aa r X i v : . [ m a t h . L O ] J a n SOLOVAY’S RELATIVE CONSISTENCY PROOF FOR FIM AND BI
JOAN RAND MOSCHOVAKISOCCIDENTAL COLLEGE (EMERITA)
Robert M. Solovay is a classical mathematician and logician whose many contributionsto set theory, proof theory, cryptography and computer science are well known. Althoughhe “thinks classically,” occasionally a question involving intuitionistic logic attracts hisattention. This historical note documents how he answered one such question and con-tributed a rather surprising result to the metamathematics of intuitionistic analysis.In 2002 Solovay wondered about the consistency strength of Kleene’s system I of in-tuitionistic analysis (cf. [2]) (which he called “ FIM ”), in relation to classical systemssuch as
ATR and Π -CA studied by Simpson in [6]. Kleene had proved in [1] that I is consistent relative to its neutral, classically correct subsystem B , which Solovay called“ BSK ”. In this exposition Solovay’s acronyms for I and B will replace Kleene’s from nowon.Solovay conjectured that FIM has the same consistency strength as a classical theory BI (“Bar-Induction”) he described as follows in an email message to the author on 30July 2002: The logic is classical. There are two sorts of variables: lower case lettersstand for the number variables; upper case letters for the set variables.There is a binary predicate = in two flavors: for equality of numbersor equality of sets; there is a binary epsilon relation; there are the usualfunction symbols from Peano: 0, S, +, ∗ .There are the usual Peano axioms. Induction is in the strong form.[Arbitrary formulas of the language are allowed.] One has extensionalityfor sets.One has arithmetic comprehension. The set of n such that Phi(n)exists [for each Phi without bound set variables.]The key axiom asserts that if R is a binary relation R which is a linearordering and has the property that for every non-empty subset of its fieldthere is an R-least element, then one has *full* R-induction. [*full* meansthat for any formula Phi expressible in the language, the corresponding“least element with respect to R” principle holds.]That completes my description of Bar-Induction.These facts are known about the consistency strength of Bar-Induction:1) It proves the consistency of ATR . [Roughly ATR says that ifR is a well-ordering of omega, and X is a set, then one can define thehyperarithmetical hierarchy relativized to X with indices chosen from R.]2) Π -Comprehension proves that Bar-Induction has a beta model [onein which all the things that the model believes are well-orderings areindeed well-orderings]. The quotations are verbatim, except for correcting trivial typographical errors (e.g. “consisten-ciency”), rendering sub- and superscripts in latex, and then replacing e.g. “Pi ” by “Π .” It looks quite routine to me to carry out the usual realizability of FIMwithin BI [henceforth, my abbreviation for Bar-Induction]. This will givea relative consistency proof for FIM relative to BI.To go the converse direction the obvious try is the “negative interpre-tation” of Godel. But I seem to get something like BI is consistent ifBSK [ = FIM - Continuity] + “Markov’s principle” is consistent.So one needs BSK + Markov’s principle is consistent relative to BSK.. . .For comparison with BI a brief description of FIM is needed here. The logic is intu-itionistic. There are two sorts of variables: lower case Latin letters a , b , c , . . . , x , y , z , a , . . . stand for the number variables; lower case Greek letters α, β, . . . stand for the (one-placenumber-theoretic) function variables (the “choice sequence” variables, in intuitionisticterminology).There are the usual function symbols 0, S, +, · , plus a finite number of symbols foradditional primitive recursive functions and functionals sufficient to formalize the theoryof recursive partial functionals (as Kleene did in [1]). There is a binary predicate = forequality of numbers; if s , t are terms (of type 0) then s = t is a prime formula. Parenthesesindicate application, e.g. α (x) is a term. Equality of functions is defined extensionallyby ( α = β ) ≡ ∀ x( α (x) = β (x)). Church’s λ , which can be eliminated, allows constructingfunctors (terms of type 1) from terms, e.g. λ x . λ -reduction schema.There are the usual Peano axioms and the defining axioms for the additional func-tion(al) symbols. Induction is in the strong form (arbitrary formulas of the language areallowed). There is an open equality axiom x = y → α (x) = α (y), and equality axioms forthe function(al) constants are provable in the subsystem IA (“two-sorted intuitionisticarithmetic”) of FIM outlined so far.Now Kleene did not restrict countable choice to arithmetical formulas. Full countablechoice for numbers AC : ∀ x ∃ yA(x , y) → ∃ β ∀ xA(x , β (x))is a theorem ( ∗ BSK = IA + AC + BI!, where AC ( x ∀ x ∃ α A(x , α ) → ∃ β ∀ xA(x , λ y .β (2 x · y ))and BI! is an axiom schema ( x IA plus thecountable comprehension schema AC !: ∀ x ∃ !yA(x , y) → ∃ β ∀ xA(x , β (x)) , where in general ∃ !yB(y) ≡ ∃ yB(y) & ∀ x ∀ y(B(x) & B(y) → x = y). Anne Troelstra ob-served (p. 73 of [7]) that quantifier-free countable choice qf-AC (AC for formulasA(x , y) containing only bounded numerical quantifiers, with parameters allowed) sufficesfor this purpose. Like Troelstra’s system EL , the subsystem IRA = IA + qf-AC of BSK precisely expresses intuitionistic recursive analysis (cf. Vafeiadou [8]).Kleene’s formalization of function-realizability culminated in Theorem 50 of [1], whichhad the simple consistency of
FIM relative to IA + AC ! + BI! as a corollary. Troel-stra’s remark on p. 208 of [7] suggested that AC ! might be weakened to qf-AC heretoo. Since arithmetic comprehension is expressible in the language of FIM by restrict-ing AC ! to arithmetic formulas A(x , y), and quantifier-free formulas are arithmetic,Solovay’s claim that FIM is consistent relative to BI was not very surprising. OLOVAY’S RELATIVE CONSISTENCY PROOF FOR FIM AND BI 3
It did not seem at all obvious, however, that arithmetic comprehension could be neg-atively interpreted in
BSK + MP. Working out the details required choosing correctlyamong Kleene’s four versions (26.3a-d in [2], described following the quotation) of thebar-induction axiom, as Solovay observed in an email message of 1 August 2002:BI is the system referred to by Simpson as Π ∞ -TI .Now the various forms of 26.3 are not equivalent in BI. The reason isthat we don’t have full comprehension or x2.2, so it’s hard to get a realas in 26.3b from a decidable predicate. So we have 26.3b in BI but *not*26.3a.Put it another way:In BI a linear-ordering coded by a real that happens to be a well-ordering one can do induction along. But if the linear ordering is justgiven by a formula of our language, BI says nothing. [Of course, in thepresence of full comprehension the distinction between “coded by a real”and “given by a formula” evaporates.]Letting w vary over sequence numbers (primitive recursive codes for finite sequenceswith a primitive recursive length function lh(w), where 1 codes the empty sequence, 2 n+1 codes the sequence ( n ), and ∗ denotes concatenation), Kleene’s x ∀ w(R(w) ∨ ¬ R(w)) & ∀ α ∃ xR( α (x)) & ∀ w(R(w) → A(w)) & ∀ w( ∀ nA(w ∗ n+1 ) → A(w)) → A(1) , while x ”) can be abbreviated by ∀ α ∃ x ρ ( α (x)) = 0 & ∀ w( ρ (w) = 0 → A(w)) & ∀ w( ∀ nA(w ∗ n+1 ) → A(w)) → A(1) , where ρ represents an arbitrary element of Baire space (“a real”). The version of Markov’s Principle referred to here as “MP” is the strong analyticalform ∀ α [ ¬∀ α ¬ ( α (x) = 0) → ∃ x α (x) = 0] . Kleene’s formalization of function-realizability in [1] established that
FIM + MP is simplyconsistent relative to a subsystem of
BSK + MP with countable choice weakened tocountable comprehension. Solovay observed that arithmetical comprehension is enough.In an email message of 20 August 2002 he outlined a proof in primitive recursivearithmetic
PRA that
BSK + MP is consistent relative to BI .I’ve thought through about 95% of the details on the following:PRA proves [Con(BI) implies Con(BSK + MP)].The proof basically is just carrying out the function realizability forBSK inside of BI. One actually proves the following in PRA.There is a primitive recursive function which takes as input a proof ofa sentence, A, in BSK + MP and outputs the following:(a) a Godel number e of a partial recursive function [mapping numbersto numbers];(b) a proof in BI that the function, call it φ n , is total.(c) a proof in BI that φ n realizes A.Of course, much less would suffice for the relative consistency proof.We could get by with the output Kleene considered two other forms of bar induction, x x ∀ w(R(w) ∨ ¬ R(w)) & ∀ α ∃ xR( α (x)) by ∀ α ∃ !xR( α (x))). BI is weaker than the other formsover IRA , but all four are equivalent over IA + AC !. JOAN RAND MOSCHOVAKIS OCCIDENTAL COLLEGE (EMERITA) [email protected] (d) A proof in BI that there is some function, α [not necessarily recur-sive] that realizes A.I remark that with realizability as presented in Kleene one can notsharpen (a) and (b) to(e) a “primitive recursive Godel number” of some primitive recursivefunction from numbers to numbers. [The reason is that if we could get(e) we could find a primitive recursive algorithm for every provably re-cursive function of the theory BI, and Ackerman’s fcn. is such a provablyrecursive function.]To complete his theorem Solovay needed to prove BI consistent relative to BSK +MP. While BI (Kleene’s x BSK + MP, andmost other axioms of BI are negatively interpretable in BSK , arithmetical comprehensionpresented a problem. After one false start, on 31 August 2002 Solovay sent the authorthe following message in which the role of the intended good classical models is explicitlyacknowledged:I think I’ve got it right. Recall that the plan is to define a classical theoryBI- such that(1) BI- has the same theorems as BI and this is finitistically provable.(2) BI- is designed to be easy to negatively interpret in BSK + MP.I have a new candidate for BI- for which I have checked (1). I’moptimistic re (2) but I have not yet thought through the details.I will consider a variant of BI where the type 1 variables range overfunctions from omega to omega.Without describing BI with utter precision, it includes:(1) The usual axioms and operations of Peano Arithmetic;(2) Full induction for any formula expressible in the language [param-eters allowed].(3) Axioms that insure that the type 1 variables are closed under Tur-ing jump and “recursive in” and contain all recursive functions.(4) The key axiom of Bar-Induction: If R is a linear ordering on omega,and there is no descending chain through R given by one of the type 1functions, then one has induction over R for arbitrary formulas of thelanguage [parameters allowed].BI- is obtained from BI by replacing (3) and (4) by suitable variants:(3 ′ ) We require that the type 1 functions contain all primitive recursivefunctions and that if α and β are type 1 functions and that γ is primitiverecursive in α and β then γ is a type 1 function.[Of course, I’m being sloppy here and implicitly describing axioms bydescribing what the intended good models of the theory are.](4 ′ ) Axiom x26.3b of Kleene’s FIM. [Caution: In the current classicalcontext, it makes quite a difference which version of 26.3 one takes.]. . .As I said, I’m claiming that BI- and BI have the same theorems andthat this can be proved in PRA. OLOVAY’S RELATIVE CONSISTENCY PROOF FOR FIM AND BI 5
Now
BI- looks like a classical version of a subsystem of
IRA + BI . In response toa request for a proof in BSK + MP of the negative interpretation of arithmetic compre-hension, Solovay responded on 2 September 2002 with a beautiful argument for the resultreferred to as “Solovay’s Lemma” in [3]. I haven’t tried for a direct proof. But perhaps the place to start isanalysing the proof of arithmetic comprehension in BI-.Here is a sketch of my argument.Let α : ω → ω . We aim to prove the existence of a β with the followingproperties:1) β (2n) = α (n);2) β (2n + 1) > ∃ yT α (n , n , y).3) If β (2n + 1) > α (n , n , y).We first define the following ρ which will be uniformly primitive recur-sive in α :1) If s is not a sequence number then ρ (s) = 1.2) Now let s be a sequence number. If for some j < lh(s), we havej = 2k and (s) j = α (k), then ρ (s) = 0;OR 3) if for some j < lh(s) we have j = 2k + 1 and (s) j = 0 and ∃ y ≤ lh(s)T α (k , k , y)then ρ (s) = 0:OR 4) if for some j < lh(s), we have j = 2k + 1 , (s) j = m + 1 and m isnot the least y such that T α (k , k , y) then ρ (s) = 0.OTHERWISE ρ (s) = 1.Now I describe the predicate A(x). [For use in the Brouwer principleaxiom.]not A(s) iff1) s is a sequence number;2) let j = 2k < lh(s). Then (s) j = α (k).3) let j = 2k + 1 , j < lh(s). Then (s) j > ∃ yT α (k , k , y). If so lettingy k be the least such y we have (s) j = y k + 1.From the fact that not A(1) we conclude by bar induction that ∃ γ ∀ n ρ ( γ (n)) > γ is our desired β .This proof, which guarantees that the range of the type 1 variables in an omega-modelof BI- is closed under the Turing jump, uses only primitive recursive comprehension andclassical bar induction. Arithmetic comprehension follows easily by formula induction.To complete his proof that Con(
BSK + MP) implies Con( BI ) Solovay confirmed inanother email message on 2 September 2002 that BI- can be negatively interpreted in
BSK + MP:The theory BI- doesn’t explicitly have arithmetic comprehension amongthe axioms. Instead it’s a theorem. But the set of things whose nega-tive interpretation is a theorem of BSK + MP is closed under classicaldeducibility. So it is enough to check the axioms of BI- have negativeinterpretations that are theorems of BSK + MP. The author has since used variations of his argument for several purposes in the context of intu-itionistic analysis, but Solovay has not published it himself. This is a LaTeX transcription of his originalproof.
JOAN RAND MOSCHOVAKIS OCCIDENTAL COLLEGE (EMERITA) [email protected]
The only problematical axioms are the instances of the version ofBrouwer’s principle that are axioms of BI-. Such an axiom has fourclauses:Hypotheses:1) Every α hits a bar.2) A holds at bars.3) A holds at s if it holds at all one step extensions of s.Conclusion:4) A holds at 1.Now the negative interpretations of clauses 2) through 4) transforminto things of the same shape. [The A gets replaced by its negativeinterpretation.]But 1) gives trouble. The inner existential number quantifier getsnegatively replaced. But MP allows us to restore this existential numberquantifier. This is crucial since the version of Brouwer available in BSKhas the existential number quantifier and not its negative transform.So roughly the negative transform puts a crimp in clause 1) and MPallows us to remove it.The result is an elegant, elementary proof of the relative consistency of formal systemsof classical and intuitionistic analysis with bar induction (and Markov’s Principle): Theorem. (Solovay) In primitive recursive arithmetic
PRA the following are equiva-lent: (a) Con( BI ).(b) Con( BSK + MP).(c) Con(
FIM + MP).A careful examination of his proof reveals the
Corollary.
In primitive recursive arithmetic
PRA the following are equivalent:(a) Con( BI ).(b) Con( IRA + BI + DNS ).(c) Con( FIM + DNS ).(d) Con( FIM + MP).Here DNS is the double-negation-shift principle ∀ α ¬¬∃ x ρ ( α (x)) = 0 → ¬¬∀ α ∃ x ρ ( α (x)) = 0 , which is weaker than MP over IRA . DNS is adequate for the negative interpretation ofBI over IRA + BI (cf. [4]), and IRA + BI is adequate for Kleene’s consistency proofof FIM as Troelstra and Solovay independently observed. Like MP, DNS is self-realizingover IRA . References [1] S. C. Kleene.
Formalized recursive functionals and formalized realizability . Number 89 in Memoirs.Amer. Math. Soc., 1969.[2] S. C. Kleene and R. E. Vesley.
The Foundations of Intuitionistic Mathematics, Especially in Relationto Recursive Functions . North Holland, 1965.[3] J. R. Moschovakis. Classical and constructive hierarchies in extended intuitionistic analysis.
Jour.Symb. Logic , 68:1015–1043, 2003. It is not certain that the equiconsistency of
BSK + MP with
BSK can be proved in
PRA , but cf.[5].
OLOVAY’S RELATIVE CONSISTENCY PROOF FOR FIM AND BI 7 [4] J. R. Moschovakis. Calibrating the negative interpretation. Extended abstract for 12th PanhellenicLogic Symposium, 2019.[5] J. R. Moschovakis. Markov’s principle and subsystems of intuitionistic analysis.
Jour. Symb. Logic ,84:870–876, 2019.[6] S. G. Simpson.
Subsystems of Second Order Arithmetic . Perspectives in logic. ASL, Cambridge Uni-versity Press, second edition, 2009.[7] A. S. Troelstra. Intuitionistic formal systems. In A. S. Troelstra, editor,
Metamathematical Investiga-tion of Intuitionistic Arithmetic and Analysis , Lecture Notes in Math. Springer-Verlag, 1973.[8] G. Vafeiadou. A comparison of minimal systems for constructive analysis. arXiv:1808.000383.
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