Note on Π 0 n + 1 - LEM , Σ 0 n + 1 - LEM and Σ 0 n + 1 - DNE

In [1] Akama, Berardi, Hayashi and Kohlenbach used a monotone modified realizability interpretation to establish the relative independence of Σ n+1-DNE from Π 0 n+1-LEM over HA, and hence the independence of Σ n+1-LEM from Π 0 n+1-LEM over HA, for all n ≥ 0. We show that the same relative independence results hold for these arithmetical principles over Kleene and Vesley’s system FIM of intuitionistic analysis [3], which extends HA and is consistent with PA but not with classical analysis. The double negations of the closures of Σ n+1-LEM, Σ n+1-DNE and Π 0 n+1-LEM are also considered, and shown to behave differently with respect to HA and FIM. Various elementary questions remain to be answered. Definitions of the Arithmetical Principles. Unless otherwise noted, “LEM” (Law of Excluded Middle), “DNE” (Double Negation Elimination), and “LLPO” (Lesser Limited Principle of Omniscience) denote the (universal closures of the) purely arithmetical schemas, without function variables. If Φ is Σ n or Π 0 n for some n ≥ 1 then (i) Φ-LEM is A ∨ ¬A where A ∈ Φ. (ii) Φ-DNE is ¬¬A → A where A ∈ Φ. (iii) Φ-LLPO is ¬(A ∨ B) → (C ∨ D) , where A,B ∈ Φ and C,D are the duals of A,B respectively. (iv) ∆n-LEM is (A ↔ B) → (B ∨ ¬B) where A ∈ Π n and B ∈ Σ n. The precise statement of ∆n-LEM is important, since Σ n+1-DNE is equivalent over HA + Σ n-LEM to the schema (¬A ↔ B) → (A ∨ ¬A) where A,B ∈ Σ n+1. Kleene used this principle for n = 0 to prove that every ∆1 relation is recursive. The corresponding observation for n ≥ 0 is the KleenePost-Mostowski Theorem. I am grateful to Ulrich Kohlenbach for pointing me to [1], and to the organizers of the 2005 Oberwolfach conference on Proof Theory and Constructive Mathematics for a terrific mathematical experience. 1 Not even with ∀α[∀x(α(x) = 0) ∨ (¬∀xα(x) = 0)]. In contrast, the extension of Markov’s Principle (Σ 1-DNE) to the two-sorted language is consistent with FIM. In FIM + MP (but not in FIM) it is possible to prove that the constructive arithmetical hierarchy is proper; cf. [5], which also shows that FIM is not conservative over HA with respect to arithmetical formulas. 100 Joan R. Moschovakis 1 Some Results of Akama, Berardi, Hayashi and Kohlenbach Extended to FIM Lemma 1. The following are equivalent, for any theory T ⊇ HA: (i) T + Π 1 -LEM proves Σ 1 -LEM. (ii) T + Π 1 -LEM proves Markov’s Principle Σ 0 1 -DNE. Proof. (i) ⇒ (ii) holds because decidable predicates are stable under double negation. (ii) ⇒ (i) holds because [∀x¬R(x) ∨ ¬∀x¬R(x)] & [¬¬∃xR(x) → ∃xR(x)] → [∃xR(x) ∨ ¬∃xR(x)] . Now let T (e, x, y) be a quantifier-free formula numeralwise expressing in HA (hence also in FIM) the Kleene T-predicate, and let z ≤ U(y) be a quantifier-free formula numeralwise expressing in HA (hence also in FIM) the relation “z ≤ U(y)” where U(y) is the value computed by the computation with gödel number y, or the gödel number of y if y is not the gödel number of a computation. With Kleene’s coding HA proves ∀e∀x∀y[T (e, x, y) → ∀z(z ≤ U(y) → ¬T (e, x, z))], and we will use this property to prove the next lemma. Lemma 2. HA (hence also FIM) proves ∀f¬∀x∃y[T (f, x, y) ∧ [∀zz≤U(y)¬T (x, x, z) → ∀y¬T (x, x, y)]] . Proof. Assume for contradiction ∀x∃y[T (f, x, y) ∧ [∀zz≤U(y)¬T (x, x, z) → ∀y¬T (x, x, y)]] . After ∀-elimination assume for ∃y-elimination: T (f, f, y) ∧ [∀zz≤U(y)¬T (f, f, z) → ∀y¬T (f, f, y)] , from which T (f, f, y) ∧ ∀y¬T (f, f, y) follows by the remark on coding. FIM satisfies the “independence of (stable) premise” rule IPR: (∗) If FIM (¬A → ∃xB(x)) then FIM ∃x(¬A → B(x)) where x is not free in A. The beautiful proof by Visser that HA is closed under IPR (cf. p. 138 of [6]) works also for FIM. If one uses the monotone form (∗27.13 in [3]) of the bar induction schema, it is straightforward to show that FIM proves the Friedman translation of each of its mathematical axioms, and the logical rules and axioms behave as usual. Lemma 3. FIM + Π 1 -LEM does not prove Σ 0 1 -LEM. Proof. We use without much comment the fact that quantifier-free formulas are decidable and stable in FIM. Since primitive recursive codes for finite sequences of natural numbers are available in HA and hence in FIM, to prove the lemma we need only derive a contradiction from the assumption that ∀x[∀y¬R(x, y) ∨ ∃yR(x, y)] is derivable in FIM from the universal closures of Note on Π n+1-LEM, Σ 0 n+1-LEM and Σ 0 n+1-DNE 101 finitely many instances ∀xPi(x, z) ∨ ¬∀xPi(x, z), 1 ≤ i ≤ k, of Π 1 -LEM, where R(x, y) is T (x, x, y) and the Pi(x, z) are quantifier-free. Assume such a derivation exists, and let D(z) abbreviate ∧k i=1(∀xPi(x, z)∨¬∀xPi(x, z)). Then by the deduction theorem, FIM proves (i) ∀zD(z) → ∀x[∀y¬R(x, y) ∨ ∃yR(x, y)] . We can construct a purely arithmetical formula E(w, z), with no ∃ and no ∨, such that FIM proves (ii) E(w, z) ↔ ¬¬E(w, z) and (iii) E(σ(k), z) ↔ [∧k i=1({∀xPi(x, z) : σ(i−̇1) > 0} ∪ {¬∀xPi(x, z) : σ(i−̇1) = 0}) ] whence (iv) ∀z [D(z) ↔ ∃σ ∈ 2 E(σ(k), z)]