A Translation Theorem For Restricted R-Formulas

The three-sorted formal system RLS described in [4] is like RLS(≺) in [5] but without the ≺. IRLS is a strictly intuitionistic subsystem of RLS. This note gives a natural, syntactically defined translation φ mapping each restricted formula E with only number and lawlike sequence variables free, to a formula φ(E) containing only number and lawlike sequence variables, such that IRLS proves E ↔ φ(E). If E contains no choice sequence variables then φ(E) is E. 1 The systems RLS, IRLS, R, IR and C 1.1 A three-sorted language L The language, extending the two-sorted language of [2] and [1], contains three sorts of variables with or without subscripts, also used as metavariables: i, j, k, l,m, n, w, x, y, z over natural numbers, a, b, c, d, e, g, h over lawlike sequences, α, β, γ, . . . over arbitrary choice sequences; finitely many constants f0 (= 0), f1 (= ′) (successor), f2 (= +), f3 (= ·), f4 (= exp), f5, . . . , fp for primitive recursive functions and functionals; the binary predicate constant = (between terms); Church’s λ denoting function abstraction; parentheses (,) denoting function application; and the logical symbols & ,∨,→,¬ and quantifiers ∀,∃ over each sort of variable. I thank Sean Walsh and Kai Wehmeyer of UC Irvine, and the organizers of the 2014 Chiemsee Summer School, for giving me new opportunities to talk about this subject, resulting in this theorem. I am also very grateful to an anonymous referee whose careful reading led to many improvements. Vol. \jvolume No. \jnumber \jyear IFCoLog Journal of Logic and its Applications Joan Rand Moschovakis Terms (of type 0) and functors (of type 1) are defined inductively. Number variables and 0 are terms. Sequence variables of both sorts, and unary function constants, are functors. If fi is a ki,mi-ary function constant, u1, . . . , uki are functors and t1, . . . , tmi are terms, then fi(u1, . . . , uki , t1, . . . , tmi) is a term. If u is a functor and t is a term then (u)(t) (also written u(t)) is a term. If t is a term and x is a number variable then λx(t) (also written λx.t) is a functor. Prime formulas are of the form s = t where s, t are terms. If u, v are functors then u = v abbreviates ∀x (u(x) = v(x)). Composite formulas are formed as usual, with parentheses determining scopes. Terms and functors with no occurrences of arbitrary choice sequence variables are R-terms and R-functors respectively. Formulas with no free occurrences of arbitrary choice sequence variables are R-formulas. 1.2 The logical axioms and rules The logical basis is intuitionistic three-sorted predicate logic, extending the rules and axiom schemas in [2] to formulas, terms and functors of L as defined above, with new rules and axiom schemas for lawlike sequence variables and R-functors: 9R. C → A(b) / C → ∀bA(b) if b is not free in C. 10R. ∀bA(b)→ A(u) if u is an R-functor free for b in A(b). 11R. A(u)→ ∃bA(b) if u is an R-functor free for b in A(b). 12R. A(b)→ C / ∃bA(b)→ C if b is not free in C. 1.3 Axioms for 3-sorted intuitionistic number theory Equality axioms assert that = is an equivalence relation and x = y → α(x) = α(y), so ∀x(a(x) = a(x)) is provable (since lawlike sequence variables are functors), so ∀a∃β∀x(a(x) = β(x)) follows by the instance ∀x(a(x) = γ(x))→ ∃β∀x(a(x) = β(x)) of axiom schema 11F from [2]. Just as Brouwer’s infinitely proceeding sequences include all the sharp arrows, every lawlike sequence is (equal to) a choice sequence. By a similar argument, if u is an R-functor in which the variable b does not occur then ∃b∀x(b(x) = u(x)) is provable, so every R-functor denotes a lawlike sequence. For terms r(x), t with t free for x in r(x), the λ-reduction axiom schema is (λx.r(x))(t) = r(t), where r(t) is the result of substituting t for all free occurrences of x in r(x). A Translation Theorem The mathematical axioms include the assertions that 0 (= f0) is not a successor and the successor function (= f1) is one-to-one, the defining equations for the primitive recursive function and functional constants f2, . . . , fp ([2], [1]) and the mathematical induction schema extended to L. For the countable axiom of choice AC01. ∀x∃αA(x, α)→ ∃α∀xA(x, λy.α(2 · 3)) the x must be distinct from y, and free for α in A(x, α). Finite sequences are coded primitive recursively as in [2], so 〈x0, . . . , xk〉 = Π0pi where pi is the ith prime with p0 = 2, and (y)i is the exponent of pi in the prime factorization of y. Let Seq(y) abbreviate ∀i < lh(y)((y)i > 0) where lh(y) is the number of nonzero exponents in the prime factorization of y. The empty sequence is coded by 〈 〉 = 1, and if k ≥ 0 then 〈x0 + 1, . . . , xk + 1〉 codes the finite sequence (x0, . . . , xk). If Seq(y) and Seq(z) then y∗z codes the concatenation of the sequences coded by y and z. The finite initial segment of length n of a choice sequence α is coded by α(n), where α(0) = 1 and α(n+ 1) = 〈α(0) + 1, . . . , α(n) + 1〉. Other useful abbreviations are α ∈ w for α(lh(w)) = w, w v y for Seq(y) & ∀i < lh(w)((w)i = (y)i), and w < y for w v y & lh(y) > lh(w). If Seq(w) then w ∗ α = β where β ∈ w and β(lh(w) + n) = α(n); if ¬Seq(w) then w ∗ α = α. Note that w ∗ α is a functor and w ∗ a is an R-functor. 1.4 Bar induction Kleene formulated Brouwer’s “bar theorem” as an axiom schema, in four versions which are all equivalent using AC01 (or even AC00!), and included it in his basic system B. The version we assume (now for the three-sorted language) is1 BI! ∀α∃!xR(α(x)) & ∀w(Seq(w) & R(w)→ A(w)) & ∀w(Seq(w) & ∀nA(w ∗ 〈n〉)→ A(w))→ A(1). This schema (for the two-sorted language without lawlike sequence variables) completed Kleene’s basic system B, which is neutral in the sense that it is correct both intuitionistically and classically. 1.5 R-lawless sequences, restricted quantification and lawlike comprehension Intuitively, a lawless sequence should not be predictable by any lawlike process, but this negative condition is not enough to satisfy Kreisel’s axioms. Instead, call 1In general, ∃!xA(x) abbreviates ∃xA(x) & ∀x∀y(A(x) & A(y) → x = y). Joan Rand Moschovakis a choice sequence β a predictor if β maps finite sequence codes to finite sequence codes, and call a choice sequence α R-lawless if every lawlike predictor correctly predicts α somewhere. Formally, RLS(α) abbreviates ∀b[Pred(b)→ ∃x α ∈ α(x) ∗ b(α(x))], where Pred(b) abbreviates ∀w(Seq(w)→ Seq(b(w))). Since each prediction affects only finitely many values, this positive condition leaves room for (indeed, insures) plenty of chaotic behavior if there are only countably many lawlike predictors. The usual diagonal argument guarantees that there is no lawlike enumeration of the lawlike sequences, but a classical model with countably many lawlike sequences is described in [5]. Troelstra’s extension principle, which claims that every continuous partial function defined on all lawless sequences has a continuous total extension, fails for Rlawless sequences, since ∀α[RLS(α) → ∃nα(n) = 1] but the function assigning to each R-lawless α the least n such that α(n) = 1 cannot be extended continuously to all choice sequences. And while Kreisel and Troelstra considered any two distinct lawless sequences to be independent, a stronger condition for independence is needed here. Two R-lawless sequences α, β will be called independent if their fair merge [α, β] is lawless, and similarly for α0, . . . , αk where [α0, . . . , αk]((k + 1)n + i) = αi(n) for 0 ≤ i ≤ k and all n. This natural notion of independence for lawless sequences was proposed by M. Fourman at the Brouwer Centenary Conference in 1981. The class of restricted formulas is defined inductively: Each formula E with no arbitrary choice sequence quantifiers is restricted. If A is restricted and contains free no arbitrary choice sequence variables other than α, then ∀α[RLS(α) → A] and ∃α[RLS(α) & A] are restricted. If k > 0 and A is restricted with no arbitrary choice sequence variables other than α0, . . . , αk occurring free, then for i = 0, . . . , k the formulas ∀αi[RLS([α0, . . . , αk])→ A] and ∃αi[RLS([α0, . . . , αk]) & A] are restricted. No other formulas are restricted. There is a lawlike function-comprehension schema AC00! ∀x∃!yA(x, y)→ ∃b∀xA(x, b(x)) where A(x, y) is any restricted R-formula and b is free for y in A(x, y). By this axiom, the lawlike sequences are closed under “recursive in.”2 2While a restricted formula can have free occurrences of arbitrary choice sequence variables, a restricted R-formula cannot. If A(x, y) is a restricted R-formula, the informal abbreviation μyA(x, y) may be allowed under either of the assumptions ∃!yA(x, y) or ∃y(A(x, y) & ∀z < y¬A(x, z)). A Translation Theorem For restricted R-formulas A(x, a) the lawlike comprehension schema entails AC01! ∀x∃!aA(x, a)→ ∃b∀xA(x, λy.b(2 · 3)), with the obvious conditions on the variables. 1.6 Axioms for R-lawless sequences These are Kreisel’s and Troelstra’s axioms from [3] and [7], adapted to Kleene’s convention for coding continuous functions, with inequality of lawless sequences replaced by independence. There are two density axioms: ∀w(Seq(w)→ ∃α[RLS(α) & α ∈ w]), RLS1. ∀w(Seq(w)→ ∀α[RLS(α)→ ∃β[RLS([α, β]) & β ∈ w]]). RLS2. Kreisel’s principle of open data is stated as follows, on condition that A(α) is restricted and has no other arbitrary choice sequence variables free, and β is free for α in A(α): RLS3. ∀α[RLS(α)→ (A(α)→ ∃w(Seq(w) & α ∈ w & ∀β[RLS(β)→ (β ∈ w → A(β))]))]. Effective continuous choice for lawless sequences is the schema RLS4. ∀α[RLS(α)→ ∃bA(α, b)]→ ∃e∃b∀α[RLS(α)→ ∃!ye(α(y)) > 0 & ∀y(e(α(y)) > 0→ A(α, λx. b(〈e(α(y))−̇1, x〉)))] where A(α, b) is restricted with no arbitrary choice sequence variables but α free, and e, y, α are free for b in A(α, b).3 1.7 The restricted law of excluded middle For A(α) restricted, with no choice sequence variables free except possibly α, RLS also has the axiom schema RLEM. ∀α[RLS(α)→ A(α) ∨ ¬A(α)]. By an easy argument, RLS3 and the restricted LEM entail the following principle of closed data with the same restrictions on A(α) as for RLS3: RLS5. ∀α[RLS(α)→ (∀w(α ∈ w → ∃β[RLS(β) & β ∈ w & A(β)])→ A(α))]. 3In general, e(α) ' n abbreviates ∃x(e(α(x)) = n+ 1 & ∀y < xe(α(y)) = 0). Joan Rand Moschovakis In a strictly intuitionistic system without RLEM, RLS5 may or may not be taken as an additional axiom schema. With RLS1, RLEM entails the law of excluded middle for all formulas with only lawlike and number variables. Observe that RLS1, RLS2, and all instances of RLS3, RLS4, RLEM and RLS5 are restricted R-formulas. 1.8 Five axiomatic systems In addition to Kleene’s basic formal system B for neutral analysis we consider five other formal systems. All but one are consistent with full intuitionistic analysis FIM as formalized in [2].4 IRLS extends B to the three-sorted language and adds axioms RLS1,2 and axiom schemas AC00! and RLS3,4. IRLS expresses a strictly intuitionistic theory of lawlike and relatively lawless sequences in the context of full intuitionistic analysis. RLS is the three-sorted semi-intuitionistic system IRLS + RLEM.5 Lawlike classical analysis R is the two-sorted subsystem of RLS obtained by restricting the language to number and lawlike sequence variables, omitting RLS14 and BI!, replacing RLEM by A ∨ ¬A for formulas of the two-sorted language, replacing AC01 and AC00! by AC01 (like AC01! but without the !) for formulas of the two-sorted language, and restating the equality axioms and primitive recursive definitions of function constants using lawlike instead of arbitrary choice sequence variables. Constructive analysis IR is the two-sorted intuitionistic subsystem of R obtained by omitting A ∨ ¬A. Note that IR has no version of bar induction. Classical analysis C is the two-sorted system obtained from Kleene’s B by strengthening the logic to classical logic. A lawlike version BI!R of BI!, with lawlike sequence variables replacing arbitrary choice sequence variables, is provable in R. Thus C and R are notational variants, as are B and IR + BI!R. 1.9 Closure properties of RLS: Lemma The three-sorted subsystem IRS of IRLS obtained by omitting RLS1-4, but retaining AC00!, proves (i) ∀α[RLS(α)↔ ∀w(Seq(w)→ RLS(w ∗ α))]. 4The relative consistency of a common extension of RLS and FIM is established in [5] under the assumption that a definably well-ordered subset of ω is countable. 5The translation theorem will show that RLS can also be axiomatized by IRLS plus the law of excluded middle for strictly lawlike formulas, so RLS is indeed semi-intuitionistic. A Translation Theorem (ii) ∀b(∀x∀y(b(x) = b(y)→ x = y) & ∀y(∃x b(x) = y ∨ ¬∃x b(x) = y)→ ∀α[RLS(α)→ RLS(α ◦ b)]). (iii) ∀b(Pred(b)→ ∀n∀α[RLS(α)→ ∃m(m ≥ n & α ∈ α(m) ∗ b(α(m)))]). Proofs. This is a formal version of Lemma 2 of [5]. For (i →) assume Seq(w) and Pred(b). Then ∀x∃!y((Seq(x)→ y = b(w ∗ x)) & (¬Seq(x)→ y = 0)), so by AC00! there is a c such that Pred(c) and ∀x(Seq(x)→ c(x) = b(w ∗x)), so if RLS(α) then ∃z α ∈ α(z) ∗ c(α(z)) and hence ∃z(w ∗ α ∈ w ∗ α(lh(w) + z) ∗ b(w ∗ α(lh(w) + z))). For (i ←) take w = 1 = 〈 〉, so w ∗ α = α. The proofs of (ii), (iii) similarly formalize the proofs of (ii), (iii) of Lemma 2 of [5]. 2 1.10 Axioms RLS1-3 reconsidered Lemma 1.9 guarantees that the following schemas RLS1′, RLS2′ and RLS3′ are equivalent over IRS to RLS1, RLS2 and RLS3 respectively. ∃αRLS(α) RLS1′. ∀α[RLS(α)↔ ∃βRLS([α, β])] RLS2′. ∀α[RLS(α)→ (A(α)↔ ∃w(Seq(w) &α ∈ w&∀β[RLS(β)→ A(w ∗ β)]))] RLS3′. under the same conditions on A(α) as for RLS3. The next section suggests a way to simplify RLS4 as well. 2 The translation theorem 2.1 Theorem Every restricted formula E of the three-sorted language with no arbitrary choice sequence variables free is equivalent in IRLS to a formula φ(E) of the two-sorted language with only number and lawlike sequence variables. The mapping φ is syntactically defined. If E contains no choice sequence variables then φ(E) is E. The proof is similar to Troelstra’s proof of the translation theorem for LS into the language without lawless sequence variables (cf. [8], 663ff), with a significant difference. Instead of the constant K0 Troelstra used to represent the class of lawlike codes of continuous total functions, we can define the condition for e to be a lawlike code of a continuous partial function defined on all the R-lawless sequences: J0(e) ≡ ∀w(Seq(w) & ∀n < lh(w)(e(w(n)) = 0)→ ∃y(Seq(y) & e(w ∗ y) > 0)), J1(e) ≡ J0(e) & ∀w[e(w) > 0→ Seq(w) & ∀y(Seq(y)→ e(w ∗ y) = e(w))]. Joan Rand Moschovakis By Lemma 1.9(i) and the next lemma, the conclusion of effective continuous choice for R-lawless sequences can be rewritten ∃e∃b(J1(e) & ∀n∀w(e(w) = n+ 1→ ∀α[RLS(α)→ A(w ∗ α, λx.b(〈n, x〉))])). 2.2 Lemma (i) IRS + RLS1 proves ∀e(J0(e)↔ ∀α[RLS(α)→ e(α) ↓]), and (ii) IRS proves ∀α[RLS(α)↔ ∀e(Jj(e)→ e(α) ↓)] for j = 0, 1, where e(α) ↓ abbreviates ∃m (e(α(m)) > 0). Proofs. (i) →: Assume J0(e). Using AC00! define a lawlike predictor g by g(w) = { 〈 〉 if ∃y v w(e(y) > 0) ∨ ¬Seq(w), μy(Seq(y) & e(w ∗ y) > 0) otherwise. If RLS(α) then α ∈ α(n) ∗ g(α(n)) for some n, so e(α) ↓. (i) ←: Assume ∀α[RLS(α) → e(α) ↓] and Seq(w). By RLS1 there is an α ∈ w with RLS(α), so for some y = α(m): e(y) > 0 & ∀n < m(e(α(n)) = 0). If also ∀n < lh(w)(e(w(n)) = 0) then w v y, so y ∗ e(y) = w ∗ z where e(w ∗ z) > 0. (ii) → follows immediately from (i) → (with the fact that ∀e(J1(e) → J0(e))). For (ii) ←: Assume ∀e(J1(e)→ e(α) ↓) and let g be a lawlike predictor. Define e as follows: e(w) = { 1 if Seq(w) & ∃y v w(y ∗ g(y) v w), 0 otherwise. Then J1(e) holds, so e(α) ↓, so g correctly predicts α somewhere. 2 The proof of the translation theorem depends on Lemmas 1.9, 2.2, and the following sequence of lemmas removing restricted existential quantifiers and reducing restricted R-formulas of the form ∀α[RLS(α)→ A] and ∀αi[RLS([α0, . . . , αk])→ A] to simpler formulas of the same kind. For the case that A is prime the reduction is complete in one step, even in IRS + RLS1. 2.3 Lemma If s(α), t(α) are terms with no arbitrary choice variables but α free, and a is free for α in both, then IRS + RLS1 proves (i) ∀α[RLS(α)→ s(α) = t(α)]↔ ∀a[s(a) = t(a)] and (ii) ∃α[RLS(α) & s(α) = t(α)]↔ ∃a[s(a) = t(a)]. A Translation Theorem Proof. By induction on the complexity of the term s(α) (expressing the value of a primitive recursive function of α and the other free variables) IRS proves ∀α∃x∃y∀β(β(x) = α(x)→ s(β) = y). Only the argument for (i) is completed here since the proof of (ii) is similar. (i)←: Assume ∀a[s(a) = t(a)] andRLS(α). Let x, y, z satisfy ∀β(β(x) = α(x)→ s(β) = y & t(β) = z] and let w = α(x). Then ∀a[a(x) = w → s(a) = y & t(a) = z], so y = z since w∗λn.0 is lawlike by AC00!, so s(α) = t(α) since s(α) = y & t(α) = z. So IRS proves ∀a[s(a) = t(a)]→ ∀α[RLS(α)→ s(α) = t(α)]. (i)→: Assume ∀α[RLS(α)→ s(α) = t(α)]. Let x, y, z satisfy ∀β[β(x) = a(x)→ s(β) = y & t(β) = z], so s(a) = y & t(a) = z. By RLS1 there is a β such that RLS(β) & β(x) = a(x), so s(β) = t(β) and y = z, so s(a) = t(a). So IRS + RLS1 proves ∀α[RLS(α)→ s(α) = t(α)]→ ∀a[s(a) = t(a)]. 2 2.4 Lemma IRLS proves (i) ∀α[RLS(α)→ A(α) & B(α)]↔ ∀α[RLS(α)→ A(α)] & ∀α[RLS(α)→ B(α)], (ii) ∀α[RLS(α)→ A(α) ∨B(α)]↔ ∃e[J1(e) & ∀w(e(w) > 0→ ∀α[RLS(α)→ A(w ∗ α)] ∨ ∀α[RLS(α)→ B(w ∗ α)])], (iii) ∀α[RLS(α)→ (A(α)→ B(α))]↔ ∀w(Seq(w)→ (∀α[RLS(α)→ A(w ∗ α)]→ ∀α[RLS(α)→ B(w ∗ α)])), (iv) ∀α[RLS(α)→ ¬A(α)]↔ ∀w(Seq(w)→ ¬∀α[RLS(α)→ A(w ∗ α)]), for A(α), B(α) restricted, with no arbitrary choice sequence variables other than α occurring free. Proofs. (ii) follows from RLS4 using Lemmas 1.9 and 2.2 with the observation ∀e(J0(e)→ ∃g[J1(g) & ∀x∀w(g(w) = x+ 1↔ ∃y v w (e(y) = x+ 1 & ∀z < y e(z) = 0))]. Using Lemma 1.9 again: (iii) follows from RLS3, (iv) → follows from RLS1, and (iv) ← is an easy consequence of RLS3. 2 Joan Rand Moschovakis 2.5 Lemma IRLS proves ∃α[RLS(α) & A(α)]↔ ∃w(Seq(w) & ∀α[RLS(α)→ A(w ∗ α)]) if A(α) is restricted and contains free no arbitrary choice sequence variable but α. Proof. Immediate from RLS3. 2 2.6 Lemma For restricted formulas A(α, x), A(α, b) containing free no arbitrary choice sequence variables other than α, IRLS proves (i) ∀α[RLS(α)→ ∃xA(α, x)]↔ ∃e[J1(e) & ∀w(e(w) > 0→ ∃x∀α[RLS(α)→ A(w ∗ α, x)])], (ii) ∀α[RLS(α)→ ∃bA(α, b)]↔ ∃e[J1(e) & ∀w(e(w) > 0→ ∃b∀α[RLS(α)→ A(w ∗ α, b)])], (iii) ∀α[RLS(α)→ ∀xA(α, x)]↔ ∀x∀α[RLS(α)→ A(α, x)], (iv) ∀α[RLS(α)→ ∀bA(α, b)]↔ ∀b∀α[RLS(α)→ A(α, b)]. Proofs. (i) and (ii) are by RLS4; (iii) and (iv) are by predicate logic. 2 2.7 Lemma For A(α, β) restricted with no arbitrary choice sequence variables free except the distinct variables α, β, IRLS proves (i) ∀α[RLS(α)→ ∀β[RLS([α, β])→ A(α, β)]]↔ ∀γ[RLS(γ)→ A([γ]0, [γ]1)], (ii) ∀α[RLS(α)→ ∃β[RLS([α, β]) & A(α, β)]]↔ ∃e[J1(e) & ∀y(e(y) > 0→ ∃w(Seq(w) & ∀γ[RLS(γ)→ A(y ∗ [γ]0, w ∗ [γ]1)]))], where in general [γ]0(n) = γ(2n) and [γ]1(n) = γ(2n+ 1). Proofs. (i) is immediate from the definitions (note that γ = [ [γ]0, [γ]1 ]) and the fact that RLS([α, β]) → RLS(α) by Lemma 1.9(ii). (ii) follows from RLS2, RLS3 and the closure properties in Lemma 1.9. 2 A Translation Theorem 2.8 Proof of the translation theorem Definition. The index of a restricted formula E of L is I(E) = 2i+ j + 2k where 1. i is the number of restricted existential sequence quantifiers occurring in E, 2. j is the number of restricted universal sequence quantifiers occurring in E and 3. k is the maximum number of logical symbols (&,∨,→,¬, ∀, ∃) occurring in any part F of a subformula of E of any of the forms ∀α[RLS(α) → F ], ∃α[RLS(α) & F ], ∀αi[RLS([α0, . . . , αk])→ F ] or ∃αi[RLS([α0, . . . , αk]) & F ]. If C ↔ D denotes any restricted R-formula of a type displayed in the statement of any of the Lemmas 2.3, 2.4, 2.5, 2.6 or 2.7, inspection shows that I(C) > I(D). The lemmas permit the reduction of a given restricted R-formula E to a formula F of the two-sorted language, with only number and lawlike sequence variables, such that IRLS proves E ↔ F . For a uniform translation, the sequence in which the lemmas are to be applied can be determined uniquely (modulo the renaming of variables) by the logical form of E, beginning with the leftmost occurrence of a restricted quantifier. The successive reductions produce a sequence E0, . . . , Eq of restricted R-formulas with I(Ei) > I(Ei+1), where E0 is E and I(Eq) = 0, so we can define φ(E) = Eq. There are two wrinkles which are best illustrated by an example. Suppose Ei is a restricted R-formula of the form ∃α[RLS(α) & ∀β[RLS([α, β]) → A(α, β)]]. Lemma 2.5 reduces this to ∃w[Seq(w) & ∀α[RLS(α)→ ∀β[RLS([w∗α, β])→ A(w∗ α, β)]]], which is not restricted but can be simplified to ∃w[Seq(w) & ∀α[RLS(α)→ ∀β[RLS([α, β]) → A(w ∗ α, β)]]] using Lemma 1.9 once. If needed, repeated uses of Lemma 1.9 reduce A(w ∗ α, β) to a restricted formula A′(w,α, β). Then Ei+1 is ∃w[Seq(w) & ∀α[RLS(α)→ ∀β[RLS([α, β])→ A′(w,α, β)]]], a restricted R-formula with I(Ei+1) < I(Ei) since A(α, β) and A′(w,α, β) have the same number of logical symbols. By Lemma 2.7(i), in the next step ∀α[RLS(α)→ ∀β[RLS([α, β])→ A′(w,α, β)]] is reduced to ∀γ[RLS(γ) → A′(w, [γ]0, [γ]1)], which may not be restricted. But if for example A′(w,α, β) is ∀δ[RLS([α, β, δ]) → B(w,α, β, δ)], Lemma 1.9 reduces A′(w, [γ]0, [γ]1) to ∀δ[RLS([γ, δ])→ B(w, [γ]0, [γ]1, δ)] and eventually to a restricted formula ∀δ[RLS([γ, δ]) → B′(w, γ, δ)]. Then Ei+2 is ∃w[Seq(w) & ∀γ[RLS(γ) → ∀δ[RLS([γ, δ])→ B′(w, γ, δ)]]], a restricted R-formula with I(Ei+2) < I(Ei+1). 2 2.9 Final remarks Evidently the translation will be unique only up to congruence (renaming of bound variables). While technically RLS1′ is not restricted, it is equivalent over IRS to Joan Rand Moschovakis ∃α[RLS(α) & 0 = 0], which reduces over IRS + RLS1 to ∃w∀α[RLS(α) → 0 = 0] and then to ∃w∀a(0 = 0), which is equivalent in IR to 0 = 0. RLS2′ permits a similar analysis over IRS + RLS2. Note that the intuitionistic system IRLS proves AC01; and if AC01 replaces AC00! then RLS4 becomes provable from the other axioms of the semi-classical system RLS. It follows that if IRS′ comes from IRS by strengthening AC00! to AC01, then RLS = IRS′ + RLS′ 1-3 + RLEM. Evidently IRLS is not a conservative extension of IRS or even IRS′, since ∃α∀a¬∀x(α(x) = a(x)) is provable in IRLS but not in IRS′. Similarly, RLS is not a conservative extension of IRS′ + RLEM. However, IRS′ is a conservative extension of its two-sorted subsystem B. I am indebted to A. S. Troelstra (cf. [6]) for the hint that prompted these observations. Two questions remain. Is IRLS a conservative extension of (two-sorted) constructive analysis IR? Is the semi-classical system RLS a conservative extension of two-sorted classical analysis R? Since R proves lawlike countable choice AC01 and a lawlike version BI!R of the bar induction schema, the translation lemma suggests that RLS may be a conservative extension of R, but this is only a conjecture.