aa r X i v : . [ m a t h . L O ] J a n CALIBRATING THE NEGATIVE INTERPRETATION
JOAN RAND MOSCHOVAKIS
Warm thanks to Garyfallia Vafeiadou for clarifying the connections among weaksubsystems of constructive analysis and for many improvements to this essay, andto Robert Solovay for the clever proof that suggested these questions. I have triedto acknowledge uses of Kleene’s, Kreisel’s and Troelstra’s ideas and methods, butsome have become second nature.1.
What this essay is about
G¨odel and Gentzen used simple negative interpretations to prove that classicalPeano arithmetic PA is equiconsistent with intuitionistic Heyting arithmetic HA .By hereditarily replacing A ∨ B by its classical equivalent ¬ ( ¬ A & ¬ B), and ∃ xA(x)by its classical equivalent ¬∀ x ¬ A(x), they were able to show that the negativefragment of HA (omitting the logical symbols ∨ , ∃ with their axioms and rules)is a faithful translation of PA . The negative interpretations of the mathematicalaxioms of PA are provable in HA and the negative interpretations of the classicallogical axioms and rules are correct by intuitionistic logic.The situation is different for intuitionistic and classical theories of numbers andnumber-theoretic functions with axioms of comprehension or countable choice. Thenegative translation ∀ x ¬ ( ¬∀ y α ( h x , y i ) = 0 & ¬¬∀ y α ( h x , y i ) = 0)of ∀ x( ∀ y α ( h x , y i ) = 0 ∨ ¬∀ y α ( h x , y i ) = 0) is provable by intuitionistic logic, but thenegative translation ¬∀ β ¬∀ x( β (x) = 0 ↔ ∀ y α ( h x , y i ) = 0)of ∃ β ∀ x( β (x) = 0 ↔ ∀ y α ( h x , y i ) = 0) is independent of Brouwer’s intuitionistic anal-ysis as formalized in [6] (cf. [13], [16], [18]). The general question is this:
Given a classically correct subsystem S of e.g.Kleene and Vesley’s formal system I for intuitionistic analysis, exactly what mustbe added to S in order to prove the G¨odel-Gentzen negative interpretation of S ?Let S + g be the minimum classical extension of S in this sense.Classical and constructive mathematics coexist in S + g exactly as far as the math-ematical axioms of S permit. By viewing the choice sequence variables α, β, . . . ofthe language L ( I ) of I alternatively as variables over classical one-place number-theoretic functions, restricting language and logic by omitting the symbols ∨ and ∃ with their axioms and rules, and replacing each mathematical axiom of S by itsnegative translation, one obtains a faithful translation S g of S + ( ¬¬ A → A) withinthe extended intuitionistic system S + g . In particular, if B is obtained from I bydropping the axiom schema of continuous choice CC (“Brouwer’s Principle for aFunction,” axiom schema x B + g includes the negative translationof a system C ≡ B + ( ¬¬ A → A) of classical analysis with countable choice.
The goal here is different from Kleene’s in [4] where he showed that I is consis-tent with all purely arithmetical formulas, and all negations of prenex formulas, ofthe full language L ( I ) which are provable in C . This essay focuses narrowly on theprecise constructive cost of expanding interesting subsystems of B to include theirclassical content. The minimum classical extension S + g of a subsystem S of B isclassically correct for the full language, but assumes only the essential intuitionis-tically dubious principles .Kleene’s informal and formal function-realizability respectively guarantee thatthe classical extension B + g + CC of I is consistent relative to B + g and satisfiesthe Church-Kleene recursive instantiation rule (5.9(iii), page 101 of [5]). Theseresults extend (relative to B + g + MP ) to B + g + CC + MP , whereMP . ∀ α ( ¬∀ x ¬ α (x) = 0 → ∃ x α (x) = 0)is a strong analytical form of Markov’s Principle. Intuitionistic logic proves thenegative translation (MP ) g of MP , so ( S + MP ) + g = S + g + MP for eachsubsystem S of B .Kleene’s proof in [6] that I MP extends to show that B + g + CC MP .Vesley’s proof in [23] that I is consistent with his schema VS, where I + VS ⊢ ¬ MP ,extends to show that B + g + CC + VS is consistent and refutes MP .A seminal analysis of double negation shift and the negative interpretation ofcountable choice, in the context of HA ω , was carried out by Berardi, Bezem andCoquand in [1]. The recent, technical [2] treats weak nonconstructive principles inthe context of EL , HA or HA ω . The bibliographies of both point to other relatedwork. For a precise comparison of EL and e.g. HA with the systems treated here,see [19], [20]. 2. What is “constructive analysis?”
Like Brouwer, Bishop worked informally, but it seems reasonable to assume thathe would not have objected to the mathematical content of any of the axiomsor axiom schemas of Kleene’s neutral basic system B except the principle of barinduction. Bishop used countable choice routinely, so Kleene’s strongest countablechoice axiom schema ( x . ∀ x ∃ α A(x , α ) → ∃ β ∀ xA(x , λ y .β ( h x , y i ))may reasonably be assumed to hold in constructive analysis, with its consequence( ∗ . ∀ x ∃ yA(x , y) → ∃ α ∀ xA(x , α (x))for all formulas A(x , α ) and A(x , y) of the language, with the appropriate conditionson the variables (e.g. for AC : α, x must be free for y in A(x , y)).Weaker subsystems of B are distinguished by restrictions on AC , which inturn determine the classical omega-models of the subsystems. Classical omega-models are important for constructive analysis because (a) Bishop’s mathematics isconsistent with classical mathematics, and (b) it is reasonable to assume that theconstructive natural numbers are standard.2.1. Two-sorted intuitionistic arithmetic IA . The weakest system treatedhere is IA , which extends the first-order intuitionistic arithmetic IA of Kleene’s[3] by adding variables α, β, γ, . . . over one-place number-theoretic functions, quan-tifiers ∀ α, ∃ α with their (intuitionistic) logical axioms and rules, and finitely many ALIBRATING THE NEGATIVE INTERPRETATION 3 constants for primitive recursive function(al)s with their defining axioms. Terms (oftype 0) and functors (of type 1) are defined inductively. Church’s lambda symbolmay be used to define functors from terms. There is an axiom schema of lambda-reduction ( λ x . t(x))(s) = t(s) (where t(x) , s are terms, and s is free for x in t(x)).Equality at type 0 is a primitive notion, and is decidable in IA . Equality attype 1 is defined extensionally by α = β ≡ ∀ x( α (x) = β (x)), and IA includes theopen equality axiom ∀ x ∀ y(x = y → α (x) = α (y)). The primitive recursive sequences provide a classical omega-model of this system,so IA can only prove the existence of primitive recursive functions.2.2. Intuitionistic recursive analysis IRA.
Vafeiadou ([20], [19]) proved thatTroelstra’s formal system EL ([16], [18], [17]) is mathematically equivalent, in thesense that the two systems share a common definitional extension, to the subsystemof Kleene’s B obtained by adding to IA the recursive comprehension axiom ∀ x ∃ y ρ ( h x , y i ) = 0 → ∃ α ∀ x ρ ( h x , α (x) i ) = 0or ∀ x ∃ y ρ ( h x , y i ) = 0 → ∃ α ∀ x[ ρ ( h x , α (x) i ) = 0 & ∀ z ≤ α (x) ρ ( h x , z i ) = 0]. Over IA recursive comprehension is equivalent to the schema qf-AC , the restriction of AC to formulas A(x , y) which are “quantifier-free” (contain no sequence quantifiers, andonly bounded number quantifiers).The general recursive sequences provide a classical omega-model of intuitionisticrecursive analysis IRA ≡ IA + qf-AC . Countable comprehension and arithmetical countable choice.
Strongerthan qf-AC over IA , but weaker than AC , is countable comprehension AC ! . ∀ x ∃ !yA(x , y) → ∃ α ∀ xA(x , α (x)) , where ∃ !yA(x , y) abbreviates ∃ yA(x , y) & ∀ y ∀ z(A(x , y) & A(x , z) → y = z). AC !entails recursive comprehension because quantifier-free formulas are decidable in IA , so the hypothesis of an instance of qf-AC provides unique least witnessesfor the corresponding instance of AC !.Vafeiadou ([20], [19]) proved that AC ! is equivalent over IRA to the schemaCF d . ∀ x(A(x) ∨ ¬ A(x)) → ∃ α ∀ x[ α (x) ≤ α (x) = 0 ↔ A(x))] . It follows that IA + AC ! and IA + AC have the same classical omega-models,including all analytically definable functions. Note that IA + AC ! proves AC !(like AC but with the hypothesis ∀ x ∃ ! α A(x , α )). Beginning in [9], [10] I proposed M = IA + AC ! as the natural base for all varieties of constructive analysis.In correspondence with the author in 2002 ([8], [12]) Robert Solovay asked andanswered a question involving arithmetical countable choice AC Ar , the restrictionof AC to formulas A(x , y) which are arithmetical in the sense of containing onlynumber quantifiers; sequence parameters are permitted. AC Ar is evidently strongerthan qf-AC but weaker than AC ! over IA .The arithmetical sequences provide a classical omega-model of IA + AC Ar . For a precise definition of HA see [20], [14]. HA is the “least subsystem” L of I in [9], [5]. Another formalization of intuitionistic recursive analysis is Veldman’s
BIM . JOAN RAND MOSCHOVAKIS Two stronger classically correct intuitionistic principles
In addition to the principles of constructive analysis, Brouwer assumed the “bartheorem” and its consequence, the “fan theorem.” Kleene included in B an axiomschema of bar induction, in four versions which are equivalent over IA + AC !.For current purposes the appropriate version isBI . ∀ α ∃ x ρ ( α (x)) = 0 & ∀ w(Seq(w) & ρ (w) = 0 → A(w))& ∀ w(Seq(w) & ∀ sA(w ∗ h s + 1 i ) → A(w)) → A( h i )( x IRA . Brouwer used bar induction to prove his “fan theorem,” which he combined witha classically false continuity principle to conclude that every completely definedfunction on the closed unit interval is uniformly continuous. A neutral form ofthe fan theorem, classically equivalent to K¨onig’s lemma for the binary tree, provesthat pointwise continuous functions are uniformly continuous on compact sets. Theversion we consider (cf. [22] Corollary 9.8) isFT . ∀ α B( α ) ∃ x ρ ( α (x)) = 0 → ∃ n ∀ α B( α ) ∃ x ≤ n ρ ( α (x)) = 0 , where B( α ) ≡ ∀ x α (x) ≤ IRA + FT but the arithmetical sequences do.4. Two families of classical principles consistent with I Double negation shift principles.
Kuroda’s double negation shift schema for numbers is.
DNS . ∀ x ¬¬ A(x) → ¬¬∀ xA(x) , for all formulas A(x) of the language. The converse of DNS is provable in IA ,so the → can be strengthened to ↔ . The restriction DNS − of DNS to negativeformulas A(x) is provable in IA since IA ⊢ ¬¬ A ↔ A for every formula A notcontaining ∨ or ∃ .The restriction of DNS to Σ formulas A(x) is a weak consequenceΣ − DNS . ∀ α [ ∀ x ¬¬∃ y α ( h x , y i ) = 0 → ¬¬∀ x ∃ y α ( h x , y i ) = 0]of MP which Brouwer used in 1918 to prove that the intuitionistic real numbersform a closed species. Van Atten [21] notes that Brouwer later formulated a strongerdefinition of “closed” in order to avoid this use of (a consequence of) Markov’sPrinciple.In [15] Scedrov and Vesley studied a principle of which Σ -DNS is a special case.They proved that B Σ -DNS by showing that Σ -DNS fails in Krol’s modelof intuitionistic analysis ([7]), and observed that B + Σ -DNS MP becauseevery theorem of B + Σ -DNS is S realizable, while MP is not. It follows that I + Σ -DNS is consistent with Vesley’s Schema VS ([23]), which proves Brouwer’screating-subject counterexamples including ¬ MP . In Kleene’s primitive recursive coding of finite sequences h i = 1, h a , . . . , a n i = Π j=nj=0 p a j j wherep j is the j th prime, and ( h a , . . . , a n i ) j = a j . Finite initial segments of infinite sequences are codedby setting lh(w) = Σ j < w sg((w) j ) where sg(n) = 1 . − (1 . − n), and Seq(w) ≡ ∀ j < lh(w) (w) j = 0; h a + 1 , . . . , a n + 1 i uniquely codes the finite sequence (a , . . . , a n ) and ∗ denotes concatenation. ALIBRATING THE NEGATIVE INTERPRETATION 5
Two double negation shift axioms for functions.
Full double negation shiftfor functions conflicts with Brouwer’s continuity principles. A limited but usefulversion, which follows trivially from MP (so is consistent with I ) but is weakerthan MP (because it is classically S realizable), isDNS . ∀ α ¬¬∃ x ρ ( α (x)) = 0 → ¬¬∀ α ∃ x ρ ( α (x)) = 0 . Scedrov and Vesley also considered a principle of which this is a special case, andobserved in effect that
IRA + DNS ⊢ Σ -DNS .G¨odel, Dyson and Kreisel proved that the weak completeness of intuitionisticpredicate logic for Beth semantics is equivalent, over IRA , to the following conse-quence of DNS :GDK . ∀ α B( α ) ¬¬∃ x ρ ( α (x)) = 0 → ¬¬∀ α B( α ) ∃ x ρ ( α (x)) = 0 . Because GDK is ∆ realizable (cf. [13]) while DNS is not, I + GDK DNS .4.2. Weak comprehension principles.
A number-theoretic relation A(x) (per-haps with sequence parameters) has a characteristic function for x only if it satisfies ∀ x(A(x) ∨ ¬ A(x)). The weak comprehension schemaWCF . ¬¬∃ ζ ∀ x( ζ (x) = 0 ↔ A(x))asserts only that it is consistent to assume that A(x) has a characteristic function.By Vafeiadou’s characterization the restriction WCF − of WCF to negative for-mulas A(x) is provable in the minimum classical extension of IA + AC !. Animportant special case is Π − WCF . ∀ α ¬¬∃ ζ ∀ x( ζ (x) = 0 ↔ ∀ y α ( h x , y i ) = 0) . Minimum classical extensions of some subsystems of B Theorem. (i) (IA ) + g = IA .(ii) (IRA) + g ≡ ( IA + qf-AC ) + g = IRA + Σ -DNS .(iii) ( IA + AC Ar ) + g = IA + AC Ar + Σ -DNS + Π -WCF .(iv) ( IA + AC !) + g = IA + AC ! + Σ -DNS + WCF − .(v) ( IA + AC ) + g = IA + AC + Σ -DNS + WCF − .(vi) ( IA + AC ) + g = IA + AC + (AC ) g .(vii) ( IA + FT ) + g = IA + FT + GDK.(viii) ( IRA + FT ) + g = IRA + FT + Σ -DNS + GDK.(ix) ( IRA + BI ) + g = ( IRA ) + g + BI + (BI ) g ⊆ IRA + BI + DNS .(x) ( IA + AC + BI ) + g = IA + AC + BI + Σ -DNS + WCF − .(xi) B + g = ( IA + AC + BI ) + g = B + (AC ) g . Proofs . (i): The G¨odel-Gentzen negative translations of the axioms of IA areprovable in IA , and the negative translations of the rules of inference are admissiblefor IA , so no additions are needed.(ii): To each quantifier-free formula A(x , y) there is by [5] a term s(x , y), withthe same free variables, such that IA proves both ∀ x ∀ y(A(x , y) ↔ s(x , y) = 0) and ∀ x ∀ y(u( h x , y i ) = 0 ↔ s(x , y) = 0) where u = λ z . s((z) , (z) ). Therefore IA proves ∃ β ∀ x ∀ y[A(x , y) ↔ β ( h x , y i ) = 0]. By intuitionistic logic the negative translation Over
IRA or EL , Π -WCF is equivalent to the principle ¬¬ Π -LEM in [2], and similarlyfor Σ -WCF and ¬¬ Σ -LEM. JOAN RAND MOSCHOVAKIS of ∀ x ∃ y β ( h x , y i ) = 0 is equivalent to ∀ x ¬¬∃ y β ( h x , y i ) = 0, and the negative trans-lation of ∃ α ∀ x β ( h x , α (x) i ) = 0 is equivalent to ¬¬∃ α ∀ x β ( h x , α (x) i ) = 0; therefore IRA + Σ -DNS ⊢ (qf-AC ) g . Conversely, Σ -DNS is equivalent over IRA tothe negative translation of an instance of qf-AC .(iii): Since qf-AC is a special case of AC Ar , IRA ⊆ IA + AC Ar . By formulainduction, IRA + Π -WCF proves ¬¬∃ η ∀ x ∀ y( η ( h x , y i ) = 0 ↔ A(x , y)) for everynegative arithmetical formula A(x , y). The negative translation of AC Ar now followsusing qf-AC and Σ -DNS as in (ii). This is a variation of Solovay’s argument;he started with Σ -WCF instead of Σ -DNS and Π -WCF , which give a precisecharacterization here. See the next theorem also.Conversely, ∀ x ∃ z(z = 0 ↔ ∀ y α ( h x , y i ) = 0) → ∃ ζ ∀ x( ζ (x) = 0 ↔ ∀ y α ( h x , y i ) = 0)is an instance of AC Ar , and ∀ x ¬¬∃ z(z = 0 ↔ ∀ y α ( h x , y i ) = 0) is provable in IA .It follows that ¬¬∃ ζ ∀ x( ζ (x) = 0 ↔ ∀ y α ( h x , y i ) = 0) is provable in IA + (AC Ar ) g .(iv): IA + AC ! = IRA + CF d by Vafeiadou’s characterization; therefore( IA + AC !) + g = ( IRA + CF d ) + g = ( IRA ) + g + CF d + (CF d ) g . Each instanceof WCF − is equivalent over IA to the conclusion of the negative translation of aninstance of CF d , and the negative translation of ∀ x(A(x) ∨ ¬ A(x)) is provable in IA for all formulas A(x), so (CF d ) g and WCF − are equivalent over IA .(v) follows from (iv) because the negative translations of AC and AC ! areequivalent over IA , and (vi) requires no comment.(vii): It is routine to show that IA + FT + GDK proves (FT ) g . The proofof GDK in IA + (FT ) g is an easy exercise. (viii) follows by (ii).(ix): It is routine to show that IRA + BI + DNS proves (BI ) g , and Σ -DNS follows from DNS in IRA . Now use (ii).(x) and (xi) follow from (v) and (vi) because IA + AC + ( ¬¬ A → A) ⊢ BI (cf. [6]). (cid:3) Corollary.
For each subsystem S of B considered in Theorem 5.1:(i) S ⊢ (MP ) g and so ( S + MP ) + g = S + g + MP .(ii) S + g + MP is consistent with strong continuous choice CC ( x S + g + CC MP .In particular, while MP is consistent with but independent of B + g + CC , itsclassical content (expressed by its negative interpretation) is contained in IA . Proofs. (i) is trivial; the rest is implicit in [6]. (ii) holds by classical function-realizability (cf. Lemma 8.4(a) of [6]). (iii) holds because every theorem of S + g +CC is S realizable but MP is not (cf. Lemma 10.7, Theorem 11.3 and Corollary11.10(a) in [6]). (cid:3) Two questions.
In general only one or two additional axioms needs to beadded to a subsystem S of B in order to prove its G¨odel-Gentzen negative inter-pretation. The unrestricted axioms of countable choice and comprehension haveresisted this treatment, requiring instead the addition of an axiom schema WCF − or (AC ) g . Is there a more elegant solution?And while DNS evidently suffices for the negative interpretation of BI , it maybe stronger than necessary. Does IRA + BI + Σ -DNS + (BI ) g ⊢ DNS ?6. Bar induction in two contexts
The next theorem sharpens Solovay’s proof that IA + AC Ar + BI + ( ¬¬ A → A)can be negatively interpreted in
IRA + BI + MP . In fact he proved the stronger ALIBRATING THE NEGATIVE INTERPRETATION 7 result that Σ -WCF (thus weak comprehension for all arithmetical formulas ) holdsin IRA + BI + MP ; but Π -WCF gives weak comprehension for all negative arithmetical formulas, which suffices with Σ -DNS and qf-AC for the negativeinterpretation of AC Ar . For the derivation of Π -WCF by bar induction and forthe negative interpretation of BI , MP is not needed; DNS suffices.The second result shows that DNS is no stronger over ( IRA + BI ) + g than thefollowing weak version of the classical Π comprehension axiom:Π − WCF . ∀ γ ¬¬∃ ζ ∀ x( ζ (x) = 0 ↔ ∀ α ∃ y γ ( α ( h x , y i )) = 0) . Is it as strong? The question appears to be open.6.1.
Theorem. (i)
IRA + (BI ) g ⊢ Π -WCF .(ii) IA + AC Ar + BI + ( ¬¬ A → A) can be negatively interpreted in (andtherefore is equiconsistent with) its subsystem
IRA + BI + DNS . Proofs. (i): Adapting Solovay’s argument that
IRA + BI + MP ⊢ Σ -WCF (as in [12], [8]), assume for contradiction (a) ∀ ζ ¬∀ x( ζ (x) = 0 ↔ ∀ y α ( h x , y i ) = 0).Then (b) ∀ ζ ¬¬∃ x[( ζ (x) = 0 & ¬¬∃ y α ( h x , y i ) = 0) ∨ ( ζ (x) = 0 & ∀ y α ( h x , y i ) = 0)]follows in IA , and this entails (c) ∀ ζ ¬¬∃ x[( ζ ((x) ) = 0 & α ( h (x) , (x) i ) = 0) ∨ ( ζ ((x) ) = 0 & ∀ y α ( h (x) , y i ) = 0)].In IRA one can define a binary sequence ρ such that ρ (w) = 0 if and only ifSeq(w) and for some j < lh(w) either (d) (w) j = 1 & ∃ y < lh(w) α ( h j , y i ) = 0, or (e) (w) j > α ( h j , ((w) j . − i ) = 0 ∨ ∃ y < (w) j . − α ( h j , y i ) = 0].Now prove (f ) ∀ ζ ¬¬∃ n ρ ( ζ (n)) = 0 by cases on (c) using (d) and (e), giving the firsthypothesis for an application of (BI ) g . The negative inductive predicate A(w) isA(w) ≡ ¬¬∃ j < lh(w)[((w) j = 1 → ¬∀ y α ( h j , y i ) = 0)& ((w) j > → [ α ( h j , ((w) j . − i ) = 0 → ∃ y < ((w) j . − α ( h j , y i ) = 0])] . Evidently (g) ∀ w(Seq(w) & ρ (w) = 0 → A(w)). In order to establish the inductivehypothesis (h) ∀ w(Seq(w) & ∀ sA(w ∗ h s + 1 i ) → A(w)), argue by contradiction asfollows, noting that in general (w ∗ h n i ) lh(w) = n.Assume Seq(w) & ∀ sA(w ∗ h s + 1 i ) & ¬ A(w). From A(w ∗ h i ) and ¬ A(w) weget (w ∗ h i ) lh(w) = 1 & ¬∀ y α ( h lh(w) , y i ) = 0. From ∀ nA(w ∗ h n + 2 i ) and ¬ A(w)we get ∀ n[(w ∗ h n + 2 i ) lh(w) > α ( h lh(w) , n i ) = 0 → ∃ y < n α ( h lh(w) , y i ) = 0)],from which it follows that ∀ n( α ( h lh(w) , n i ) = 0 → ∃ y < n α ( h lh(w) , y i ) = 0), con-tradicting ¬∀ y α ( h lh(w) , y i ) = 0. This completes the proof of (h).By (BI ) g conclude A( h i ), which is impossible because lh( h i ) = 0. ThereforeΠ -WCF holds in IRA + (BI ) g .(ii): (BI ) g was treated in Theorem 5.1(ix), and IRA + (BI ) g ⊢ (AC Ar ) g byformula induction from (i) (cf. the proof of Theorem 5.1(iii)). Observe that IRA ⊆ IA + AC Ar , and IA proves ( ¬¬ A → A) g for all formulas A. (cid:3) Theorem. (i) IA ⊢ (DNS ) g . Therefore, if S is any system such that IA ⊆ S ⊆ B ,then ( S + DNS ) + g = S + g + DNS .(ii) IRA + (BI ) g + Π -WCF ⊢ DNS . JOAN RAND MOSCHOVAKIS
Proof.
Only (ii) requires comment. Assume (a) ∀ α ¬¬∃ x ρ ( α (x)) = 0. The goalis to prove ¬¬∀ α ∃ x ρ ( α (x)) = 0 in IA + Π -WCF + (BI ) g . First define in IA afunction γ such that (b) ∀ x(Seq(x) → ∀ y( ρ (x ∗ α (y)) = 0 ↔ γ ( α ( h x , y i )) = 0)). Asthe desired conclusion is negative, assume (c) ∀ x( ζ (x) = 0 ↔ ∀ α ∃ y γ ( α ( h x , y i )) = 0)for “ ¬¬∃ -elimination” from the appropriate instance of Π -WCF . Then in partic-ular (d) ∀ w(Seq(w) → ( ζ (w) = 0 ↔ ∀ α ∃ y ρ (w ∗ α (y)) = 0)), so (a) holds.From (d) follow the other hypotheses (e) ∀ w(Seq(w) & ρ (w) = 0 → ζ (w) = 0)and (f ) ∀ w(Seq(w) & ∀ s ζ (w ∗ h s + 1 i ) = 0 → ζ (w) = 0) of the instance of (BI ) g with ζ (w) = 0 as the inductive predicate, so (g) ζ ( h i ) = 0, hence ∀ α ∃ x ρ ( α (x)) = 0.Discharging hypothesis (c) by ¬¬∃ -elimination, (h) ¬¬∀ α ∃ x ρ ( α (x)) = 0. (cid:3) Alternative varieties of constructive analysis
Markov’s recursive mathematical analysis.
Troelstra has shown thatMarkov’s recursive analysis RUSS, with the Kreisel-Lacombe-Shoenfield-TsejtlinTheorem, can be expressed in the language of arithmetic (cf.
CRM , [18]) using“extended Church’s Thesis” ECT and an arithmetical form of Markov’s Principle.In the two-sorted language Markov’s recursive analysis can be expressed by MRA ≡ IRA + CT + MP whereCT . ∀ α ∃ e[ ∀ x ∃ yT(e , x , y) & ∀ x ∀ y(T(e , x , y) → U(y) = α (x))](abbreviated ∀ α GR( α )) expresses a strong version of Church’s Thesis which isrefutable both in C and in I . Its negative interpretation, however, is consistentwith I (but not with C ).7.1.1. Theorem. (i)
MRA + g = MRA .(ii)
MRA is negatively interpretable in its subsystem
IRA + Σ -DNS + ∀ α ¬¬ GR( α ).(iii) I + Σ -DNS + ∀ α ¬¬ GR( α ) + VS is consistent classically, and proves ¬ MP . Proofs. (i) holds by IA + Σ -DNS ⊢ ( ∀ α ¬¬ GR( α ) ↔ (CT ) g ) with Theo-rem 5.1(ii) and the trivial facts that IRA + MP ⊢ Σ -DNS and IA + CT ⊢ ∀ α ¬¬ GR( α ). (ii) follows by Theorems 5.1(ii) and 5.2(i),(ii). (iii) holds by G realizability (cf. [11]). (cid:3) Bishop’s constructive mathematical analysis.
To the extent that it canbe identified with the common subsystem IA + AC of I and C, by Theorem5.1(vi) and the fact that IA + AC + ( ¬¬ A → A) ⊢ BI Bishop’s constructiveanalysis BISH has the same classical content as Kleene’s B . Evidently the classicalcontents (as expressed by the G¨odel-Gentzen negative interpretation) of classicalanalysis with countable choice, Bishop’s constructive analysis, and Markov’s re-cursive analysis are individually consistent with Kleene’s and Vesley’s versions ofintuitionistic analysis.While the perceived conflicts among CLASS, INT, BISH and RUSS do reflectthe different ways the full language is used in these four varieties of mathematicalpractice, those differences are not just a matter of intuitionistic versus classicallogic. They are essentially informed by fundamentally different ideas about whatconstitutes an infinite sequence of natural numbers. ALIBRATING THE NEGATIVE INTERPRETATION 9
References [1] S. Berardi, M. Bezem, and T Coquand. On the computational content of the Axiom of Choice.
Jour. Symb. Logic , 63(2):600–622, 1998.[2] M. Fujiwara and U. Kohlenbach. Interrelation between weak fragments of double negationshift and related principles.
Jour. Symb. Logic , 81(3):991–1012, 2018.[3] S. C. Kleene.
Introduction to Metamathematics . D. van Nostrand Company, Inc., 1952.[4] S. C. Kleene. Classical extensions of intuitionistic mathematics. In Y. Bar-Hillel, editor,
Logic,Methodology and Philosophy of Science , pages 31–44. North-Holland, 1965.[5] S. C. Kleene.
Formalized recursive functionals and formalized realizability . Number 89 inMemoirs. Amer. Math. Soc., 1969.[6] S. C. Kleene and R. E. Vesley.
The Foundations of Intuitionistic Mathematics, Especially inRelation to Recursive Functions . North Holland, 1965.[7] M. D. Krol’. A topological model for intuitionistic analysis with Kripke’s scheme.
Z. math.Logik und Grundlagen der Mathematik , 24:427–436, 1978.[8] J. R. Moschovakis. Solovay’s relative consistency proof for FIM and BI. arXiv:2101.05878v1.[9] J. R. Moschovakis.
Disjunction, existence and λ -eliminability in formalized intuitionisticanalysis . PhD thesis, University of Wisconsin, 1965.[10] J. R. Moschovakis. Disjunction and existence in formalized intuitionistic analysis. In J. N.Crossley, editor, Sets, Models and Recursion Theory , pages 309–331. North-Holland, 1967.[11] J. R. Moschovakis. Can there be no nonrecursive functions?
Jour. Symb. Logic , 36:309–315,1971.[12] J. R. Moschovakis. Classical and constructive hierarchies in extended intuitionistic analysis.
Jour. Symb. Logic , 68:1015–1043, 2003.[13] J. R. Moschovakis. Unavoidable sequences in constructive analysis.
Math. Logic Quarterly ,56(2):205–215, 2010.[14] J. R. Moschovakis and G. Vafeiadou. Some axioms for constructive analysis.
Arch. Math.Logik , 51:443–459, 2012.[15] A. Scedrov and R. Vesley. On a weakening of Markov’s Principle.
Arch. Math. Logik , 23:153–160, 1983.[16] A. S. Troelstra, editor.
Metamathematical Investigations of Intuitionistic Arithmetic andAnalysis . Number 344 in Lecture notes in mathematics. Springer-Verlag, 1973.[17] A. S. Troelstra. Corrections to some publications. University of Amsterdam, December 10,2018. https://eprints.illc.uva.nl/1650/1/CombiCorr101218.pdf.[18] A. S. Troelstra and D. van Dalen.
Constructivism in Mathematics , volume I and II. North-Holland, 1988. (cf. Corrections).[19] G. Vafeiadou. A comparison of minimal systems for constructive analysis. arXiv:1808.000383.[20] G. Vafeiadou.
Formalizing Constructive Analysis: a comparison of minimal systems and astudy of uniqueness principles . PhD thesis, University of Athens, 2012.[21] M. van Atten. The creating subject, the Brouwer-Kripke schema, and infinite proofs.
Indag.Math. , 29:1565–1636, 2018.[22] W. Veldman. Brouwer’s Fan Theorem as an axiom and as a contrast to Kleene’s alternative.
Arch. Math. Logic , 53:621–693, 2014.[23] R. E. Vesley. A palatable substitute for Kripke’s schema. In A. Kino, J. Myhill, and R. E.Vesley, editors,
Intuitionism and Proof Theory , pages 197–207. North-Holland, 1970.
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