| } ≤ { |β| } in ΨC(A) { |=A| } = =A { |P(f )(α)| } = ΨC(f ){ |α| } { |α ∧ β| } = { |α| } ∧ { |β| } { |α → β| } = { |α| } → { |β| } { | A

f φ|

} = Πf { |φ| }

Full comprehension and (TCT), (CT)

P

⊥ { |−| }

ΨC

E

• (RC) iff

{ |−| } E = idΨC α ≤ β in P(A) iff { |α| } ≤ { |β| } in ΨC(A) { |=A| } = =A { |P(f )(α)| } = ΨC(f ){ |α| } { |α ∧ β| } = { |α| } ∧ { |β| } { |α → β| } = { |α| } → { |β| } { | A

f φ|

} = Πf { |φ| } { |α ∨ β| } = { |−| } E [{ |α| } ∨ { |β| }] { | E

f φ|

} = { |−| } E [Σf { |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:BR(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:BR(x, y) ⊢ R(x, f (x)) Theorem: if R has a Skolem arrow for B { | E

πφ|

} = { |−| } E [Σπ{ |φ| }] = Σπ{ |φ| } where π: A × B → A, i.e. { |∃y:Bφ(x, y)| } = Σy:B{ |φ| }(x, y)

Full comprehension and (TCT), (CT)

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

Full comprehension and (TCT), (CT)

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

Full comprehension and (TCT), (CT)

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

Full comprehension and (TCT), (CT)

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

• Suppose P has Skolem arrows:

NN × N × N → N NN → N

Full comprehension and (TCT), (CT)

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

• Suppose P has Skolem arrows:

NN × N × N → N NN → N

• P

SubCex/lex

• P

{ |−| } ⊥

• Ieqc
• ΨC

Ieqc

• E

Full comprehension and (TCT), (CT)

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

• Suppose P has Skolem arrows:

NN × N × N → N NN → N

• P

SubCex/lex

• (TCT)

P

{ |−| } ⊥

• Ieqc
• ΨC

Ieqc

• E

Full comprehension and (TCT), (CT)

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

• Suppose P has Skolem arrows:

NN × N × N → N NN → N (TCT)

• P

SubCex/lex

• (TCT)

(TCT) P

{ |−| } ⊥

• Ieqc
• ΨC

Ieqc

• E
• (TCT)

Full comprehension and (TCT), (CT)

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

• Suppose P has Skolem arrows:

NN × N × N → N NN → N (TCT)

• P

SubCex/lex

• (TCT)

(TCT) P

{ |−| } ⊥

• Ieqc
• ΨC

Ieqc

• E
• (TCT) (CT)

Full comprehension and (TCT), (CT)

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

• Suppose P has Skolem arrows:

NN × N × N → N NN → N (TCT)

• P

SubCex/lex

• (TCT) (CT)

(TCT) P

{ |−| } ⊥

• Ieqc
• ΨC

Ieqc

• E
• (TCT) (CT)

Application to assemblies

Application to assemblies

The category of Asm

Application to assemblies

The category of Asm

• bjects: (A, α) where α: A → Pow ∗(N)

Application to assemblies

The category of Asm

• bjects: (A, α) where α: A → Pow ∗(N)

arrows: f : (A, α) → (B, β) where f : A → B has a track

Application to assemblies

The category of Asm

• bjects: (A, α) where α: A → Pow ∗(N)

arrows: f : (A, α) → (B, β) where f : A → B has a track, i.e. there exists n ∈ N, such that for all a ∈ A and all p ∈ α(a) ϕn(p) ↓ and ϕn(p) ∈ β(f (a))

Application to assemblies

The category of Asm

• bjects: (A, α) where α: A → Pow ∗(N)

arrows: f : (A, α) → (B, β) where f : A → B has a track, i.e. there exists n ∈ N, such that for all a ∈ A and all p ∈ α(a) ϕn(p) ↓ and ϕn(p) ∈ β(f (a)) P Asm ⊆full Asm on those (A, α) where α(a) is a singleton, i.e. α: A → N

Application to assemblies

Application to assemblies

Asmop

SAsm

• Forget
• Heyt

Set op

Pow

Application to assemblies

Asmop

SAsm

• Forget
• Heyt

Set op

Pow

• P

Asmop

SP

Asm

• Forget
• Heyt

Set op

Pow

Application to assemblies

Asmop

SAsm

• Forget
• Heyt

Set op

Pow

• P

Asmop

SP

Asm

• Forget
• Heyt

Set op

Pow

• Theorem:

SP

Asm ≡ SAsm

Application to assemblies (TCT), (CT)

SP

Asm: P

Asmop − → Heyt N = (N, idN)

Application to assemblies (TCT), (CT)

SP

Asm: P

Asmop − → Heyt N = (N, idN)

◮ (CT) fails to hold

∀x:N∃y:NR(x, y) → ∃e:N∀x:N∃y:N [T(e, x, y) = 1 ∧ R(x, U(y))] take for R the graph of a non-computable function

Application to assemblies (TCT), (CT)

SP

Asm: P

Asmop − → Heyt N = (N, idN)

◮ (CT) fails to hold

∀x:N∃y:NR(x, y) → ∃e:N∀x:N∃y:N [T(e, x, y) = 1 ∧ R(x, U(y))] take for R the graph of a non-computable function

◮ (TCT) holds

∀f :NN∃e:N∀x:N∃y:N [T(e, x, y) = 1 ∧ U(y) = f (x)] definition of arrow in P Asm

Application to assemblies (TCT), (CT)

SP

Asm: P

Asmop − → Heyt N = (N, idN)

◮ (CT) fails to hold

∀x:N∃y:NR(x, y) → ∃e:N∀x:N∃y:N [T(e, x, y) = 1 ∧ R(x, U(y))] take for R the graph of a non-computable function

◮ (TCT) holds

∀f :NN∃e:N∀x:N∃y:N [T(e, x, y) = 1 ∧ U(y) = f (x)] definition of arrow in P Asm + Skolem arrows

Application to assemblies (TCT), (CT)

SAsm

SubEff

• SP

Asm { |−| } ⊥

• Ieqc
• ΨP

Asm Ieqc

• E

Application to assemblies (TCT), (CT)

SAsm

SubEff

• (TCT)

Skolem arrows

SP

Asm { |−| } ⊥

• Ieqc
• ΨP

Asm Ieqc

• E

Application to assemblies (TCT), (CT)

(TCT) SAsm

SubEff

• (CT)

(TCT)

Skolem arrows

SP

Asm { |−| } ⊥

• Ieqc
• ΨP

Asm Ieqc

• E
• (CT)

Thank you

References

• A. Carboni.

Some free constructions in realizability and proof theory.

• J. Pure Appl. Algebra, 103 117–148 1995.
• A. Carboni, P.J. Freyd and A. Scedrov.

A categorical approach to realizability and polymorphic types. Mathematical Foundations of Programming Language Semantics 298 23–42 1988.

• J. M. E. Hyland.

The effective topos. The L.E.J. Brouwer Centenary Symposium. 1982.

• F. W. Lawvere.

Equality in hyperdoctrines and comprehension schema as an adjoint functor.

• A. Heller, editor, Proc. New York Symposium on Application of Categorical Algebra. 1970.

References

M.E. Maietti, F. Pasquali. and G. Rosolini. Triposes, exact completions, and Hilbert’s ǫ-operator. Tbilisi Mathematical Journal. 2017. M.E. Maietti. and G. Rosolini. Relating quotient completions via categorical logic. Dieter Probst and Peter Schuster (eds.), ”Concepts of Proof in Mathematics, Philosophy, and Computer Science”. De Gruyter. 2016. M.E. Maietti. and G. Rosolini. Unifying exact completions. Applied Categorical Structures. 2013. M.E. Maietti. and G. Rosolini. Elementary quotient completion. Theory and Applications of Categories. 2013. M.E. Maietti. and G. Rosolini. Quotient completion for the foundation of constructive mathematics. Logica Universalis. 2013.