[00:00] Lecture I: Constructive mathematics ─────────────────────────────────── [02:12] Thm.: There are irrational numbers x and y such that x^y is rational. [03:25] Proof: Either √2^√2 is rational or not. [03:47] In the first case, we are done by x ≔ √2 and y ≔ √2. [04:14] In the second case, we are done by x ≔ √2^√2 and y ≔ √2. (Then x^y = √2^(√2·√2) = √2^2 = 2 ∈ ℚ.) [06:04] This proof is *nonconstructive*. It doesn't actually present a (guaranteed to be correct) example for such a pair of irrational numbers. [07:15] Proof: Set x ≔ √2 and y ≔ log_{√2} 3. (Then x^y = 3 ∈ ℚ.) [08:00] This alternative proof is constructive. It explicitly presents a witness of the claim. [09:00] Thm.: Given any natural number n ∈ ℕ, there is a prime p > n. [09:30] Proof (constructive): Any prime factor of n! + 1 will do. [12:30] Thm.: Every function α : ℕ → ℕ is good in the sense that for some i < j, α(i) ≤ α(j). [13:30] Proof (unconstructive): There is a minimal value α(i). Set j ≔ i + 1. [18:40] "Def.": Constructive mathematics = a flavor of mathematics centered around informative constructions [19:50] = the same as classical mathematics, but without LEM, DNE and AC [21:40] = mathematics built on intuitionistic logic instead of classical logic [23:00] LEM: φ ∨ ¬φ (for any formula φ) [24:50] DNE: ¬¬φ ⇒ φ (for any formula φ) [26:00] AC: axiom of choice [28:00] Note: This characterization of constructive mathematics is misleading! As we will learn in Lecture 3, constructive mathematics is actually an expansion of classical mathematics, not a restriction. [30:10] We need to distinguish the following two figures of proof: [30:40] 1. Claim: φ. Proof: Assume for the sake of contradiction that ¬φ. Then ..., ↯. Hence φ. [32:13] This is a proper proof by contradiction. Constructively, such an argument only shows ¬¬φ. We would need DNE or LEM to conclude φ from ¬¬φ. [32:30] 2. Claim: ¬ψ. Proof: Assume ψ. Then ..., ↯. Hence ¬ψ. [33:30] This is a constructively acceptable proof of a negated statement. [34:10] (By definition, ¬ψ :≡ (ψ ⇒ ↯).) [35:00] Thm.: √2 is not rational. Proof (constructive): Assume that √2 is rational. Then √2 = a/b, ..., ↯. [39:50] In constructive mathematics, we use the same logical symbols as in classical mathematics, but we interpret them differently (BHK interpretation): [40:50] statement classical meaning constructive meaning ───────── ───────────────── ────────────────────────── [41:30] φ φ is true. We have a witness for φ. [42:40] α ∧ β α and β are both true. We have a witness for α and one for β. [43:30] α ∨ β α is true or β is true. We have a witness for α, or we have one for β. [45:00] α ⇒ β If α is true, then so is β. We have a uniform procedure for turning witnesses for α into witnesses for β. [47:10] ¬α α is false. There is no witness for α. [47:45] ∀x:X. φ(x) For all x : X, φ(x) is true. We have a uniform procedure which inputs an arbitrary x : X and outputs a witness for φ(x). [50:20] ∃x:X. φ(x) For at least one x : X, We have an x : X and a witness for φ(x). φ(x) is true. [51:45] Ex.: ¬(α ∧ β) ⇒ ((¬α) ∨ (¬β)). [53:55] unconstructive! [55:10] Ex.: ¬¬φ ⇒ φ. unconstructive [56:50] Ex.: φ ⇒ ¬¬φ. [57:30] constructive [58:00] Ex.: ¬¬∃x. the key is at position x vs. ∃x. the key is at position x. [60:30] Constructive mathematics supports finer distinctions than classical mathematics. [64:00] What more do we know if we have proved a theorem by restricted means than if we merely know the theorem is true? ─Georg Kreisel What are uses of constructive mathematics? [66:00] - fun [66:55] - philosophy [67:10] - mental hygiene [68:15] - better appreciation for classical logic [69:20] - elegance assistance [72:50] - program extraction [74:45] - automatic parameter-dependence [78:00] - axiomatic freedom, internal language of toposes [81:50] - bounds, growth rates, ... [86:00] Prop.: 1. Every inhabited detachable set of natural numbers contains a minimum. [87:00] 2. Every inhabited set of natural numbers does *not not* contain a minimum. [87:30] 3. If every inhabited set of natural numbers contains a minimum, then LEM. [89:10] Def.: A set X is *inhabited* if and only if ∃x:X. [89:20] A set X is *nonempty* if and only if ¬(X = ∅), i.e. if and only if ¬¬∃x:X. [92:00] Def.: A set X ⊆ ℕ is *detachable* if and only if ∀n:ℕ. n ∈ X ∨ ¬(n ∈ X).