Primitive Recursive Bars Are Inductive Jon Sterling Carnegie - - PowerPoint PPT Presentation

Primitive Recursive Bars Are Inductive

Jon Sterling

Carnegie Mellon University

August 17, 2016

𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

“All bars on a spread can be coded as inductive trees.” ⋆ ⇒ the Fan Theorem (intuitionistic König’s Lemma) ⋆ ⇒ all functions on the interval 𝕁 ≜ [0, 1] are uniformly continuous ⋆ ⇒ completeness of intuitionistic first-order logic (Bickford, Constable) In general, implies termination for a class of functional programs on infinite trees.

𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

“All bars on a spread can be coded as inductive trees.” ⋆ ⇒ the Fan Theorem (intuitionistic König’s Lemma) ⋆ ⇒ all functions on the interval 𝕁 ≜ [0, 1] are uniformly continuous ⋆ ⇒ completeness of intuitionistic first-order logic (Bickford, Constable) In general, implies termination for a class of functional programs on infinite trees.

𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

“All bars on a spread can be coded as inductive trees.” ⋆ ⇒ the Fan Theorem (intuitionistic König’s Lemma) ⋆ ⇒ all functions on the interval 𝕁 ≜ [0, 1] are uniformly continuous ⋆ ⇒ completeness of intuitionistic first-order logic (Bickford, Constable) In general, implies termination for a class of functional programs on infinite trees.

𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

“All bars on a spread can be coded as inductive trees.” ⋆ ⇒ the Fan Theorem (intuitionistic König’s Lemma) ⋆ ⇒ all functions on the interval 𝕁 ≜ [0, 1] are uniformly continuous ⋆ ⇒ completeness of intuitionistic first-order logic (Bickford, Constable) In general, implies termination for a class of functional programs on infinite trees.

𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

“All bars on a spread can be coded as inductive trees.” ⋆ ⇒ the Fan Theorem (intuitionistic König’s Lemma) ⋆ ⇒ all functions on the interval 𝕁 ≜ [0, 1] are uniformly continuous ⋆ ⇒ completeness of intuitionistic first-order logic (Bickford, Constable) In general, implies termination for a class of functional programs on infinite trees.

𝕿𝖖𝖘𝖋𝖇𝖊𝖙 𝖇𝖙 𝖙𝖖𝖇𝖉𝖋𝖙 𝖇𝖔𝖊 𝖚𝖘𝖋𝖋𝖙

A 𝓽𝓺𝓼𝓯𝓫𝓮 is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate 𝔗 on lists of natural numbers subject to some laws:

• 1. If

⃗ 𝑣 ∈ 𝔗, then there exists an 𝑦 ∈ ℕ such that ⃗ 𝑣⌢𝑦 ∈ 𝔗.

• 2. If

⃗ 𝑣⌢𝑦 ∈ 𝔗, then also ⃗ 𝑣 ∈ 𝔗.

• 3. Finally, ⟨⟩ ∈ 𝔗.

The predicate 𝔗 either defines the finite prefixes of paths down an infinite tree, or it defines the lattice of open sets of a topological space.

𝕿𝖖𝖘𝖋𝖇𝖊𝖙 𝖇𝖙 𝖙𝖖𝖇𝖉𝖋𝖙 𝖇𝖔𝖊 𝖚𝖘𝖋𝖋𝖙

A 𝓽𝓺𝓼𝓯𝓫𝓮 is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate 𝔗 on lists of natural numbers subject to some laws:

• 1. If

⃗ 𝑣 ∈ 𝔗, then there exists an 𝑦 ∈ ℕ such that ⃗ 𝑣⌢𝑦 ∈ 𝔗.

• 2. If

⃗ 𝑣⌢𝑦 ∈ 𝔗, then also ⃗ 𝑣 ∈ 𝔗.

• 3. Finally, ⟨⟩ ∈ 𝔗.

The predicate 𝔗 either defines the finite prefixes of paths down an infinite tree, or it defines the lattice of open sets of a topological space.

𝕿𝖖𝖘𝖋𝖇𝖊𝖙 𝖇𝖙 𝖙𝖖𝖇𝖉𝖋𝖙 𝖇𝖔𝖊 𝖚𝖘𝖋𝖋𝖙

A 𝓽𝓺𝓼𝓯𝓫𝓮 is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate 𝔗 on lists of natural numbers subject to some laws:

• 1. If

⃗ 𝑣 ∈ 𝔗, then there exists an 𝑦 ∈ ℕ such that ⃗ 𝑣⌢𝑦 ∈ 𝔗.

• 2. If

⃗ 𝑣⌢𝑦 ∈ 𝔗, then also ⃗ 𝑣 ∈ 𝔗.

• 3. Finally, ⟨⟩ ∈ 𝔗.

The predicate 𝔗 either defines the finite prefixes of paths down an infinite tree, or it defines the lattice of open sets of a topological space.

𝕿𝖖𝖘𝖋𝖇𝖊𝖙 𝖇𝖙 𝖙𝖖𝖇𝖉𝖋𝖙 𝖇𝖔𝖊 𝖚𝖘𝖋𝖋𝖙

A 𝓽𝓺𝓼𝓯𝓫𝓮 is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate 𝔗 on lists of natural numbers subject to some laws:

• 1. If

⃗ 𝑣 ∈ 𝔗, then there exists an 𝑦 ∈ ℕ such that ⃗ 𝑣⌢𝑦 ∈ 𝔗.

• 2. If

⃗ 𝑣⌢𝑦 ∈ 𝔗, then also ⃗ 𝑣 ∈ 𝔗.

• 3. Finally, ⟨⟩ ∈ 𝔗.

The predicate 𝔗 either defines the finite prefixes of paths down an infinite tree, or it defines the lattice of open sets of a topological space.

𝕿𝖖𝖘𝖋𝖇𝖊𝖙 𝖇𝖙 𝖙𝖖𝖇𝖉𝖋𝖙 𝖇𝖔𝖊 𝖚𝖘𝖋𝖋𝖙

A 𝓽𝓺𝓼𝓯𝓫𝓮 is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate 𝔗 on lists of natural numbers subject to some laws:

• 1. If

⃗ 𝑣 ∈ 𝔗, then there exists an 𝑦 ∈ ℕ such that ⃗ 𝑣⌢𝑦 ∈ 𝔗.

• 2. If

⃗ 𝑣⌢𝑦 ∈ 𝔗, then also ⃗ 𝑣 ∈ 𝔗.

• 3. Finally, ⟨⟩ ∈ 𝔗.

The predicate 𝔗 either defines the finite prefixes of paths down an infinite tree, or it defines the lattice of open sets of a topological space.

𝕿𝖖𝖘𝖋𝖇𝖊𝖙 𝖇𝖙 𝖙𝖖𝖇𝖉𝖋𝖙 𝖇𝖔𝖊 𝖚𝖘𝖋𝖋𝖙

A 𝓽𝓺𝓼𝓯𝓫𝓮 is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate 𝔗 on lists of natural numbers subject to some laws:

• 1. If

⃗ 𝑣 ∈ 𝔗, then there exists an 𝑦 ∈ ℕ such that ⃗ 𝑣⌢𝑦 ∈ 𝔗.

• 2. If

⃗ 𝑣⌢𝑦 ∈ 𝔗, then also ⃗ 𝑣 ∈ 𝔗.

• 3. Finally, ⟨⟩ ∈ 𝔗.

The predicate 𝔗 either defines the finite prefixes of paths down an infinite tree, or it defines the lattice of open sets of a topological space.

𝕺𝖋𝖏𝖍𝖎𝖈𝖕𝖘𝖎𝖕𝖕𝖊𝖙 𝖇𝖔𝖊 𝕼𝖕𝖏𝖔𝖚𝖙

A list of naturals ⃗ 𝑣 ∈ ℕ∗ can be thought of as a neighborhood around a point, or as a prefix of a path through an infinite tree. A stream of naturals 𝛽 ∈ ℕℕ can be thought of as an ideal point in the spread (space), or as a path through the spread’s infinite tree. ⃗ 𝑣 ≺ 𝛽 ( ⃗ 𝑣 approximates 𝛽) 𝛽 ∈ ⃗ 𝑣 ( ⃗ 𝑣 is a neighborhood around 𝛽)

𝖀𝖕 𝕮𝖇𝖘 𝕭 𝕺𝖕𝖊𝖋

A bar 𝔆 is a predicate on neighborhoods such that every point “hits it”. More generally, 𝔆 bars a neighborhood ⃗ 𝑣 when every path through ⃗ 𝑣 ends up in 𝔆. ∀𝛽 ∈ ⃗ 𝑣. ∃𝑜 ∈ ℕ. 𝛽[𝑜] ∈ 𝔆 ⃗ 𝑣 ◁ 𝔆

𝖀𝖕 𝕮𝖇𝖘 𝕭 𝕺𝖕𝖊𝖋

A bar 𝔆 is a predicate on neighborhoods such that every point “hits it”. More generally, 𝔆 bars a neighborhood ⃗ 𝑣 when every path through ⃗ 𝑣 ends up in 𝔆. ∀𝛽 ∈ ⃗ 𝑣. ∃𝑜 ∈ ℕ. 𝛽[𝑜] ∈ 𝔆 ⃗ 𝑣 ◁ 𝔆

𝕵𝖔𝖊𝖛𝖉𝖚𝖏𝖜𝖋 𝕮𝖇𝖘𝖎𝖕𝖕𝖊

“All demonstrations of barhood can be analyzed into inductive mental constructions.” Let 𝔗♮( ⃗ 𝑣) ≜ { 𝑦 ∈ ℕ ∣ ⃗ 𝑣⌢𝑦 ∈ 𝔗 }. Presupposing ⃗ 𝑣 ∈ 𝔗 and 𝔆 monotone: ⃗ 𝑣 ∈ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 𝜃 ∀𝑦 ∈ 𝔗♮( ⃗ 𝑣). ⃗ 𝑣⌢𝑦 ◀𝑗𝑜𝑒 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 ϝ Admissible (by monotonicity): ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 ⃗ 𝑣⌢𝑦 ◀𝑗𝑜𝑒 𝔆 𝜂

𝕵𝖔𝖊𝖛𝖉𝖚𝖏𝖜𝖋 𝕮𝖇𝖘𝖎𝖕𝖕𝖊

“All demonstrations of barhood can be analyzed into inductive mental constructions.” Let 𝔗♮( ⃗ 𝑣) ≜ { 𝑦 ∈ ℕ ∣ ⃗ 𝑣⌢𝑦 ∈ 𝔗 }. Presupposing ⃗ 𝑣 ∈ 𝔗 and 𝔆 monotone: ⃗ 𝑣 ∈ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 𝜃 ∀𝑦 ∈ 𝔗♮( ⃗ 𝑣). ⃗ 𝑣⌢𝑦 ◀𝑗𝑜𝑒 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 ϝ Admissible (by monotonicity): ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 ⃗ 𝑣⌢𝑦 ◀𝑗𝑜𝑒 𝔆 𝜂

𝕵𝖔𝖊𝖛𝖉𝖚𝖏𝖜𝖋 𝕮𝖇𝖘𝖎𝖕𝖕𝖊

“All demonstrations of barhood can be analyzed into inductive mental constructions.” Let 𝔗♮( ⃗ 𝑣) ≜ { 𝑦 ∈ ℕ ∣ ⃗ 𝑣⌢𝑦 ∈ 𝔗 }. Presupposing ⃗ 𝑣 ∈ 𝔗 and 𝔆 monotone: ⃗ 𝑣 ∈ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 𝜃 ∀𝑦 ∈ 𝔗♮( ⃗ 𝑣). ⃗ 𝑣⌢𝑦 ◀𝑗𝑜𝑒 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 ϝ Admissible (by monotonicity): ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 ⃗ 𝑣⌢𝑦 ◀𝑗𝑜𝑒 𝔆 𝜂

𝕵𝖔𝖊𝖛𝖉𝖚𝖏𝖜𝖋 𝕮𝖇𝖘𝖎𝖕𝖕𝖊

“All demonstrations of barhood can be analyzed into inductive mental constructions.” Let 𝔗♮( ⃗ 𝑣) ≜ { 𝑦 ∈ ℕ ∣ ⃗ 𝑣⌢𝑦 ∈ 𝔗 }. Presupposing ⃗ 𝑣 ∈ 𝔗 and 𝔆 monotone: ⃗ 𝑣 ∈ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 𝜃 ∀𝑦 ∈ 𝔗♮( ⃗ 𝑣). ⃗ 𝑣⌢𝑦 ◀𝑗𝑜𝑒 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 ϝ Admissible (by monotonicity): ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 ⃗ 𝑣⌢𝑦 ◀𝑗𝑜𝑒 𝔆 𝜂

𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

Recall ⃗ 𝑣 ◁ 𝔆 ≜ ∀𝛽 ∈ ⃗ 𝑣. ∃𝑜 ∈ ℕ. 𝛽[𝑜] ∈ 𝔆 . Brouwer’s Bar Thesis is the 𝓫𝓮𝓯𝓻𝓿𝓫𝓭𝔃 of the inductive coding of barhood:

𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

Recall ⃗ 𝑣 ◁ 𝔆 ≜ ∀𝛽 ∈ ⃗ 𝑣. ∃𝑜 ∈ ℕ. 𝛽[𝑜] ∈ 𝔆 . Brouwer’s Bar Thesis is the 𝓫𝓮𝓯𝓻𝓿𝓫𝓭𝔃 of the inductive coding of barhood: ⃗ 𝑣 ◁ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 BT?

𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

Recall ⃗ 𝑣 ◁ 𝔆 ≜ ∀𝛽 ∈ ⃗ 𝑣. ∃𝑜 ∈ ℕ. 𝛽[𝑜] ∈ 𝔆 . Brouwer’s Bar Thesis is the 𝓫𝓮𝓯𝓻𝓿𝓫𝓭𝔃 of the inductive coding of barhood: ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 ⃗ 𝑣 ◁ 𝔆 soundness ⃗ 𝑣 ◁ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness?

𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 ⃗ 𝑣 ◁ 𝔆 soundness ⃗ 𝑣 ◁ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness? ⋆ soundness is easy! Just count up the ϝ-nodes. ⋆ completeness does not (generally) hold: procedure exists, but its termination requires Brouwer’s Thesis!

𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 ⃗ 𝑣 ◁ 𝔆 soundness ⃗ 𝑣 ◁ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness? ⋆ soundness is easy! Just count up the ϝ-nodes. ⋆ completeness does not (generally) hold: procedure exists, but its termination requires Brouwer’s Thesis!

𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 ⃗ 𝑣 ◁ 𝔆 soundness ⃗ 𝑣 ◁ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness? ⋆ soundness is easy! Just count up the ϝ-nodes. ⋆ completeness does not (generally) hold: procedure exists, but its termination requires Brouwer’s Thesis!

𝕿𝖚𝖇𝖚𝖛𝖙 𝖕𝖌 𝖚𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

⋆ Classically: valid ⋆ Constructively: not valid ⋆ Intuitionistically: valid(*)

𝕿𝖚𝖇𝖚𝖛𝖙 𝖕𝖌 𝖚𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

⋆ Classically: valid ⋆ Constructively: not valid ⋆ Intuitionistically: valid(*)

𝕿𝖚𝖇𝖚𝖛𝖙 𝖕𝖌 𝖚𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

⋆ Classically: valid ⋆ Constructively: not valid ⋆ Intuitionistically: valid(*)

Let’s instantiate the Brouwer-Heyting-Kolmogorov interpretation at a simple theory of constructions!

𝕿𝖟𝖙𝖚𝖋𝖓 𝕌 𝖇𝖙 𝖇 𝖚𝖎𝖋𝖕𝖘𝖟 𝖕𝖌 𝖉𝖕𝖔𝖙𝖚𝖘𝖛𝖉𝖚𝖏𝖕𝖔𝖙

Gödel’s System 𝕌 of primitive recursive functionals of finite type. Atomic Types nat atype Types 𝜅 atype 𝜅 type 𝜏 type 𝜐 type 𝜏 → 𝜐 type Contexts ⋅ ctx Γ ctx 𝜏 type Γ, 𝑦 ∶ 𝜏 ctx (𝑦 ∉ Γ)

𝕿𝖟𝖙𝖚𝖋𝖓 𝕌 𝖇𝖙 𝖇 𝖚𝖎𝖋𝖕𝖘𝖟 𝖕𝖌 𝖉𝖕𝖔𝖙𝖚𝖘𝖛𝖉𝖚𝖏𝖕𝖔𝖙

Gödel’s System 𝕌 of primitive recursive functionals of finite type. Atomic Types nat atype Types 𝜅 atype 𝜅 type 𝜏 type 𝜐 type 𝜏 → 𝜐 type Contexts ⋅ ctx Γ ctx 𝜏 type Γ, 𝑦 ∶ 𝜏 ctx (𝑦 ∉ Γ)

𝕿𝖟𝖙𝖚𝖋𝖓 𝕌 𝖇𝖙 𝖇 𝖚𝖎𝖋𝖕𝖘𝖟 𝖕𝖌 𝖉𝖕𝖔𝖙𝖚𝖘𝖛𝖉𝖚𝖏𝖕𝖔𝖙

Γ, 𝑦 ∶ 𝜏, Δ ⊢ 𝑦 ∶ 𝜏 var Γ ⊢ z ∶ nat zero Γ ⊢ 𝑛 ∶ nat Γ ⊢ s(𝑛) ∶ nat succ Γ, 𝑦 ∶ nat, 𝑧 ∶ 𝜏 ⊢ 𝑡[𝑦, 𝑧] ∶ 𝜏 Γ ⊢ 𝑨 ∶ 𝜏 Γ ⊢ 𝑜 ∶ nat Γ ⊢ rec𝜏([𝑦, 𝑧].𝑡[𝑦, 𝑧]; 𝑨; 𝑜) ∶ 𝜏 rec Γ, 𝑦 ∶ 𝜏 ⊢ 𝑛[𝑦] ∶ 𝜐 Γ ⊢ 𝜇𝑦.𝑛[𝑦] ∶ 𝜏 → 𝜐 lam Γ ⊢ 𝑛 ∶ 𝜏 → 𝜐 Γ ⊢ 𝑜 ∶ 𝜏 Γ ⊢ 𝑛 •𝜏 𝑜 ∶ 𝜐 ap

𝕾𝖋𝖇𝖒𝖏𝖠𝖇𝖈𝖏𝖒𝖏𝖚𝖟 𝖕𝖌 𝖈𝖇𝖘𝖎𝖕𝖕𝖊

⋆ Use System 𝕌 as a theory of constructions for primitive recursive arithmetic ⋆ ⃗ 𝑣 ◁ 𝔆 should be realized by a functional ⋅ ⊢ 𝜚 ∶ (nat → nat) → nat ⋆ Idea: construct a model for System 𝕌 in which we can read from the interpretation of 𝜚 a proof of ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆

𝕾𝖋𝖇𝖒𝖏𝖠𝖇𝖈𝖏𝖒𝖏𝖚𝖟 𝖕𝖌 𝖈𝖇𝖘𝖎𝖕𝖕𝖊

⋆ Use System 𝕌 as a theory of constructions for primitive recursive arithmetic ⋆ ⃗ 𝑣 ◁ 𝔆 should be realized by a functional ⋅ ⊢ 𝜚 ∶ (nat → nat) → nat ⋆ Idea: construct a model for System 𝕌 in which we can read from the interpretation of 𝜚 a proof of ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆

𝕾𝖋𝖇𝖒𝖏𝖠𝖇𝖈𝖏𝖒𝖏𝖚𝖟 𝖕𝖌 𝖈𝖇𝖘𝖎𝖕𝖕𝖊

⋆ Use System 𝕌 as a theory of constructions for primitive recursive arithmetic ⋆ ⃗ 𝑣 ◁ 𝔆 should be realized by a functional ⋅ ⊢ 𝜚 ∶ (nat → nat) → nat ⋆ Idea: construct a model for System 𝕌 in which we can read from the interpretation of 𝜚 a proof of ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆

𝕿𝖚𝖇𝖔𝖊𝖇𝖘𝖊 𝖙𝖋𝖓𝖇𝖔𝖚𝖏𝖉𝖙 𝖕𝖌 𝕿𝖟𝖙𝖚𝖋𝖓 𝕌

Atomic Types 𝒲 [nat] ≜ ℕ Types 𝒲 𝜅 ≜ 𝒲 [𝜅] 𝒲 𝜏 → 𝜐 ≜ 𝒲 𝜏 → 𝒲 𝜐 Contexts 𝒣 Γ ≜ ∏

𝑦∈|Γ|

𝒲 Γ(𝑦)

𝕿𝖚𝖇𝖔𝖊𝖇𝖘𝖊 𝖙𝖋𝖓𝖇𝖔𝖚𝖏𝖉𝖙 𝖕𝖌 𝕿𝖟𝖙𝖚𝖋𝖓 𝕌

Presupposing 𝜍 ∈ 𝒣 Γ and Γ ⊢ 𝑛 ∶ 𝜏, define Γ ⊢ 𝑛 ∶ 𝜏𝜍 ∈ 𝒲 𝜏 by recursion

• n 𝑛:

Γ ⊢ 𝑦 ∶ 𝜏𝜍 ≜ 𝜍(𝑦) Γ ⊢ z ∶ nat𝜍 ≜ 0 Γ ⊢ s(𝑛) ∶ nat𝜍 ≜ 1 + Γ ⊢ 𝑛 ∶ nat𝜍 Γ ⊢ rec𝜏([𝑦, 𝑧].𝑡[𝑦, 𝑧]; 𝑨; 𝑜) ∶ 𝜏𝜍 ≜ PrimRec(𝑇, 𝑎, 𝑂) where 𝑇(𝑏, 𝑐) ≜ Γ, 𝑦 ∶ nat, 𝑧 ∶ 𝜏 ⊢ 𝑡[𝑦, 𝑧] ∶ 𝜏𝜍,𝑦↦𝑏,𝑧↦𝑐 𝑎 ≜ Γ ⊢ 𝑨 ∶ 𝜏𝜍 𝑂 ≜ Γ ⊢ 𝑜 ∶ nat𝜍

𝕿𝖚𝖇𝖔𝖊𝖇𝖘𝖊 𝖙𝖋𝖓𝖇𝖔𝖚𝖏𝖉𝖙 𝖕𝖌 𝕿𝖟𝖙𝖚𝖋𝖓 𝕌

⋮ Γ ⊢ 𝜇𝑦.𝑛[𝑦] ∶ 𝜏 → 𝜐𝜍 ≜ 𝑏 ↦ Γ, 𝑦 ∶ 𝜏 ⊢ 𝑛[𝑦] ∶ 𝜐𝜍,𝑦↦𝑏 Γ ⊢ 𝑛 •𝜏 𝑜 ∶ 𝜐𝜍 ≜ (Γ ⊢ 𝑛 ∶ 𝜏 → 𝜐𝜍) (Γ ⊢ 𝑜 ∶ 𝜏𝜍)

𝕾𝖋𝖜𝖏𝖙𝖏𝖔𝖍 𝖚𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

A functional ⋅ ⊢ 𝜚 ∶ (nat → nat) → nat can be applied to a meta-level sequence 𝛽 as follows: 𝜚 ⟨𝛽⟩ ≜ ⋅ ⊢ 𝜚 ∶ (nat → nat) → nat (𝛽) Redefine barhood as follows:

𝕾𝖋𝖜𝖏𝖙𝖏𝖔𝖍 𝖚𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

A functional ⋅ ⊢ 𝜚 ∶ (nat → nat) → nat can be applied to a meta-level sequence 𝛽 as follows: 𝜚 ⟨𝛽⟩ ≜ ⋅ ⊢ 𝜚 ∶ (nat → nat) → nat (𝛽) Redefine barhood as follows: ∀𝛽 ∈ ⃗ 𝑣. ∃𝑜 ∈ ℕ. 𝛽[𝑜] ∈ 𝔆 ⃗ 𝑣 ◁ 𝔆

𝕾𝖋𝖜𝖏𝖙𝖏𝖔𝖍 𝖚𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

A functional ⋅ ⊢ 𝜚 ∶ (nat → nat) → nat can be applied to a meta-level sequence 𝛽 as follows: 𝜚 ⟨𝛽⟩ ≜ ⋅ ⊢ 𝜚 ∶ (nat → nat) → nat (𝛽) Redefine barhood as follows: ∃⋅ ⊢ 𝜚 ∶ (nat → nat) → nat. ∀𝛽 ∈ ⃗ 𝑣. 𝛽[𝜚 ⟨𝛽⟩] ∈ 𝔆 ⃗ 𝑣 ◁𝕌 𝔆

𝕱𝖙𝖉𝖇𝖘𝖊𝖕 𝖊𝖏𝖇𝖒𝖕𝖍𝖛𝖋𝖙

An inductive encoding of functionals ℕℕ → 𝑎: 𝑨 ∈ 𝑎 𝜃(𝑨) ∈ ⦃ℕℕ, 𝑎⦄ return 𝑦 ∈ ℕ 𝑓 ∈ ℕ → ⦃ℕℕ, 𝑎⦄ ϙ⟨𝑦⟩(𝑓) ∈ ⦃ℕℕ, 𝑎⦄ query ⋆ ⦃ℕℕ, −⦄ is an endofunctor on Set ⋆ ⦃ℕℕ, −⦄ is a monad on Set souped up neighborhood functions!

𝕱𝖙𝖉𝖇𝖘𝖊𝖕 𝖊𝖏𝖇𝖒𝖕𝖍𝖛𝖋𝖙

An inductive encoding of functionals ℕℕ → 𝑎: 𝑨 ∈ 𝑎 𝜃(𝑨) ∈ ⦃ℕℕ, 𝑎⦄ return 𝑦 ∈ ℕ 𝑓 ∈ ℕ → ⦃ℕℕ, 𝑎⦄ ϙ⟨𝑦⟩(𝑓) ∈ ⦃ℕℕ, 𝑎⦄ query ⋆ ⦃ℕℕ, −⦄ is an endofunctor on Set ⋆ ⦃ℕℕ, −⦄ is a monad on Set souped up neighborhood functions!

𝕱𝖙𝖉𝖇𝖘𝖊𝖕 𝖊𝖏𝖇𝖒𝖕𝖍𝖛𝖋𝖙

An inductive encoding of functionals ℕℕ → 𝑎: 𝑨 ∈ 𝑎 𝜃(𝑨) ∈ ⦃ℕℕ, 𝑎⦄ return 𝑦 ∈ ℕ 𝑓 ∈ ℕ → ⦃ℕℕ, 𝑎⦄ ϙ⟨𝑦⟩(𝑓) ∈ ⦃ℕℕ, 𝑎⦄ query ⋆ ⦃ℕℕ, −⦄ is an endofunctor on Set ⋆ ⦃ℕℕ, −⦄ is a monad on Set souped up neighborhood functions!

𝕱𝖙𝖉𝖇𝖘𝖊𝖕 𝖊𝖏𝖇𝖒𝖕𝖍𝖛𝖋𝖙

An inductive encoding of functionals ℕℕ → 𝑎: 𝑨 ∈ 𝑎 𝜃(𝑨) ∈ ⦃ℕℕ, 𝑎⦄ return 𝑦 ∈ ℕ 𝑓 ∈ ℕ → ⦃ℕℕ, 𝑎⦄ ϙ⟨𝑦⟩(𝑓) ∈ ⦃ℕℕ, 𝑎⦄ query ⋆ ⦃ℕℕ, −⦄ is an endofunctor on Set ⋆ ⦃ℕℕ, −⦄ is a monad on Set souped up neighborhood functions!

𝕱𝖙𝖉𝖇𝖘𝖊𝖕 𝖊𝖏𝖇𝖒𝖕𝖍𝖛𝖋𝖙

We can execute dialogue trees against a sequence 𝛽 ∈ ℕℕ: 𝜃(𝑨) ⋄ 𝛽 ≜ 𝑨 ϙ⟨𝑦⟩(𝑓) ⋄ 𝛽 ≜ 𝑓(𝛽(𝑦)) ⋄ 𝛽

𝕱𝖙𝖉𝖇𝖘𝖊𝖕 𝖊𝖏𝖇𝖒𝖕𝖍𝖛𝖋𝖙

We can execute dialogue trees against a sequence 𝛽 ∈ ℕℕ: 𝜃(𝑨) ⋄ 𝛽 ≜ 𝑨 ϙ⟨𝑦⟩(𝑓) ⋄ 𝛽 ≜ 𝑓(𝛽(𝑦)) ⋄ 𝛽 ⦃ℕℕ, 𝐵⦄ ⦃ℕℕ, 𝐶⦄ 𝐵 𝐶 ⦃ℕℕ, 𝑔⦄ − ⋄ 𝛽 − ⋄ 𝛽 𝑔

𝕱𝖙𝖉𝖇𝖘𝖊𝖕 𝖊𝖏𝖇𝖒𝖕𝖍𝖛𝖋𝖙

We can execute dialogue trees against a sequence 𝛽 ∈ ℕℕ: 𝜃(𝑨) ⋄ 𝛽 ≜ 𝑨 ϙ⟨𝑦⟩(𝑓) ⋄ 𝛽 ≜ 𝑓(𝛽(𝑦)) ⋄ 𝛽 ⦃ℕℕ, 𝐵⦄ ⦃ℕℕ, 𝐶⦄ 𝐵 ⦃ℕℕ, 𝐶⦄ 𝐶 𝑔⋆ − ⋄ 𝛽 − ⋄ 𝛽 𝑔 − ⋄ 𝛽

𝕰𝖏𝖇𝖒𝖋𝖉𝖚𝖏𝖉𝖇𝖒 𝖙𝖋𝖓𝖇𝖔𝖚𝖏𝖉𝖙 𝖕𝖌 𝕿𝖟𝖙𝖚𝖋𝖓 𝕌

Atomic Types 𝒲 [nat] ≜ ℕ Types 𝒲 ⟪𝜅⟫ ≜ ⦃ℕℕ, 𝒲 [𝜅]⦄ 𝒲 ⟪𝜏 → 𝜐⟫ ≜ 𝒲 ⟪𝜏⟫ → 𝒲 ⟪𝜐⟫ Contexts 𝒣 ⟪Γ⟫ ≜ ∏

𝑦∈|Γ|

𝒲 ⟪Γ(𝑦)⟫

𝕰𝖏𝖇𝖒𝖋𝖉𝖚𝖏𝖉𝖇𝖒 𝖙𝖋𝖓𝖇𝖔𝖚𝖏𝖉𝖙 𝖕𝖌 𝕿𝖟𝖙𝖚𝖋𝖓 𝕌

Presupposing 𝜍 ∈ 𝒣 ⟪Γ⟫ and Γ ⊢ 𝑛 ∶ 𝜏, we define the interpretation ⟪Γ ⊢ 𝑛 ∶ 𝜏⟫𝜍 ∈ 𝒲 ⟪𝜏⟫: ⟪Γ ⊢ 𝑦 ∶ 𝜏⟫𝜍 ≜ 𝜍(𝑦) ⟪Γ ⊢ z ∶ nat⟫𝜍 ≜ 𝜃(0) ⟪Γ ⊢ s(𝑛) ∶ nat⟫𝜍 ≜ ⦃ℕℕ, 1 + −⦄ (⟪Γ ⊢ 𝑛 ∶ nat⟫𝜍) ⟪Γ ⊢ rec𝜏([𝑦, 𝑧].𝑡[𝑦, 𝑧]; 𝑨; 𝑜) ∶ 𝜏⟫𝜍 ≜ PrimRec(𝑇, 𝑎, −)✪

𝜏 (𝑂)

where 𝑇(𝑏, 𝑐) ≜ ⟪Γ, 𝑦 ∶ nat, 𝑧 ∶ 𝜏 ⊢ 𝑡[𝑦, 𝑧] ∶ 𝜏⟫𝜍,𝑦↦𝑏,𝑧↦𝑐 𝑎 ≜ ⟪Γ ⊢ 𝑨 ∶ 𝜏⟫𝜍 𝑂 ≜ ⟪Γ ⊢ 𝑜 ∶ nat⟫𝜍

𝕰𝖏𝖇𝖒𝖋𝖉𝖚𝖏𝖉𝖇𝖒 𝖙𝖋𝖓𝖇𝖔𝖚𝖏𝖉𝖙 𝖕𝖌 𝕿𝖟𝖙𝖚𝖋𝖓 𝕌

⋮ ⟪Γ ⊢ 𝜇𝑦.𝑛[𝑦] ∶ 𝜏 → 𝜐⟫𝜍 ≜ 𝑏 ↦ ⟪Γ, 𝑦 ∶ 𝜏 ⊢ 𝑛[𝑦] ∶ 𝜐⟫𝜍,𝑦↦𝑏 ⟪Γ ⊢ 𝑛 •𝜏 𝑜 ∶ 𝜐⟫𝜍 ≜ (⟪Γ ⊢ 𝑛 ∶ 𝜏 → 𝜐⟫𝜍) (⟪Γ ⊢ 𝑜 ∶ 𝜏⟫𝜍)

𝕯𝖕𝖎𝖋𝖘𝖋𝖔𝖉𝖋 𝖜𝖏𝖇 𝕸𝖕𝖍𝖏𝖉𝖇𝖒 𝕾𝖋𝖒𝖇𝖚𝖏𝖕𝖔𝖙

We need to show that the two interpretations cohere at base type, i.e. that the following diagram commutes: ⋅ ⊢ 𝜅 𝒲 ⟪𝜅⟫ 𝒲 𝜅 ⟪⋅ ⊢ − ∶ 𝜅⟫ ⋅ ⊢ − ∶ 𝜅 − ⋄ 𝛽 Slogan: Strengthen The Inductive Hypothesis With Logical Relations As A Weapon!

𝕯𝖕𝖎𝖋𝖘𝖋𝖔𝖉𝖋 𝖜𝖏𝖇 𝕸𝖕𝖍𝖏𝖉𝖇𝖒 𝕾𝖋𝖒𝖇𝖚𝖏𝖕𝖔𝖙

We need to show that the two interpretations cohere at base type, i.e. that the following diagram commutes: ⋅ ⊢ 𝜅 𝒲 ⟪𝜅⟫ 𝒲 𝜅 ⟪⋅ ⊢ − ∶ 𝜅⟫ ⋅ ⊢ − ∶ 𝜅 − ⋄ 𝛽 Slogan: Strengthen The Inductive Hypothesis With Logical Relations As A Weapon!

𝕯𝖕𝖎𝖋𝖘𝖋𝖔𝖉𝖋 𝖜𝖏𝖇 𝕸𝖕𝖍𝖏𝖉𝖇𝖒 𝕾𝖋𝖒𝖇𝖚𝖏𝖕𝖔𝖙

“logical relations” = family of relations defined by compositionally on

• bject-level types

For all 𝛽 ∈ ℕℕ, define: ℛ𝛽

𝜏 ⊆ 𝒲 𝜏 × 𝒲 ⟪𝜏⟫

ℛ𝛽

Γ ⊆ 𝒣 Γ × 𝒣 ⟪Γ⟫

𝐺 = 𝑒 ⋄ 𝛽 𝐺 ℛ𝛽

𝜅 𝑒

∀𝐻 ∈ 𝒲 𝜏 , 𝑓 ∈ 𝒲 ⟪𝜏⟫. 𝐻 ℛ𝛽

𝜏 𝑓 ⟹ 𝐺(𝐻) ℛ𝛽 𝜐 𝑒(𝑓)

𝐺 ℛ𝛽

𝜏→𝜐 𝑒

∀𝑦 ∈ |Γ|. 𝜍0(𝑦) ℛ𝛽

Γ(𝑦) 𝜍1(𝑦)

𝜍0 ℛ𝛽

Γ 𝜍1

𝕯𝖕𝖎𝖋𝖘𝖋𝖔𝖉𝖋 𝖜𝖏𝖇 𝕸𝖕𝖍𝖏𝖉𝖇𝖒 𝕾𝖋𝖒𝖇𝖚𝖏𝖕𝖔𝖙

“logical relations” = family of relations defined by compositionally on

• bject-level types

For all 𝛽 ∈ ℕℕ, define: ℛ𝛽

𝜏 ⊆ 𝒲 𝜏 × 𝒲 ⟪𝜏⟫

ℛ𝛽

Γ ⊆ 𝒣 Γ × 𝒣 ⟪Γ⟫

𝐺 = 𝑒 ⋄ 𝛽 𝐺 ℛ𝛽

𝜅 𝑒

∀𝐻 ∈ 𝒲 𝜏, 𝑓 ∈ 𝒲 ⟪𝜏⟫. 𝐻 ℛ𝛽

𝜏 𝑓 ⟹ 𝐺(𝐻) ℛ𝛽 𝜐 𝑒(𝑓)

𝐺 ℛ𝛽

𝜏→𝜐 𝑒

∀𝑦 ∈ |Γ|. 𝜍0(𝑦) ℛ𝛽

Γ(𝑦) 𝜍1(𝑦)

𝜍0 ℛ𝛽

Γ 𝜍1

𝖀𝖎𝖋 𝕳𝖋𝖔𝖋𝖘𝖏𝖉 𝕼𝖕𝖏𝖔𝖚

In the dialogue model, we can define a so-called “generic point” which is not definable in System 𝕌: generic ∈ 𝒲 ⟪nat → nat⟫ generic ≜ (ϙ⟨−⟩(𝜃))⋆ The Generic Point Is The Magic Weapon To Victoriously Trace A Functional's Interaction With The Ambient Choice Sequence! — Quotations From Chairman Thierry Coquand

𝖀𝖎𝖋 𝕳𝖋𝖔𝖋𝖘𝖏𝖉 𝕼𝖕𝖏𝖔𝖚

In the dialogue model, we can define a so-called “generic point” which is not definable in System 𝕌: generic ∈ 𝒲 ⟪nat → nat⟫ generic ≜ (ϙ⟨−⟩(𝜃))⋆ The Generic Point Is The Magic Weapon To Victoriously Trace A Functional's Interaction With The Ambient Choice Sequence! — Quotations From Chairman Thierry Coquand

𝖀𝖇𝖑𝖏𝖔𝖍 𝖙𝖚𝖕𝖉𝖑

⋆ For any functional ⋅ ⊢ 𝜚 ∶ (nat → nat) → nat, we can compute a dialogue tree ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) ∈ ⦃ℕℕ, ℕ⦄. ⋆ Similar to a Brouwerian mental construction of ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆, but not the same! ⋆ Escardó’s trees represent 𝓺𝓯𝓼𝓽𝓳𝓽𝓾𝓯𝓸𝓾 𝓳𝓸𝓽𝓺𝓯𝓭𝓾𝓳𝓹𝓸 of a choice sequence; Brouwer’s trees represent 𝓯𝓺𝓲𝓯𝓷𝓯𝓼𝓫𝓶 𝓭𝓹𝓸𝓽𝓿𝓷𝓺𝓾𝓳𝓹𝓸 of a choice sequence. ⋆ Idea: Normalize dialogue trees into Brouwerian mental constructions.

𝖀𝖇𝖑𝖏𝖔𝖍 𝖙𝖚𝖕𝖉𝖑

⋆ For any functional ⋅ ⊢ 𝜚 ∶ (nat → nat) → nat, we can compute a dialogue tree ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) ∈ ⦃ℕℕ, ℕ⦄. ⋆ Similar to a Brouwerian mental construction of ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆, but not the same! ⋆ Escardó’s trees represent 𝓺𝓯𝓼𝓽𝓳𝓽𝓾𝓯𝓸𝓾 𝓳𝓸𝓽𝓺𝓯𝓭𝓾𝓳𝓹𝓸 of a choice sequence; Brouwer’s trees represent 𝓯𝓺𝓲𝓯𝓷𝓯𝓼𝓫𝓶 𝓭𝓹𝓸𝓽𝓿𝓷𝓺𝓾𝓳𝓹𝓸 of a choice sequence. ⋆ Idea: Normalize dialogue trees into Brouwerian mental constructions.

𝖀𝖇𝖑𝖏𝖔𝖍 𝖙𝖚𝖕𝖉𝖑

⋆ For any functional ⋅ ⊢ 𝜚 ∶ (nat → nat) → nat, we can compute a dialogue tree ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) ∈ ⦃ℕℕ, ℕ⦄. ⋆ Similar to a Brouwerian mental construction of ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆, but not the same! ⋆ Escardó’s trees represent 𝓺𝓯𝓼𝓽𝓳𝓽𝓾𝓯𝓸𝓾 𝓳𝓸𝓽𝓺𝓯𝓭𝓾𝓳𝓹𝓸 of a choice sequence; Brouwer’s trees represent 𝓯𝓺𝓲𝓯𝓷𝓯𝓼𝓫𝓶 𝓭𝓹𝓸𝓽𝓿𝓷𝓺𝓾𝓳𝓹𝓸 of a choice sequence. ⋆ Idea: Normalize dialogue trees into Brouwerian mental constructions.

𝖀𝖇𝖑𝖏𝖔𝖍 𝖙𝖚𝖕𝖉𝖑

⋆ For any functional ⋅ ⊢ 𝜚 ∶ (nat → nat) → nat, we can compute a dialogue tree ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) ∈ ⦃ℕℕ, ℕ⦄. ⋆ Similar to a Brouwerian mental construction of ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆, but not the same! ⋆ Escardó’s trees represent 𝓺𝓯𝓼𝓽𝓳𝓽𝓾𝓯𝓸𝓾 𝓳𝓸𝓽𝓺𝓯𝓭𝓾𝓳𝓹𝓸 of a choice sequence; Brouwer’s trees represent 𝓯𝓺𝓲𝓯𝓷𝓯𝓼𝓫𝓶 𝓭𝓹𝓸𝓽𝓿𝓷𝓺𝓾𝓳𝓹𝓸 of a choice sequence. ⋆ Idea: Normalize dialogue trees into Brouwerian mental constructions.

𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝖋𝖖𝖎𝖋𝖓𝖋𝖘𝖇𝖒 𝖊𝖏𝖇𝖒𝖋𝖉𝖚𝖏𝖉𝖙

Recall Escardó’s dialogues: 𝑨 ∈ 𝑎 𝜃(𝑨) ∈ ⦃ℕℕ, 𝑎⦄ return 𝑦 ∈ ℕ 𝑓 ∈ ℕ → ⦃ℕℕ, 𝑎⦄ ϙ⟨𝑦⟩(𝑓) ∈ ⦃ℕℕ, 𝑎⦄ query Brouwer’s take: 𝑨 ∈ 𝑎 𝜃(𝑨) ∈ ℕℕ, 𝑎 spit 𝑐 ∈ ℕ → ℕℕ, 𝑎 ϝ(𝑐) ∈ ℕℕ, 𝑎 bite Ephemeral execution: 𝜃(𝑨) ⟐ 𝛽 ≜ 𝑨 ϝ(𝑐) ⟐ 𝛽 ≜ 𝑐(head(𝛽)) ⟐ tail(𝛽)

𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝖋𝖖𝖎𝖋𝖓𝖋𝖘𝖇𝖒 𝖊𝖏𝖇𝖒𝖋𝖉𝖚𝖏𝖉𝖙

Recall Escardó’s dialogues: 𝑨 ∈ 𝑎 𝜃(𝑨) ∈ ⦃ℕℕ, 𝑎⦄ return 𝑦 ∈ ℕ 𝑓 ∈ ℕ → ⦃ℕℕ, 𝑎⦄ ϙ⟨𝑦⟩(𝑓) ∈ ⦃ℕℕ, 𝑎⦄ query Brouwer’s take: 𝑨 ∈ 𝑎 𝜃(𝑨) ∈ ℕℕ, 𝑎 spit 𝑐 ∈ ℕ → ℕℕ, 𝑎 ϝ(𝑐) ∈ ℕℕ, 𝑎 bite Ephemeral execution: 𝜃(𝑨) ⟐ 𝛽 ≜ 𝑨 ϝ(𝑐) ⟐ 𝛽 ≜ 𝑐(head(𝛽)) ⟐ tail(𝛽)

𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝖋𝖖𝖎𝖋𝖓𝖋𝖘𝖇𝖒 𝖊𝖏𝖇𝖒𝖋𝖉𝖚𝖏𝖉𝖙

Recall Escardó’s dialogues: 𝑨 ∈ 𝑎 𝜃(𝑨) ∈ ⦃ℕℕ, 𝑎⦄ return 𝑦 ∈ ℕ 𝑓 ∈ ℕ → ⦃ℕℕ, 𝑎⦄ ϙ⟨𝑦⟩(𝑓) ∈ ⦃ℕℕ, 𝑎⦄ query Brouwer’s take: 𝑨 ∈ 𝑎 𝜃(𝑨) ∈ ℕℕ, 𝑎 spit 𝑐 ∈ ℕ → ℕℕ, 𝑎 ϝ(𝑐) ∈ ℕℕ, 𝑎 bite Ephemeral execution: 𝜃(𝑨) ⟐ 𝛽 ≜ 𝑨 ϝ(𝑐) ⟐ 𝛽 ≜ 𝑐(head(𝛽)) ⟐ tail(𝛽)

𝕺𝖕𝖘𝖓𝖇𝖒𝖏𝖠𝖏𝖔𝖍 𝖊𝖏𝖇𝖒𝖕𝖍𝖛𝖋𝖙

Presupposing 𝑢 ∈ ⦃ℕℕ, 𝑎⦄, define total normalization relation ⃗ 𝑣 ⊩ 𝑢 𝑐 with 𝑐 ∈ ℕℕ, 𝑎. Then, define, ⃗ 𝑣 ⊩ 𝑢 𝑐 norm ⃗

𝑣(𝑢) ≜ 𝑐

Structurally recursive definition easy, but bureaucratic. See paper.

𝖀𝖎𝖋 𝕮𝖏𝖘𝖊𝖙'-𝕱𝖟𝖋 𝖂𝖏𝖋𝖝

⋅ ⊢ 𝜅 𝒲 [𝜅] ⦃ℕℕ, 𝒲 [𝜅]⦄ ℕℕ, 𝒲 [𝜅] ⋅ ⊢ − ∶ 𝜅 ⟪⋅ ⊢ − ∶ 𝜅⟫ − ⋄ 𝛽 norm⟨⟩(−) − ⟐ 𝛽

𝖀𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙

It suffices to show the validity of the following completeness rule: ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness

• Proof. Let 𝑢 ≜ ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ 𝑢

𝑐. By coherence, we have ⃗ 𝑣 ⊕ 𝛽[𝑐 ⟐ 𝛽 + | ⃗ 𝑣|] ∈ 𝔆 for all 𝛽. Proceed by induction on 𝑐 ∈ ℕℕ, ℕ . ⋆ Case 𝑐 ≡ 𝜃(0). Then we already have ⃗ 𝑣 ∈ 𝔆, so apply 𝜃 rule. ⋆ Case 𝑐 ≡ 𝜃(𝑙 + 1). Apply ϝ rule, and use IH at 𝜃(𝑙). ⋆ Case 𝑐 ≡ ϝ(𝑔). Apply ϝ rule, fixing 𝑧 ∈ ℕ; use IH at 𝑔(𝑧).

𝖀𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖕𝖘𝖋𝖓

It suffices to show the validity of the following completeness rule: ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness

• Proof. Let 𝑢 ≜ ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ 𝑢

𝑐. By coherence, we have ⃗ 𝑣 ⊕ 𝛽[𝑐 ⟐ 𝛽 + | ⃗ 𝑣|] ∈ 𝔆 for all 𝛽. Proceed by induction on 𝑐 ∈ ℕℕ, ℕ . ⋆ Case 𝑐 ≡ 𝜃(0). Then we already have ⃗ 𝑣 ∈ 𝔆, so apply 𝜃 rule. ⋆ Case 𝑐 ≡ 𝜃(𝑙 + 1). Apply ϝ rule, and use IH at 𝜃(𝑙). ⋆ Case 𝑐 ≡ ϝ(𝑔). Apply ϝ rule, fixing 𝑧 ∈ ℕ; use IH at 𝑔(𝑧).

𝖀𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖕𝖘𝖋𝖓

It suffices to show the validity of the following completeness rule: ⃗ 𝑣 ◁𝕌 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness

• Proof. Let 𝑢 ≜ ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ 𝑢

𝑐. By coherence, we have ⃗ 𝑣 ⊕ 𝛽[𝑐 ⟐ 𝛽 + | ⃗ 𝑣|] ∈ 𝔆 for all 𝛽. Proceed by induction on 𝑐 ∈ ℕℕ, ℕ . ⋆ Case 𝑐 ≡ 𝜃(0). Then we already have ⃗ 𝑣 ∈ 𝔆, so apply 𝜃 rule. ⋆ Case 𝑐 ≡ 𝜃(𝑙 + 1). Apply ϝ rule, and use IH at 𝜃(𝑙). ⋆ Case 𝑐 ≡ ϝ(𝑔). Apply ϝ rule, fixing 𝑧 ∈ ℕ; use IH at 𝑔(𝑧).

𝖀𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖕𝖘𝖋𝖓

It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ 𝜚 ∶ (nat → nat) → nat. ∀𝛽 ∈ ⃗ 𝑣. 𝛽[𝜚 ⟨𝛽⟩] ∈ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness

• Proof. Let 𝑢 ≜ ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ 𝑢

𝑐. By coherence, we have ⃗ 𝑣 ⊕ 𝛽[𝑐 ⟐ 𝛽 + | ⃗ 𝑣|] ∈ 𝔆 for all 𝛽. Proceed by induction on 𝑐 ∈ ℕℕ, ℕ . ⋆ Case 𝑐 ≡ 𝜃(0). Then we already have ⃗ 𝑣 ∈ 𝔆, so apply 𝜃 rule. ⋆ Case 𝑐 ≡ 𝜃(𝑙 + 1). Apply ϝ rule, and use IH at 𝜃(𝑙). ⋆ Case 𝑐 ≡ ϝ(𝑔). Apply ϝ rule, fixing 𝑧 ∈ ℕ; use IH at 𝑔(𝑧).

𝖀𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖕𝖘𝖋𝖓

It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ 𝜚 ∶ (nat → nat) → nat. ∀𝛽 ∈ ⟨⟩. ⃗ 𝑣 ⊕ 𝛽[𝜚 ⟨𝛽⟩ + | ⃗ 𝑣|] ∈ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness

• Proof. Let 𝑢 ≜ ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ 𝑢

𝑐. By coherence, we have ⃗ 𝑣 ⊕ 𝛽[𝑐 ⟐ 𝛽 + | ⃗ 𝑣|] ∈ 𝔆 for all 𝛽. Proceed by induction on 𝑐 ∈ ℕℕ, ℕ . ⋆ Case 𝑐 ≡ 𝜃(0). Then we already have ⃗ 𝑣 ∈ 𝔆, so apply 𝜃 rule. ⋆ Case 𝑐 ≡ 𝜃(𝑙 + 1). Apply ϝ rule, and use IH at 𝜃(𝑙). ⋆ Case 𝑐 ≡ ϝ(𝑔). Apply ϝ rule, fixing 𝑧 ∈ ℕ; use IH at 𝑔(𝑧).

𝖀𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖕𝖘𝖋𝖓

It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ 𝜚 ∶ (nat → nat) → nat. ∀𝛽 ∈ ⟨⟩. ⃗ 𝑣 ⊕ 𝛽[𝜚 ⟨𝛽⟩ + | ⃗ 𝑣|] ∈ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness

• Proof. Let 𝑢 ≜ ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ 𝑢 𝑐.

By coherence, we have ⃗ 𝑣 ⊕ 𝛽[𝑐 ⟐ 𝛽 + | ⃗ 𝑣|] ∈ 𝔆 for all 𝛽. Proceed by induction on 𝑐 ∈ ℕℕ, ℕ . ⋆ Case 𝑐 ≡ 𝜃(0). Then we already have ⃗ 𝑣 ∈ 𝔆, so apply 𝜃 rule. ⋆ Case 𝑐 ≡ 𝜃(𝑙 + 1). Apply ϝ rule, and use IH at 𝜃(𝑙). ⋆ Case 𝑐 ≡ ϝ(𝑔). Apply ϝ rule, fixing 𝑧 ∈ ℕ; use IH at 𝑔(𝑧).

𝖀𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖕𝖘𝖋𝖓

It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ 𝜚 ∶ (nat → nat) → nat. ∀𝛽 ∈ ⟨⟩. ⃗ 𝑣 ⊕ 𝛽[𝜚 ⟨𝛽⟩ + | ⃗ 𝑣|] ∈ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness

• Proof. Let 𝑢 ≜ ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ 𝑢 𝑐. By

coherence, we have ⃗ 𝑣 ⊕ 𝛽[𝑐 ⟐ 𝛽 + | ⃗ 𝑣|] ∈ 𝔆 for all 𝛽. Proceed by induction on 𝑐 ∈ ℕℕ, ℕ . ⋆ Case 𝑐 ≡ 𝜃(0). Then we already have ⃗ 𝑣 ∈ 𝔆, so apply 𝜃 rule. ⋆ Case 𝑐 ≡ 𝜃(𝑙 + 1). Apply ϝ rule, and use IH at 𝜃(𝑙). ⋆ Case 𝑐 ≡ ϝ(𝑔). Apply ϝ rule, fixing 𝑧 ∈ ℕ; use IH at 𝑔(𝑧).

𝖀𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖕𝖘𝖋𝖓

It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ 𝜚 ∶ (nat → nat) → nat. ∀𝛽 ∈ ⟨⟩. ⃗ 𝑣 ⊕ 𝛽[𝜚 ⟨𝛽⟩ + | ⃗ 𝑣|] ∈ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness

• Proof. Let 𝑢 ≜ ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ 𝑢 𝑐. By

coherence, we have ⃗ 𝑣 ⊕ 𝛽[𝑐 ⟐ 𝛽 + | ⃗ 𝑣|] ∈ 𝔆 for all 𝛽. Proceed by induction on 𝑐 ∈ ℕℕ, ℕ. ⋆ Case 𝑐 ≡ 𝜃(0). Then we already have ⃗ 𝑣 ∈ 𝔆, so apply 𝜃 rule. ⋆ Case 𝑐 ≡ 𝜃(𝑙 + 1). Apply ϝ rule, and use IH at 𝜃(𝑙). ⋆ Case 𝑐 ≡ ϝ(𝑔). Apply ϝ rule, fixing 𝑧 ∈ ℕ; use IH at 𝑔(𝑧).

𝖀𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖕𝖘𝖋𝖓

It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ 𝜚 ∶ (nat → nat) → nat. ∀𝛽 ∈ ⟨⟩. ⃗ 𝑣 ⊕ 𝛽[𝜚 ⟨𝛽⟩ + | ⃗ 𝑣|] ∈ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness

• Proof. Let 𝑢 ≜ ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ 𝑢 𝑐. By

coherence, we have ⃗ 𝑣 ⊕ 𝛽[𝑐 ⟐ 𝛽 + | ⃗ 𝑣|] ∈ 𝔆 for all 𝛽. Proceed by induction on 𝑐 ∈ ℕℕ, ℕ. ⋆ Case 𝑐 ≡ 𝜃(0). Then we already have ⃗ 𝑣 ∈ 𝔆, so apply 𝜃 rule. ⋆ Case 𝑐 ≡ 𝜃(𝑙 + 1). Apply ϝ rule, and use IH at 𝜃(𝑙). ⋆ Case 𝑐 ≡ ϝ(𝑔). Apply ϝ rule, fixing 𝑧 ∈ ℕ; use IH at 𝑔(𝑧).

𝖀𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖕𝖘𝖋𝖓

It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ 𝜚 ∶ (nat → nat) → nat. ∀𝛽 ∈ ⟨⟩. ⃗ 𝑣 ⊕ 𝛽[𝜚 ⟨𝛽⟩ + | ⃗ 𝑣|] ∈ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness

• Proof. Let 𝑢 ≜ ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ 𝑢 𝑐. By

coherence, we have ⃗ 𝑣 ⊕ 𝛽[𝑐 ⟐ 𝛽 + | ⃗ 𝑣|] ∈ 𝔆 for all 𝛽. Proceed by induction on 𝑐 ∈ ℕℕ, ℕ. ⋆ Case 𝑐 ≡ 𝜃(0). Then we already have ⃗ 𝑣 ∈ 𝔆, so apply 𝜃 rule. ⋆ Case 𝑐 ≡ 𝜃(𝑙 + 1). Apply ϝ rule, and use IH at 𝜃(𝑙). ⋆ Case 𝑐 ≡ ϝ(𝑔). Apply ϝ rule, fixing 𝑧 ∈ ℕ; use IH at 𝑔(𝑧).

𝖀𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖕𝖘𝖋𝖓

It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ 𝜚 ∶ (nat → nat) → nat. ∀𝛽 ∈ ⟨⟩. ⃗ 𝑣 ⊕ 𝛽[𝜚 ⟨𝛽⟩ + | ⃗ 𝑣|] ∈ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness

• Proof. Let 𝑢 ≜ ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ 𝑢 𝑐. By

coherence, we have ⃗ 𝑣 ⊕ 𝛽[𝑐 ⟐ 𝛽 + | ⃗ 𝑣|] ∈ 𝔆 for all 𝛽. Proceed by induction on 𝑐 ∈ ℕℕ, ℕ. ⋆ Case 𝑐 ≡ 𝜃(0). Then we already have ⃗ 𝑣 ∈ 𝔆, so apply 𝜃 rule. ⋆ Case 𝑐 ≡ 𝜃(𝑙 + 1). Apply ϝ rule, and use IH at 𝜃(𝑙). ⋆ Case 𝑐 ≡ ϝ(𝑔). Apply ϝ rule, fixing 𝑧 ∈ ℕ; use IH at 𝑔(𝑧).

𝖀𝖎𝖋 𝕮𝖇𝖘 𝖀𝖎𝖋𝖕𝖘𝖋𝖓

It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ 𝜚 ∶ (nat → nat) → nat. ∀𝛽 ∈ ⟨⟩. ⃗ 𝑣 ⊕ 𝛽[𝜚 ⟨𝛽⟩ + | ⃗ 𝑣|] ∈ 𝔆 ⃗ 𝑣 ◀𝑗𝑜𝑒 𝔆 completeness

• Proof. Let 𝑢 ≜ ⟪⋅ ⊢ 𝜚 ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ 𝑢 𝑐. By

coherence, we have ⃗ 𝑣 ⊕ 𝛽[𝑐 ⟐ 𝛽 + | ⃗ 𝑣|] ∈ 𝔆 for all 𝛽. Proceed by induction on 𝑐 ∈ ℕℕ, ℕ. ⋆ Case 𝑐 ≡ 𝜃(0). Then we already have ⃗ 𝑣 ∈ 𝔆, so apply 𝜃 rule. ⋆ Case 𝑐 ≡ 𝜃(𝑙 + 1). Apply ϝ rule, and use IH at 𝜃(𝑙). ⋆ Case 𝑐 ≡ ϝ(𝑔). Apply ϝ rule, fixing 𝑧 ∈ ℕ; use IH at 𝑔(𝑧).

𝕿𝖛𝖓𝖓𝖇𝖘𝖟 𝖕𝖌 𝕾𝖋𝖙𝖛𝖒𝖚𝖙

⋆ The Bar Theorem is 𝓭𝓹𝓸𝓽𝓾𝓼𝓿𝓭𝓾𝓳𝔀𝓯𝓶𝔃 𝔀𝓫𝓶𝓳𝓮 in primitive recursive realizability (with correct/full interpretation of functional types). ⋆ Thence, we have the 𝓒𝓫𝓼 𝓙𝓸𝓮𝓿𝓭𝓾𝓳𝓹𝓸 𝓠𝓼𝓳𝓸𝓭𝓳𝓺𝓶𝓯 and the 𝓖𝓫𝓸 𝓤𝓲𝓯𝓹𝓼𝓯𝓷 (constructive König’s Lemma). ⋆ What is the status of Brouwer’s Thesis as a 𝓽𝓭𝓳𝓯𝓸𝓾𝓳𝓰𝓳𝓭 𝓲𝔃𝓺𝓹𝓾𝓲𝓯𝓽𝓳𝓽?

𝕯𝖕𝖓𝖖𝖛𝖚𝖇𝖈𝖏𝖒𝖏𝖚𝖟 𝖇𝖚 𝕴𝖏𝖍𝖎𝖋𝖘 𝖀𝖟𝖖𝖋

𝖜𝖛𝖒𝖍𝖇𝖘 𝖓𝖋𝖉𝖎𝖇𝖔𝖏𝖉𝖇𝖒𝖏𝖙𝖓 vs. 𝓺𝓲𝓯𝓸𝓹𝓷𝓯𝓸𝓹𝓶𝓹𝓱𝔃

𝖜𝖛𝖒𝖍𝖇𝖘 𝖓𝖋𝖉𝖎𝖇𝖔𝖏𝖉𝖇𝖒𝖏𝖙𝖓 vs. 𝓺𝓲𝓯𝓸𝓹𝓷𝓯𝓸𝓹𝓶𝓹𝓱𝔃

𝕯𝖕𝖓𝖖𝖛𝖚𝖇𝖈𝖏𝖒𝖏𝖚𝖟 𝖇𝖚 𝕴𝖏𝖍𝖎𝖋𝖘 𝖀𝖟𝖖𝖋

Brouwer’s Thesis and Church’s Thesis express a 𝓽𝓭𝓳𝓯𝓸𝓾𝓳𝓰𝓳𝓭 hypothesis about the nature of higher-type computation. 𝓓𝓲𝓿𝓼𝓭𝓲'𝓽 𝓤𝓲𝓯𝓽𝓳𝓽 𝓒𝓼𝓹𝓿𝔁𝓯𝓼'𝓽 𝓤𝓲𝓯𝓽𝓳𝓽 general recursive 𝜇-calculus wellfounded mental construction + oracles Russian constructivism intuitionism vulgar/mechanicalist materialism subjective idealism uniform continuity fails on 𝕁 uniform continuity obtains

𝕯𝖕𝖓𝖖𝖛𝖚𝖇𝖈𝖏𝖒𝖏𝖚𝖟 𝖇𝖚 𝕴𝖏𝖍𝖎𝖋𝖘 𝖀𝖟𝖖𝖋

Brouwer’s Thesis and Church’s Thesis express a 𝓽𝓭𝓳𝓯𝓸𝓾𝓳𝓰𝓳𝓭 hypothesis about the nature of higher-type computation. 𝓓𝓲𝓿𝓼𝓭𝓲'𝓽 𝓤𝓲𝓯𝓽𝓳𝓽 𝓒𝓼𝓹𝓿𝔁𝓯𝓼'𝓽 𝓤𝓲𝓯𝓽𝓳𝓽 general recursive 𝜇-calculus wellfounded mental construction + oracles Russian constructivism intuitionism vulgar/mechanicalist materialism subjective idealism uniform continuity fails on 𝕁 uniform continuity obtains

𝕯𝖕𝖓𝖖𝖛𝖚𝖇𝖈𝖏𝖒𝖏𝖚𝖟 𝖇𝖚 𝕴𝖏𝖍𝖎𝖋𝖘 𝖀𝖟𝖖𝖋

Brouwer’s Thesis and Church’s Thesis express a 𝓽𝓭𝓳𝓯𝓸𝓾𝓳𝓰𝓳𝓭 hypothesis about the nature of higher-type computation. 𝓓𝓲𝓿𝓼𝓭𝓲'𝓽 𝓤𝓲𝓯𝓽𝓳𝓽 𝓒𝓼𝓹𝓿𝔁𝓯𝓼'𝓽 𝓤𝓲𝓯𝓽𝓳𝓽 general recursive 𝜇-calculus wellfounded mental construction + oracles Russian constructivism intuitionism vulgar/mechanicalist materialism subjective idealism uniform continuity fails on 𝕁 uniform continuity obtains

𝕯𝖕𝖓𝖖𝖛𝖚𝖇𝖈𝖏𝖒𝖏𝖚𝖟 𝖇𝖚 𝕴𝖏𝖍𝖎𝖋𝖘 𝖀𝖟𝖖𝖋

Brouwer’s Thesis and Church’s Thesis express a 𝓽𝓭𝓳𝓯𝓸𝓾𝓳𝓰𝓳𝓭 hypothesis about the nature of higher-type computation. 𝓓𝓲𝓿𝓼𝓭𝓲'𝓽 𝓤𝓲𝓯𝓽𝓳𝓽 𝓒𝓼𝓹𝓿𝔁𝓯𝓼'𝓽 𝓤𝓲𝓯𝓽𝓳𝓽 general recursive 𝜇-calculus wellfounded mental construction + oracles Russian constructivism intuitionism vulgar/mechanicalist materialism subjective idealism uniform continuity fails on 𝕁 uniform continuity obtains

𝕯𝖕𝖓𝖖𝖛𝖚𝖇𝖈𝖏𝖒𝖏𝖚𝖟 𝖇𝖚 𝕴𝖏𝖍𝖎𝖋𝖘 𝖀𝖟𝖖𝖋

Brouwer’s Thesis and Church’s Thesis express a 𝓽𝓭𝓳𝓯𝓸𝓾𝓳𝓰𝓳𝓭 hypothesis about the nature of higher-type computation. 𝓓𝓲𝓿𝓼𝓭𝓲'𝓽 𝓤𝓲𝓯𝓽𝓳𝓽 𝓒𝓼𝓹𝓿𝔁𝓯𝓼'𝓽 𝓤𝓲𝓯𝓽𝓳𝓽 general recursive 𝜇-calculus wellfounded mental construction + oracles Russian constructivism intuitionism vulgar/mechanicalist materialism subjective idealism uniform continuity fails on 𝕁 uniform continuity obtains