Primitive Recursive Bars Are Inductive

The Church-Turing Thesis is a scientific hypothesis as to the nature of effective computation at lower type; the nature of higher type computability, however, is far from settled and admits many possible models. In the 1920s, L.E.J. Brouwer (the father of Intuitionism) gave a mysterious "proof" of an induction principle for infinite trees called the Bar Theorem, which has many consequences, including a constructive contrapositive to König's Lemma (the Fan Theorem).
We situate the Bar Theorem more generally as "Brouwer's Thesis", a scientific claim about the nature of higher-type computability which may be contrasted with competing explanations, including recursive realizability and Turing machines. Then, we demonstrate using a technique called Effectful Forcing that Brouwer's Bar Theorem is constructively valid for any bar which can be written in Gödel's System T of primitive recursive arithmetic of finite type, a very simple programming language.

Transcript

1. Primitive Recursive Bars Are Inductive Jon Sterling Carnegie Mellon University August 17, 2016
2. ' “All bars on a spread can be coded as inductive trees.”
3. ' “All bars on a spread can be coded as inductive trees.” ⋆ ⇒ the Fan Theorem (intuitionistic König’s Lemma)
4. ' “All bars on a spread can be coded as inductive trees.” ⋆ ⇒ the Fan Theorem (intuitionistic König’s Lemma) ⋆ ⇒ all functions on the interval ≜ [0, 1] are uniformly continuous
5. ' “All bars on a spread can be coded as inductive trees.” ⋆ ⇒ the Fan Theorem (intuitionistic König’s Lemma) ⋆ ⇒ all functions on the interval ≜ [0, 1] are uniformly continuous ⋆ ⇒ completeness of intuitionistic first-order logic (Bickford, Constable)
6. ' “All bars on a spread can be coded as inductive trees.” ⋆ ⇒ the Fan Theorem (intuitionistic König’s Lemma) ⋆ ⇒ all functions on the interval ≜ [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.
7. A is at the same time a topological space and a non-wellfounded tree.
8. A is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate on lists of natural numbers subject to some laws:
9. A is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate on lists of natural numbers subject to some laws: 1. If ⃗ ∈ , then there exists an ∈ ℕ such that ⃗ ⌢ ∈ .
10. A is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate on lists of natural numbers subject to some laws: 1. If ⃗ ∈ , then there exists an ∈ ℕ such that ⃗ ⌢ ∈ . 2. If ⃗ ⌢ ∈ , then also ⃗ ∈ .
11. A is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate on lists of natural numbers subject to some laws: 1. If ⃗ ∈ , then there exists an ∈ ℕ such that ⃗ ⌢ ∈ . 2. If ⃗ ⌢ ∈ , then also ⃗ ∈ . 3. Finally, ⟨⟩ ∈ .
12. A is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate on lists of natural numbers subject to some laws: 1. If ⃗ ∈ , then there exists an ∈ ℕ such that ⃗ ⌢ ∈ . 2. If ⃗ ⌢ ∈ , then also ⃗ ∈ . 3. Finally, ⟨⟩ ∈ . The predicate either defines the finite prefixes of paths down an infinite tree, or it defines the lattice of open sets of a topological space.
14. A list of naturals ⃗ ∈ 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 ∈ ℕℕ can be thought of as an ideal point in the spread (space), or as a path through the spread’s infinite tree. ⃗ ≺ ( ⃗ approximates ) ∈ ⃗ ( ⃗ is a neighborhood around )
15. A bar is a predicate on neighborhoods such that every point “hits it”. More generally, bars a neighborhood ⃗ when every path through ⃗ ends up in .
16. A bar is a predicate on neighborhoods such that every point “hits it”. More generally, bars a neighborhood ⃗ when every path through ⃗ ends up in . ∀ ∈ ⃗ . ∃ ∈ ℕ. [] ∈ ⃗ ◁
19. “All demonstrations of barhood can be analyzed into inductive mental constructions.”
20. “All demonstrations of barhood can be analyzed into inductive mental constructions.” Let ♮( ⃗ ) ≜ { ∈ ℕ ∣ ⃗ ⌢ ∈ }. Presupposing ⃗ ∈ and monotone:
21. “All demonstrations of barhood can be analyzed into inductive mental constructions.” Let ♮( ⃗ ) ≜ { ∈ ℕ ∣ ⃗ ⌢ ∈ }. Presupposing ⃗ ∈ and monotone: ⃗ ∈ ⃗ ◀ ∀ ∈ ♮( ⃗ ). ⃗ ⌢ ◀ ⃗ ◀ ϝ
22. “All demonstrations of barhood can be analyzed into inductive mental constructions.” Let ♮( ⃗ ) ≜ { ∈ ℕ ∣ ⃗ ⌢ ∈ }. Presupposing ⃗ ∈ and monotone: ⃗ ∈ ⃗ ◀ ∀ ∈ ♮( ⃗ ). ⃗ ⌢ ◀ ⃗ ◀ ϝ Admissible (by monotonicity): ⃗ ◀ ⃗ ⌢ ◀
23. ' Recall ⃗ ◁ ≜ ∀ ∈ ⃗ . ∃ ∈ ℕ. [] ∈ .
24. ' Recall ⃗ ◁ ≜ ∀ ∈ ⃗ . ∃ ∈ ℕ. [] ∈ . Brouwer’s Bar Thesis is the of the inductive coding of barhood: ⃗ ◁ ⃗ ◀ BT?
25. ' Recall ⃗ ◁ ≜ ∀ ∈ ⃗ . ∃ ∈ ℕ. [] ∈ . Brouwer’s Bar Thesis is the of the inductive coding of barhood: ⃗ ◀ ⃗ ◁ soundness ⃗ ◁ ⃗ ◀ completeness?
26. ' ⃗ ◀ ⃗ ◁ soundness ⃗ ◁ ⃗ ◀ completeness?
27. ' ⃗ ◀ ⃗ ◁ soundness ⃗ ◁ ⃗ ◀ completeness? ⋆ soundness is easy! Just count up the ϝ-nodes.
28. ' ⃗ ◀ ⃗ ◁ soundness ⃗ ◁ ⃗ ◀ completeness? ⋆ soundness is easy! Just count up the ϝ-nodes. ⋆ completeness does not (generally) hold: procedure exists, but its termination requires Brouwer’s Thesis!
29. ⋆ Classically: valid
30. ⋆ Classically: valid ⋆ Constructively: not valid
31. ⋆ Classically: valid ⋆ Constructively: not valid ⋆ Intuitionistically: valid(*)
32. Let’s instantiate the Brouwer-Heyting-Kolmogorov interpretation at a simple theory of constructions!
33. Gödel’s System of primitive recursive functionals of finite type.
34. Gödel’s System of primitive recursive functionals of finite type. Atomic Types nat atype Types atype type type type → type Contexts ⋅ ctx Γ ctx type Γ, ∶ ctx ( ∉ Γ)
35. Γ, ∶ , Δ ⊢ ∶ var Γ ⊢ z ∶ nat zero Γ ⊢ ∶ nat Γ ⊢ s() ∶ nat succ Γ, ∶ nat, ∶ ⊢ [, ] ∶ Γ ⊢ ∶ Γ ⊢ ∶ nat Γ ⊢ rec ([, ].[, ]; ; ) ∶ rec Γ, ∶ ⊢ [] ∶ Γ ⊢ .[] ∶ → lam Γ ⊢ ∶ → Γ ⊢ ∶ Γ ⊢ • ∶ ap
36. ⋆ Use System as a theory of constructions for primitive recursive arithmetic
37. ⋆ Use System as a theory of constructions for primitive recursive arithmetic ⋆ ⃗ ◁ should be realized by a functional ⋅ ⊢ ∶ (nat → nat) → nat
38. ⋆ Use System as a theory of constructions for primitive recursive arithmetic ⋆ ⃗ ◁ should be realized by a functional ⋅ ⊢ ∶ (nat → nat) → nat ⋆ Idea: construct a model for System in which we can read from the interpretation of a proof of ⃗ ◀
39. Atomic Types [nat] ≜ ℕ Types ≜ [] → ≜ → Contexts Γ ≜ ∏ ∈|Γ| Γ()
40. Presupposing ∈ Γ and Γ ⊢ ∶ , define Γ ⊢ ∶ ∈ by recursion on : Γ ⊢ ∶ ≜ () Γ ⊢ z ∶ nat ≜ 0 Γ ⊢ s() ∶ nat ≜ 1 + Γ ⊢ ∶ nat Γ ⊢ rec ([, ].[, ]; ; ) ∶ ≜ PrimRec(, , ) where (, ) ≜ Γ, ∶ nat, ∶ ⊢ [, ] ∶ ,↦,↦ ≜ Γ ⊢ ∶ ≜ Γ ⊢ ∶ nat
41. ⋮ Γ ⊢ .[] ∶ → ≜ ↦ Γ, ∶ ⊢ [] ∶ ,↦ Γ ⊢ • ∶ ≜ ( Γ ⊢ ∶ → ) ( Γ ⊢ ∶ )
42. A functional ⋅ ⊢ ∶ (nat → nat) → nat can be applied to a meta-level sequence as follows: ⟨⟩ ≜ ⋅ ⊢ ∶ (nat → nat) → nat ()
43. A functional ⋅ ⊢ ∶ (nat → nat) → nat can be applied to a meta-level sequence as follows: ⟨⟩ ≜ ⋅ ⊢ ∶ (nat → nat) → nat () Redefine barhood as follows: ∀ ∈ ⃗ . ∃ ∈ ℕ. [] ∈ ⃗ ◁
44. A functional ⋅ ⊢ ∶ (nat → nat) → nat can be applied to a meta-level sequence as follows: ⟨⟩ ≜ ⋅ ⊢ ∶ (nat → nat) → nat () Redefine barhood as follows: ∃⋅ ⊢ ∶ (nat → nat) → nat. ∀ ∈ ⃗ . [ ⟨⟩] ∈ ⃗ ◁
45. An inductive encoding of functionals ℕℕ → : ∈ () ∈ ⦃ℕℕ, ⦄ return ∈ ℕ ∈ ℕ → ⦃ℕℕ, ⦄ ϙ⟨⟩() ∈ ⦃ℕℕ, ⦄ query
46. An inductive encoding of functionals ℕℕ → : ∈ () ∈ ⦃ℕℕ, ⦄ return ∈ ℕ ∈ ℕ → ⦃ℕℕ, ⦄ ϙ⟨⟩() ∈ ⦃ℕℕ, ⦄ query ⋆ ⦃ℕℕ, −⦄ is an endofunctor on Set
47. An inductive encoding of functionals ℕℕ → : ∈ () ∈ ⦃ℕℕ, ⦄ return ∈ ℕ ∈ ℕ → ⦃ℕℕ, ⦄ ϙ⟨⟩() ∈ ⦃ℕℕ, ⦄ query ⋆ ⦃ℕℕ, −⦄ is an endofunctor on Set ⋆ ⦃ℕℕ, −⦄ is a monad on Set
48. An inductive encoding of functionals ℕℕ → : ∈ () ∈ ⦃ℕℕ, ⦄ return ∈ ℕ ∈ ℕ → ⦃ℕℕ, ⦄ ϙ⟨⟩() ∈ ⦃ℕℕ, ⦄ query ⋆ ⦃ℕℕ, −⦄ is an endofunctor on Set ⋆ ⦃ℕℕ, −⦄ is a monad on Set souped up neighborhood functions!
49. We can execute dialogue trees against a sequence ∈ ℕℕ: () ⋄ ≜ ϙ⟨⟩() ⋄ ≜ (()) ⋄
50. We can execute dialogue trees against a sequence ∈ ℕℕ: () ⋄ ≜ ϙ⟨⟩() ⋄ ≜ (()) ⋄ ⦃ℕℕ, ⦄ ⦃ℕℕ, ⦄ ⦃ℕℕ, ⦄ − ⋄ − ⋄
51. We can execute dialogue trees against a sequence ∈ ℕℕ: () ⋄ ≜ ϙ⟨⟩() ⋄ ≜ (()) ⋄ ⦃ℕℕ, ⦄ ⦃ℕℕ, ⦄ ⦃ℕℕ, ⦄ ⋆ − ⋄ − ⋄ − ⋄
52. Atomic Types [nat] ≜ ℕ Types ⟪⟫ ≜ ⦃ℕℕ, []⦄ ⟪ → ⟫ ≜ ⟪⟫ → ⟪⟫ Contexts ⟪Γ⟫ ≜ ∏ ∈|Γ| ⟪Γ()⟫
53. Presupposing ∈ ⟪Γ⟫ and Γ ⊢ ∶ , we define the interpretation ⟪Γ ⊢ ∶ ⟫ ∈ ⟪⟫: ⟪Γ ⊢ ∶ ⟫ ≜ () ⟪Γ ⊢ z ∶ nat⟫ ≜ (0) ⟪Γ ⊢ s() ∶ nat⟫ ≜ ⦃ℕℕ, 1 + −⦄ (⟪Γ ⊢ ∶ nat⟫ ) ⟪Γ ⊢ rec ([, ].[, ]; ; ) ∶ ⟫ ≜ PrimRec(, , −)✪ () where (, ) ≜ ⟪Γ, ∶ nat, ∶ ⊢ [, ] ∶ ⟫ ,↦,↦ ≜ ⟪Γ ⊢ ∶ ⟫ ≜ ⟪Γ ⊢ ∶ nat⟫
54. ⋮ ⟪Γ ⊢ .[] ∶ → ⟫ ≜ ↦ ⟪Γ, ∶ ⊢ [] ∶ ⟫ ,↦ ⟪Γ ⊢ • ∶ ⟫ ≜ (⟪Γ ⊢ ∶ → ⟫ ) (⟪Γ ⊢ ∶ ⟫ )
55. We need to show that the two interpretations cohere at base type, i.e. that the following diagram commutes: ⋅ ⊢ ⟪⟫ ⟪⋅ ⊢ − ∶ ⟫ ⋅ ⊢ − ∶ − ⋄
56. We need to show that the two interpretations cohere at base type, i.e. that the following diagram commutes: ⋅ ⊢ ⟪⟫ ⟪⋅ ⊢ − ∶ ⟫ ⋅ ⊢ − ∶ − ⋄ Slogan: Strengthen The Inductive Hypothesis With Logical Relations As A Weapon!
57. “logical relations” = family of relations defined by compositionally on object-level types
58. “logical relations” = family of relations defined by compositionally on object-level types For all ∈ ℕℕ, define: ℛ ⊆ × ⟪⟫ ℛ Γ ⊆ Γ × ⟪Γ⟫ = ⋄ ℛ ∀ ∈ , ∈ ⟪⟫. ℛ ⟹ () ℛ () ℛ → ∀ ∈ |Γ|. 0 () ℛ Γ() 1 () 0 ℛ Γ 1
59. In the dialogue model, we can define a so-called “generic point” which is not definable in System : generic ∈ ⟪nat → nat⟫ generic ≜ (ϙ⟨−⟩())⋆
60. In the dialogue model, we can define a so-called “generic point” which is not definable in System : generic ∈ ⟪nat → nat⟫ generic ≜ (ϙ⟨−⟩())⋆ The Generic Point Is The Magic Weapon To Victoriously Trace A Functional's Interaction With The Ambient Choice Sequence! — Quotations From Chairman Thierry Coquand
61. ⋆ For any functional ⋅ ⊢ ∶ (nat → nat) → nat, we can compute a dialogue tree ⟪⋅ ⊢ ∶ (nat → nat) → nat⟫ (generic) ∈ ⦃ℕℕ, ℕ⦄.
62. ⋆ For any functional ⋅ ⊢ ∶ (nat → nat) → nat, we can compute a dialogue tree ⟪⋅ ⊢ ∶ (nat → nat) → nat⟫ (generic) ∈ ⦃ℕℕ, ℕ⦄. ⋆ Similar to a Brouwerian mental construction of ⃗ ◀ , but not the same!
63. ⋆ For any functional ⋅ ⊢ ∶ (nat → nat) → nat, we can compute a dialogue tree ⟪⋅ ⊢ ∶ (nat → nat) → nat⟫ (generic) ∈ ⦃ℕℕ, ℕ⦄. ⋆ Similar to a Brouwerian mental construction of ⃗ ◀ , but not the same! ⋆ Escardó’s trees represent of a choice sequence; Brouwer’s trees represent of a choice sequence.
64. ⋆ For any functional ⋅ ⊢ ∶ (nat → nat) → nat, we can compute a dialogue tree ⟪⋅ ⊢ ∶ (nat → nat) → nat⟫ (generic) ∈ ⦃ℕℕ, ℕ⦄. ⋆ Similar to a Brouwerian mental construction of ⃗ ◀ , but not the same! ⋆ Escardó’s trees represent of a choice sequence; Brouwer’s trees represent of a choice sequence. ⋆ Idea: Normalize dialogue trees into Brouwerian mental constructions.
65. ' Recall Escardó’s dialogues: ∈ () ∈ ⦃ℕℕ, ⦄ return ∈ ℕ ∈ ℕ → ⦃ℕℕ, ⦄ ϙ⟨⟩() ∈ ⦃ℕℕ, ⦄ query
66. ' Recall Escardó’s dialogues: ∈ () ∈ ⦃ℕℕ, ⦄ return ∈ ℕ ∈ ℕ → ⦃ℕℕ, ⦄ ϙ⟨⟩() ∈ ⦃ℕℕ, ⦄ query Brouwer’s take: ∈ () ∈ ℕℕ, spit ∈ ℕ → ℕℕ, ϝ() ∈ ℕℕ, bite
67. ' Recall Escardó’s dialogues: ∈ () ∈ ⦃ℕℕ, ⦄ return ∈ ℕ ∈ ℕ → ⦃ℕℕ, ⦄ ϙ⟨⟩() ∈ ⦃ℕℕ, ⦄ query Brouwer’s take: ∈ () ∈ ℕℕ, spit ∈ ℕ → ℕℕ, ϝ() ∈ ℕℕ, bite Ephemeral execution: () ⟐ ≜ ϝ() ⟐ ≜ (head()) ⟐ tail()
68. Presupposing ∈ ⦃ℕℕ, ⦄, define total normalization relation ⃗ ⊩ with ∈ ℕℕ, .
69. Presupposing ∈ ⦃ℕℕ, ⦄, define total normalization relation ⃗ ⊩ with ∈ ℕℕ, . Naïve Algorithm: norm ⃗ (()) ≜ () norm ⃗ (ϙ⟨⟩()) ≜ { ⃗ if < | ⃗ | ϝ( ↦ norm ⃗ ⌢ (ϙ⟨⟩())) if ≥ | ⃗ | norm ⃗ () ≡ ⃗ ⊩ Structurally recursive definition easy, but bureaucratic. See paper.
70. '- ⋅ ⊢ [] ⦃ℕℕ, []⦄ ℕℕ, [] ⋅ ⊢ − ∶ ⟪⋅ ⊢ − ∶ ⟫ − ⋄ norm⟨⟩ (−) − ⟐
73. It suffices to show the validity of the following completeness rule: ⃗ ◁ ⃗ ◀ completeness
74. It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ ∶ (nat → nat) → nat. ∀ ∈ ⃗ . [ ⟨⟩] ∈ ⃗ ◀ completeness
75. It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ ∶ (nat → nat) → nat. ∀ ∈ ⟨⟩. ⃗ ⊕ [ ⟨⟩ + | ⃗ |] ∈ ⃗ ◀ completeness
76. It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ ∶ (nat → nat) → nat. ∀ ∈ ⟨⟩. ⃗ ⊕ [ ⟨⟩ + | ⃗ |] ∈ ⃗ ◀ completeness Proof. Let ≜ ⟪⋅ ⊢ ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ .
77. It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ ∶ (nat → nat) → nat. ∀ ∈ ⟨⟩. ⃗ ⊕ [ ⟨⟩ + | ⃗ |] ∈ ⃗ ◀ completeness Proof. Let ≜ ⟪⋅ ⊢ ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ . By coherence, we have ⃗ ⊕ [ ⟐ + | ⃗ |] ∈ for all .
78. It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ ∶ (nat → nat) → nat. ∀ ∈ ⟨⟩. ⃗ ⊕ [ ⟨⟩ + | ⃗ |] ∈ ⃗ ◀ completeness Proof. Let ≜ ⟪⋅ ⊢ ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ . By coherence, we have ⃗ ⊕ [ ⟐ + | ⃗ |] ∈ for all . Proceed by induction on ∈ ℕℕ, ℕ .
79. It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ ∶ (nat → nat) → nat. ∀ ∈ ⟨⟩. ⃗ ⊕ [ ⟨⟩ + | ⃗ |] ∈ ⃗ ◀ completeness Proof. Let ≜ ⟪⋅ ⊢ ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ . By coherence, we have ⃗ ⊕ [ ⟐ + | ⃗ |] ∈ for all . Proceed by induction on ∈ ℕℕ, ℕ . ⋆ Case ≡ (0). Then we already have ⃗ ∈ , so apply rule.
80. It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ ∶ (nat → nat) → nat. ∀ ∈ ⟨⟩. ⃗ ⊕ [ ⟨⟩ + | ⃗ |] ∈ ⃗ ◀ completeness Proof. Let ≜ ⟪⋅ ⊢ ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ . By coherence, we have ⃗ ⊕ [ ⟐ + | ⃗ |] ∈ for all . Proceed by induction on ∈ ℕℕ, ℕ . ⋆ Case ≡ (0). Then we already have ⃗ ∈ , so apply rule. ⋆ Case ≡ ( + 1). Apply ϝ rule, and use IH at ().
81. It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ ∶ (nat → nat) → nat. ∀ ∈ ⟨⟩. ⃗ ⊕ [ ⟨⟩ + | ⃗ |] ∈ ⃗ ◀ completeness Proof. Let ≜ ⟪⋅ ⊢ ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ . By coherence, we have ⃗ ⊕ [ ⟐ + | ⃗ |] ∈ for all . Proceed by induction on ∈ ℕℕ, ℕ . ⋆ Case ≡ (0). Then we already have ⃗ ∈ , so apply rule. ⋆ Case ≡ ( + 1). Apply ϝ rule, and use IH at (). ⋆ Case ≡ ϝ(). Apply ϝ rule, fixing ∈ ℕ; use IH at ().
82. It suffices to show the validity of the following completeness rule: ∃⋅ ⊢ ∶ (nat → nat) → nat. ∀ ∈ ⟨⟩. ⃗ ⊕ [ ⟨⟩ + | ⃗ |] ∈ ⃗ ◀ completeness Proof. Let ≜ ⟪⋅ ⊢ ∶ (nat → nat) → nat⟫ (generic) such that ⟨⟩ ⊩ . By coherence, we have ⃗ ⊕ [ ⟐ + | ⃗ |] ∈ for all . Proceed by induction on ∈ ℕℕ, ℕ . ⋆ Case ≡ (0). Then we already have ⃗ ∈ , so apply rule. ⋆ Case ≡ ( + 1). Apply ϝ rule, and use IH at (). ⋆ Case ≡ ϝ(). Apply ϝ rule, fixing ∈ ℕ; use IH at ().
83. ⋆ The Bar Theorem is in primitive recursive realizability (with correct/full interpretation of functional types). ⋆ Thence, we have the and the (constructive König’s Lemma). ⋆ What is the status of Brouwer’s Thesis as a ?
85. vs.
86. Brouwer’s Thesis and Church’s Thesis express a hypothesis about the nature of higher-type computation.
87. Brouwer’s Thesis and Church’s Thesis express a hypothesis about the nature of higher-type computation. ' ' general recursive -calculus wellfounded mental construction + oracles
88. Brouwer’s Thesis and Church’s Thesis express a hypothesis about the nature of higher-type computation. ' ' general recursive -calculus wellfounded mental construction + oracles Russian constructivism intuitionism
89. Brouwer’s Thesis and Church’s Thesis express a hypothesis about the nature of higher-type computation. ' ' general recursive -calculus wellfounded mental construction + oracles Russian constructivism intuitionism vulgar/mechanicalist materialism subjective idealism
90. Brouwer’s Thesis and Church’s Thesis express a hypothesis about the nature of higher-type computation. ' ' general recursive -calculus wellfounded mental construction + oracles Russian constructivism intuitionism vulgar/mechanicalist materialism subjective idealism uniform continuity fails on uniform continuity obtains