On the computational content of Zorn's lemma
aa r X i v : . [ c s . L O ] A p r On the computational content of Zorn’s lemma
Thomas Powell
Abstract
We give a computational interpretation to an abstract instance ofZorn’s lemma formulated as a wellfoundedness principle in the language ofarithmetic in all finite types. This is achieved through G¨odel’s functionalinterpretation, and requires the introduction of a novel form of recursionover non-wellfounded partial orders whose existence in the model of totalcontinuous functionals is proven using domain theoretic techniques. Weshow that a realizer for the functional interpretation of open inductionover the lexicographic ordering on sequences follows as a simple applica-tion of our main results.
Keywords.
Zorn’s lemma, G¨odel’s functional interpretation, domain the-ory, continuous functionals, higher-order computability
The correspondence between proofs and programs is one of the most fundamen-tal ideas in computer science. Initially connecting intuitionistic logic with thetyped lambda calculus, it has since been extended to incorporate a wide rangeof theories and programming languages.A challenging problem in this area is to give a computational interpretationto the axiom of choice in the setting of classical logic. A number of ingenioussolutions have been proposed, ranging from Spector’s fundamental consistencyproof of classical analysis using bar recursion [25] to more modern approaches,which include the
Berardi-Bezem-Coquand functional [2], optimal strategies insequential games [9], and Krivine’s ‘quote’ and ‘clock’ [16].In this paper, we introduce both a new form of recursion and a new com-putational interpretation of a choice axiom. In contrast to the aforementionedworks, which all focus on variants of countable choice, we give a direct com-putational interpretation to an axiomatic formulation of Zorn’s lemma. Ourwork is closest in spirit to Berger’s realizability interpretation of open inductionon the lexicographic ordering via open recursion [4] - an idea which was latertransferred to the setting of G¨odel’s functional interpretation in [22]. However,a crucial difference here is that we do not work with a concrete order, buta general parametrised variant of Zorn’s lemma, from which induction on thelexicographic ordering can be considered a special case.After formulating an axiomatic version of Zorn’s lemma in the language ofPeano arithmetic in all finite types, we study related forms of recursion on chain1ounded partial orders. In particular, we introduce a new recursive schemebased on the notion of a ‘truncation’, and give precise domain theoretic condi-tions under which the resulting fixpoint in the partial continuous functionals istotal (Theorem 4.9).We then demonstrate that we can use our new form of recursion to solvethe functional interpretation of our variant of Zorn’s lemma. Our approachcompletely separates the issues of correctness (that our program does what it’ssupposed to do) with that of totality (that our program is well-defined). Themain correctness result (Theorem 5.6) is extremely general, and its proof shortand direct, suggesting that our realizing terms are natural in a fundamental way.To establish totality we make use of our earlier domain theoretic results, andagain provide conditions which ensure that our computational interpretation issatisfied in the continuous functionals. We conclude with a concrete examplewhich ties everything together, demonstrating that the functional interpretationof open induction over the lexicographic ordering can be given as a special caseof our general result.This work aims to achieve several things. Our new recursive schemes onchain bounded partial orders form a contribution to higher-order computabilitytheory, which we believe is of interest in its own right. The subsequent compu-tational interpretation of Zorn’s lemma is a new result in proof theory, which wehope will lead to novel applications in future work. Finally, through our generaland abstract setting we provide some fresh insights into known computationalinterpretations of variants of the axiom of choice, particularly open recursion[4] and Spector’s original bar recursion [25].
We begin by presenting some essential background material. G¨odel’s functionalinterpretation, which only appears from Section 5 onwards, will be introducedlater.
Zorn’s lemma is central to this article, and features not only as a proof techniquebut also in the guise of an axiomatic principle. In what follows, < will alwaysdenote a strict partial order, and ≤ its reflexive closure. Definition . We call a partially ordered set (
S, < ) chain bounded if everynonempty chain γ ⊆ S (i.e. nonempty totally ordered subset of S ) has an upperbound in S , that is an element u ∈ S such that x ≤ u for all x ∈ γ . Theorem 2.2 (Zorn’s lemma) . Let ( S, < ) be a nonempty partially ordered setwhich is chain bounded. Then S contains at least one maximal element, that isan element x ∈ S such that ¬ ( x < y ) for all y ∈ S . The following well-known application of Zorn’s lemma will form a runningillustration throughout the paper: 2 xample . Let R be some nontrivial ring with unity, and define ( S, ⊂ ) to bethe set of all proper ideals of R partially ordered by the strict subset relation.Then S is nonempty since { } ∈ S , and is also chain bounded since for anynonempty chain γ , the set S x ∈ γ x is also a proper ideal of R and thus anelement of S . Therefore by Zorn’s lemma, S has a maximal element, or in otherwords, R has a maximal ideal.Our ability to apply Zorn’s lemma to establish the existence of maximalideals relies crucially on the fact that the upper bound S x ∈ γ x is also a properideal. This in turn is due to the fact that x being a proper ideal is a ‘piecewise’property, in that it can be reduced to an infinite conjunction ranging over finitepieces of information about x . We now make this intuition precise, leading toa modification of Zorn’s lemma (Theorem 2.8) close in spirit to open induction as studied by Raoult [24]. This will form the basis of our syntactic version ofZorn’s lemma presented in Section 3. Definition . An approximation function on the set X relative to some sets D and U is taken to be a mapping [ · ] ( · ) : X × D → U , where the sets D and U will play the following intuitive roles: • D is an index set of ‘sizes’, • U is a set of ‘approximations’ of elements of X .We call [ x ] d ∈ U the approximation of x of size d . Definition . We say that (
X, < ) is chain bounded with respect to the ap-proximation function [ · ] : X × D → U if any nonempty chain γ ⊆ X has anupper bound ˜ γ ∈ X satisfying the additional property that for all d ∈ D thereis some x ∈ γ such that [˜ γ ] d = [ x ] d . Example . Let (2 R , ⊂ ) be the powerset of some set R , and D the set of all finite subsets of R . Let U := { f : d → { , } | d ∈ D } and define [ x ] d : d → { , } by[ x ] d ( a ) = 1 ⇔ a ∈ x. Then (2 R , ⊂ ) is chain bounded with respect to [ · ]. To see this, given a chain γ let ˜ γ := S x ∈ γ x and suppose that a ∈ ˜ γ . Then there must be some x a ∈ γ suchthat a ∈ x a . For d ∈ D define x := max ⊂ { x a | a ∈ d ∩ ˜ γ } ∈ γ , and note that x is well defined since γ is totally ordered. Now, if [˜ γ ] d ( a ) = 1 then a ∈ d ∩ ˜ γ and thus a ∈ x , and so [ x ] d ( a ) = 1. On the other hand, if [˜ γ ] d = 0 then a / ∈ ˜ γ and so a / ∈ x (since a ∈ x trivially implies a ∈ ˜ γ , hence [ x ] d ( a ) = 0. Therefore[˜ γ ] d = [ x ] d . Definition . We call a predicate P ( x ) on X piecewise with respect to theapproximation function [ · ] : X × D → U if P ( x ) ⇔ ( ∀ d ∈ D ) Q ([ x ] d ) for somepredicate Q ( u ) on U . 3 heorem 2.8. Let ( X, < ) be a partially ordered set which is chain boundedw.r.t. the approximation function [ · ] : X × D → U , and P ( x ) a predicate on X which is piecewise w.r.t the same function. Then whenever P ( x ) holds for some x ∈ X , there exists y ∈ X such that P ( y ) holds but ¬ P ( z ) whenever y < z .Proof. Let S := { x ∈ X | P ( x ) } , and take some nonempty chain γ ⊆ S . Ourfirst step is to show that ˜ γ ∈ S , from which it follows that ( S, < ) is chainbounded. Since P ( x ) ⇔ ( ∀ d ∈ D ) Q ([ x ] d ) for some predicate Q ( u ), it sufficesto show that Q ([˜ γ ] d ) for all d ∈ D . But using that for any d there exists some x ∈ γ ⊆ S with [˜ γ ] d = [ x ] d we’re done, since Q ([ x ] d ) follows from P ( x ). Now,suppose that P ( x ) holds for some x ∈ X , and thus S is nonempty. By Zorn’slemma, S contains a maximal element y . We clearly have P ( y ), and if y < z then z / ∈ S and thus ¬ P ( z ). Example . Let (2 R , ⊂ ) be the powerset of some nontrivial ring R , with [ x ] d defined as in Example 2.6, and let P ( x ) be denote the predicate ‘ x is a properideal of R ’. Then this is a piecewise predicate w.r.t. [ · ], since each condition ofbeing a proper ideal can be formulated in a piecewise way. For instance, 0 ∈ x is equivalent to ∀ d (0 ∈ d ⇒ [ x ] d (0) = 1)and analogously for 1 / ∈ x . Similarly, closure of x under addition can be formu-lated in a piecewise way as ∀ d, r, r ′ ( { r, r ′ , r + r ′ } ⊆ d ∧ [ x ] d ( r ) = [ x ] d ( r ′ ) = 1 ⇒ [ x ] d ( r + r ′ ) = 1)and analogously for closure under left and right sided multiplication. Thereforethe existence of a maximal ideal also follows from Theorem 2.8 above. Note thatsince in r, r ′ above are always elements of the finite set d , ∀ r, r ′ can be treatedas a bounded quantifier, and so ‘ x is a proper ideal of R ’ is piecewise even withrespect to some quantifier-free Q ( u ). In the remainder of this article, our definitions and results typically take placein one of the following settings: • Within a formal theory of arithmetic in higher-types (syntactic); • Within a type structure of continuous functionals, either the total or par-tial (semantic).We now outline both of these settings in turn. Our basic formal system will bethe standard theories of Peano (resp. Heyting) arithmetic in all finite types PA ω (HA ω ). For us, the finite types T will be generated by the following grammar: ρ, τ ::= B | N | ρ × τ | ρ ∗ | ρ → τ B and natural numbers N , and in addi-tion to the usual function type ρ → τ include cartesian products ρ × τ and finitesequence types ρ ∗ as primitives. Note that alternatively, we could work over aminimal type structure N | ρ → τ and code up products and finite sequences asderived constructions.For full definitions of PA ω resp. HA ω the reader is directed to e.g. [1, 14, 26],bearing in mind that officially we would need to extend the canonical theoriespresented there with additional constants and axioms for dealing with cartesianproducts and list operations, which is nevertheless entirely standard (for detailssee e.g. [26, Chapter I.8] and [27]).Terms of PA ω resp. HA ω are those of G¨odel’s System T (with product andsequence types). We denote by 0 ρ : ρ a canonical zero object of type ρ . For-mulas of PA ω (resp. HA ω ) include atomic formulas = B and = N for equalityat base types, and are built using the usual logical connectives, together withquantifiers for each type. Axioms and rules include those of full classical (resp.intuitionistic) logic, non-logical axioms for the constants symbols together withequality axioms and the axiom of induction. Equality at higher types is definedinductively e.g. f = ρ → τ g := ∀ x ρ ( f x = gx ), and we include axioms for exten-sionality, so that our formulation of PA ω corresponds to the fully extensionalE-PA ω of [14].The canonical models for PA ω include the type structures of all set-theoreticfunctionals S ω together with total continuous functional C ω . However, the ma-jority of recursive schemes which have been used to interpret the axiom of choice(including essentially all known variants of bar recursion) are no longer satisfi-able in S ω , and instead have C ω as their canonical model. In the remainder ofthis section, we outline some key facts about this model. In one sentence, the type structure C ω of continuous functionals consists offunctionals which only require a finite piece of information about their inputto compute a finite piece of information about their output. Over the years,they have turned out to form an elegant and robust class of functionals, and inparticular are the standard model for bar recursive extensions of the primitiverecursive functionals.There are various ways of characterising the continuous functionals, datingback to Kleene [13] (whose construction was based on associates) and Kreisel[15] (who instead used formal neighbourhoods). However, here we follow thedomain theoretic approach of Ershov [8], who demonstrated that the continuousfunctionals can be constructed as the extensional collapse of the total objectsin the type structure P ω of partial continuous functionals. This in particularprovides us with a simple method for showing that our new recursive schemes aresatisfied in C ω , namely proving that the corresponding fixpoints in P ω representtotal objects. For accomprehensive account of all this, the reader is encouragedto consult [18] or the recent book [17]. Here we provide no more than a briefoverview of the relevant theory. 5or each finite type σ , we define the domain P σ of partial continuous func-tionals of that type as follows: P B := B ⊥ and P N := N ⊥ where B ⊥ resp. N ⊥ are the usual flat domains of booleans and natural numbers, P ρ × τ := P ρ × P τ , P σ ∗ := { [ x , . . . , x n − ] | n ∈ N and x i ∈ P σ } ∪ {⊥} and finally P ρ → τ := [ P ρ → P τ ] where [ D → E ] denotes the domain of all functions between X and Y whichare continuous in the domain theoretic sense (i.e. are monotone and preservelubs of chains). We write P ω := { P σ } σ ∈ T for this type structure of partialcontinuous functionals.For each type σ , we define the set T σ ⊂ P σ of total objects in the usual way as T B := B and T N := N , T ρ × τ := T ρ × T τ , T σ ∗ := { [ x , . . . , x n − ] | n ∈ N and x i ∈ T σ } and finally T ρ → τ := { f ∈ P ρ → τ : ∀ x ( x ∈ T σ ⇒ f x ∈ T τ ) } . Furthermore,we define an equivalence relation ≈ σ on T σ to equate total objects that agreeon total inputs: x ≈ B y iff x = y and similarly for ≈ N , ( x, x ′ ) ≈ ρ × τ ( y, y ′ ) iff x ≈ ρ y and y ≈ τ y ′ , [ x , . . . , x n − ] ≈ σ ∗ [ y , . . . , y m − ] iff n = m and x i ≈ σ y i for all i < n and finally f ≈ ρ → τ g iff f x ≈ τ gx for all x ∈ T ρ .It turns out that all total objects are hereditarily extensional, in the sensethat if f ∈ T ρ → τ and x ≈ ρ y then f x ≈ τ f y , and therefore the extensional col-lapse C σ := T σ / ≈ σ of the total objects constitutes a hierarchy C ω := { C σ } σ ∈ T of functionals in its own right. We call this hierarchy the total continuous func-tionals, and as shown by Ershov, C ω is in fact isomorphic to the constructionsof Kleene and Kreisel.It is well known that C ω is a model of PA ω , and so in particular, any closedterm e : σ of System T has a canonical interpretation e C ∈ C σ , which can inturn be represented by some element e P ∈ T σ of the corresponding equivalenceclass in P ω . Suppose now that we extend System T with some new constantsymbol Φ : σ which satisfies a recursive defining axiom( ∗ ) Φ( x , . . . , x n ) = r (Φ , x , . . . , x n )where r is a closed term of System T. We can equivalently express ( ∗ ) as Φ = e (Φ) for e : σ → σ defined by e ( f ) := λx , . . . , x n .r ( f, x , . . . , x n ) . Now, since e is primitive recursive, it has a total representation e P ∈ T σ → σ ⊂ [ P σ → P σ ], and it is a basic fact of domain theory that Φ can be given aninterpretation Φ P in P ω as a least fixed point of e P i.e.Φ P := G n ∈ N e nP ( ⊥ σ )satisfies Φ P = e P (Φ P ). If we can now show that Φ P is in fact total, in otherwords that Φ P ( x , . . . , x n ) is total for all total inputs x , . . . , x n , then definingΦ C := [Φ P ] ≈ σ ∈ C σ we haveΦ C = [Φ P ] ≈ σ = [ e P (Φ P )] ≈ σ = [ e P ] ≈ σ → σ [Φ P ] ≈ σ = e C (Φ C )and therefore the object Φ C satisfies the defining axiom ( ∗ ) in C ω . In otherwords, C ω is a model of the theory PA ω + Φ, where by the latter we mean theextension of PA ω with the new constant Φ and axiom ( ∗ ).6n short, in order to show that the extension of System T with some new formof recursion Φ is satisfied in C ω , it suffices to show that the natural interpretationof Φ as a fixpoint in P ω is total. This approach has been widely used in thepast to show that various forms of strong recursion arising from the axiom ofchoice have C ω as a model (see e.g. [4, Proposition 5.1] or [5, Theorem 1]), andwill be fundamental for us as well in Section 4.In addition to showing that extensions of System T have a model, we mustalso confirm that they represent programs , in the sense that any object of type N can be effectively reduced to a numeral. This follows by appealing to Plotkin’sadequacy theorem [20]: We observe that terms of System T plus our new recur-sor Φ can be viewed as terms in PCF (recursion being dealt with by using thefixpoint combinator), which in addition inherit the usual call-by-value reductionsemantics, with the defining axiom ( ∗ ) being interpreted as a rewrite rule. Byshowing that Φ represents a total object in the semantics of PCF within P ω , itfollows that any closed term e : N in our extended calculus is denoted by somenatural number i.e. [ e ] ∈ N , and by the adequacy theorem e must then reduceto the numeral n . Remark.
In order to avoid burdening ourselves with too many subscripts, inthe remainder of this paper we use the same notation for e : σ in PA ω , itscanonical interpretation e ∈ C σ and some suitable representation e ∈ P σ , ratherthan laboriously writing e C resp. e P whenever we are working in continuousmodels. Where there is any ambiguity, we make absolutely clear which systemwe are working in, and in the case of e P for primitive recursive e we writeexplicitly how e can be represented as a partial object unless this is obvious. In this short section, we present a general axiomatic formulation of Zorn’slemma. This will be based on Theorem 2.8, and is close in spirit to the ax-iom of open induction as studied in [4]. Like open induction, our axiom is ofcourse weaker than the full statement of Zorn’s lemma. Nevertheless, as we willsee in Section 6, it in fact generalises open induction, and so in particular canbe used to formalize highly non-trivial proofs such as Nash-Williams’ minimalbad-sequence construction (cf. [4, 22]). To be more specific, our axiom schemawill take the shape of a maximum principle of the form ∃ xP ( x ) → ∃ y ( P ( y ) ∧ ∀ z > y ¬ P ( z ))where P ( x ) will range over formulas which are piecewise in the sense of Defini-tion 2.7 and < denotes some chain bounded partial order. However, our preciseformulation of the axiom will be within the language of PA ω , and therefore boththe notion of a piecewise formula and the relation < need to be represented ina suitable way. Remark.
From now all we annotate important definitions and results with thetheory or model in which they take place, which will usually be some extensionof PA ω resp. HA ω or one of C ω or P ω . 7 efinition ω /HA ω ) . Suppose that [ · ] ( · ) : σ × δ → ν is a closed term ofSystem T, and Q ( u ν ) is a formula in the language of PA ω /HA ω . Then we saythat the formula P ( x σ ) : ≡ ∀ d δ Q ([ x ] d ) is piecewise w.r.t. [ · ].Now, while it is too restrictive to demand that < be represented by someprimitive recursive functional σ × σ → B , for all applications we are interestedin it suffices that < can be expressed as a Σ formula as follows: x < y : ≡ ∃ a ρ ( y = σ x ⊕ a ∧ x ≺ a )where now ⊕ : σ × ρ → σ and ≺ : σ × ρ → B are closed terms of System T forsome type ρ (we use x ⊕ a to denote ⊕ ( x, a ) and x ≺ a to denote ≺ ( x, a ) = 1,and similarly a ≻ x to denote x ≺ a ). Definition ω /HA ω ) . Let [ · ] ( · ) : σ × δ → ν , ⊕ : σ × ρ → σ and ≺ : σ × ρ → B be closed terms of System T. The axiom schema ZL [] , ⊕ , ≺ is given by ∃ x σ ∀ d δ Q ([ x ] d ) →∃ y σ ( ∀ d Q ([ y ] d ) ∧ ∀ a ≻ y ∃ d ¬ Q ([ y ⊕ a ] d ))where Q ( u ν ) ranges over arbitrary formulas of PA ω (and does not contain x, y, a, d free).Note that our axiomatic formulation no longer mentions a main ordering < , which is instead induced by ⊕ and ≺ . Note also that chain boundednessof < is not formulated as a part of the axiom itself, and as such, validity ofZL [] , ⊕ , ≺ in some given type structure will depend on the interpretation of < being chain bounded in that model. We could of course seek to incorporatechain boundedness into the syntactic definition of Zorn’s lemma and give acomputational interpretation to the axiom as a whole. This would lead to afascinating but extremely complex computational problem which would steerus in a quite different direction to the current article, and so we leave this tofuture work (cf. Section 7). We now illustrate our new principle by continuingour example from Section 2.1, whose computational content has already beenstudied in in [23]. Example . Let σ := N → B , δ := N , ν := B ∗ and ρ := N × ( N → B ) anddefine [ x ] d := [ x (0) , . . . , x ( d − x ⊕ ( n, y ) := x ∪ yx ≺ ( n, y ) := x ( n ) · y ( n )where b represents the negation of the boolean b and( x ∪ y )( n ) := 1 if x ( n ) = 1 or y ( n ) = 1 else 0 . These are all clearly definable as closed terms of System T, and in this caseZL [] , ⊕ , ≺ is equivalent to ∃ xP ( x ) →∃ y ( P ( y ) ∧ ∀ ( n, z )( y ( n ) = z ( n ) = 1 → ¬ P ( y ∪ z )))8or P ( x ) : ≡ ∀ d Q ([ x (0) , . . . , x ( d − x : N → B as characteristic functions for subsets of the natural numbers. Moreover, givensome countable ring R whose elements can be coded up as natural numbers andwhose operations + R and · R represented as primitive recursive functions N × N → N , the existence of a maximal ideal in R would be provable in PA ω + ZL [] , ⊕ , ≺ .We do not give full details of this (an outline of the formalisation can be foundin [23]). Instead we simply sketch why both S ω and C ω satisfy ZL [] , ⊕ , ≺ and arethus models of PA ω + ZL [] , ⊕ , ≺ .Working in C ω (the same argument is also valid for S ω ) we apply Theorem 2.8for X := C N → B ∼ = B N which via the identification of sets with their characteristicfunction is isomorphic to the powerset of N , together with the proper subsetrelation, observing that x ⊂ y ⇔ ∃ ( n, z ) ∈ N × B N ( y = x ∪ z ∧ n / ∈ x ∧ n ∈ z )where the right hand side is just the interpretation of the formula ∃ ( n, z )( y = x ⊕ ( n, z ) ∧ y ≺ ( n, z )) in C ω . Clearly ( X, ⊂ ) is chain bounded w.r.t. [ x ] d :=[ x (0) , . . . , x ( d − Q ([ u (0) , . . . , u ( k − C ω on finite sequences ofnatural numbers the resulting formula P ( x ) : ≡ ∀ d Q ([ x (0) , . . . , x ( d − X in C ω is piecewise w.r.t. [ · ], and thus by Theorem 2.8 whenever ∃ xP ( x ) issatisfied there exists some y ∈ X such that P ( y ), and also ¬ P ( z ) whenever y < z (or alternatively ∀ ( n, z )( n / ∈ x ∧ n ∈ z ⇒ ¬ P ( y ∪ z )). Thus ZL [] , ⊕ , ≺ isvalid in C ω . We now come to our first main contribution, in which we study modes of re-cursion over chain bounded partial orders that form an analogue to the axiomZL [] , ⊕ , ≺ . A precise connection between a restricted form of ZL [] , ⊕ , ≺ and oursecond mode of recursion will be presented in Section 5, but the results of thissection are more general, and we consider them to be of interest in their ownright. As such, this section could be read as a short, self-contained study inwhich we explore different recursion schemes over orderings induced by the pa-rameters ( ⊕ , ≺ ). Totality of our recursors will be justified using a variant ofTheorem 2.8, and the two main modes of recursion considered here will primar-ily differ in how we achieve ‘piecewise-ness’ of the totality predicate. The first,which we characterise as ‘simple’ recursion, uses a sequential continuity princi-ple but is valid only for discrete output types, whereas the second, which we call‘controlled’ recursion, is total for arbitrary output type but uses an auxiliaryparameter in the recursor itself to ensure wellfoundedness.For the remainder of this section, we fix types σ, ρ, δ and ν , together withclosed terms [ · ] : σ × δ → ν , ⊕ : σ × ρ → σ and ≺ : σ × ρ → B of System T, whichare analogous to those in Section 3. For definitions and results below whichtake place in the model P ω , note that [ · ] ∈ T σ × δ → ν denotes some canonical9epresentation of the corresponding term of System T as a total continuousfunctional, and similarly for ⊕ ∈ T σ × ρ → σ and ≺∈ T σ × ρ → B (cf. Section 2.3). ( ⊕ , ≺ )The first recursion scheme we consider is represented by the constant Φ θ ⊕ , ≺ equipped with defining equationΦ f x = θ f x ( λa . Φ f ( x ⊕ a ) if a ≻ x else 0 θ ) (1)where f : σ → ( ρ → θ ) → θ and x : σ , and we recall that 0 θ is a canonical zeroterm of type θ . Note that in the defining equation we suppressed the parameterson Φ - and we will continue to do this whenever there is no risk of ambiguity.In what follows, it will be helpful to use the abbreviationΦ f,x := λa . Φ f ( x ⊕ a ) if a ≻ x else 0 θ so that the defining equation can then be expressed asΦ f x = f x Φ f,x . Definition P ω ) . Let L ⊆ T σ . We say that a functional ψ ∈ P σ → θ is piecewisecontinuous with respect to [ · ] and L , if for any x ∈ L such that ψx ∈ T θ thereexists some d ∈ T δ such that ∀ y ∈ T σ ([ x ] d = [ y ] d ⇒ ψy ∈ T θ ) . Definition P ω ) . A partial order < on T σ is compatible with ( ⊕ , ≺ ) if x S, < ) is chain bounded in the usual sense. Taking somenonempty chain γ ⊆ S , by chain boundedness in the sense of Definition 4.3 thishas some upper bound ˜ γ ∈ L . Suppose for contradiction that Φ f ˜ γ ∈ T θ . Bypiecewise continuity of Φ f there exists some d ∈ T δ such that [˜ γ ] d = [ y ] d impliesΦ f y ∈ T θ for any y ∈ T σ . But then there exists some x ∈ γ with [˜ γ ] d = [ x ] d and thus Φ f x ∈ T θ , contradicting x ∈ S . Therefore Φ f ˜ γ / ∈ T θ and thus ˜ γ ∈ S .To prove the main result, suppose for contradiction that Φ f / ∈ T σ → θ , whichimplies that S = ∅ . Then by Zorn’s lemma, S has some maximal element x .But for any a ∈ T ρ with x ≺ a we have x < x ⊕ a by compatibility, andthus Φ f,x ( a ) = Φ f ( x ⊕ a ) ∈ T θ . It follows that Φ f,x ∈ T ρ → θ , since in theother case ¬ ( x ≺ a ) we have Φ f,x ( a ) = 0 θ ∈ T θ . But then by totality of f we have Φ f x = f x Φ f,x ∈ T θ , contradicting x ∈ S . Therefore S = ∅ and soΦ f ∈ T σ → θ .The technique we have used in this proof is a generalisation of the proofof Theorem 0.3 from [3], which uses Zorn’s lemma to show that the so-calledBerardi-Bezem-Coquand functional defined in [2] is total. We now give a con-crete example of how the result can be applied, but first we state and prove asequential continuity lemma (cf. also [3, Lemma 0.1]), which will also be usefulin later sections. Lemma 4.5 ( P ω ) . Let θ be a discrete type i.e. one which does not containfunction types. Suppose that ψ ∈ P ( N → σ ) → θ where σ is some arbitrary type, that x ∈ T N → σ satisfies x ( ⊥ ) = ⊥ σ and that ψx ∈ T θ . Then there is some d ∈ N such that for any y ∈ P N → σ , whenever x ( i ) = y ( i ) for all i < d then ψx = ψy .Proof. We use a simple adaptation of the proof of Lemma 0.1 of [3]. Since T θ is open in the Scott topology whenever θ is discrete, there is some compact x ⊑ x such that ψx ∈ T θ , and since ψx ⊑ ψx we must in fact have ψx = ψx (that y ⊑ z implies y = z for y ∈ T θ is evidently true for θ = N ⊥ or θ = B ⊥ ,and holds for arbitrary discrete θ by induction over its structure). Now, since x is compact (i.e. contains only a finite amount of information) there is some d ∈ N such that x ( i ) = ⊥ σ for all i ≥ d . Suppose now that y ∈ P N → σ satisfies x ( i ) = y ( i ) for all i < d . We claim that x ⊑ y . To see this, note that for i < d we have x ( i ) ⊑ x ( i ) = y ( i ), for i ≥ d we have x ( i ) = ⊥ σ ⊑ y ( i ), and for i = ⊥ since x ( ⊥ ) = ⊥ σ we must also have x ( ⊥ ) = ⊥ σ ⊑ y ( ⊥ ). Therefore ψx ⊑ ψy and since ψx ∈ T θ we must have ψy = ψx = ψx . Example . Let [ · ], ⊕ and ≺ be the obvious total representatives of the prim-itive recursive functions defined in Example 3.3 i.e. extensions that are definedalso on non-total input, for example[ x ] d := [ x (0) , . . . , x ( d − d ∈ N else ⊥ T σ = T N → B is the set of all functions x : N ⊥ → B ⊥ which are monotone (in the domain theoretic sense) and satisfy x ( n ) ∈ B whenever n ∈ N . We define L ⊂ T N → B to consist of those functions which are strict , in that they satisfy in addition x ( ⊥ ) = ⊥ .Now suppose that θ is discrete. Then any function ψ ∈ P ( N → B ) → θ is piece-wise continuous w.r.t. [ · ] and L . To see this, take any strict x such that ψx ∈ T θ . Then by Lemma 4.5 there exists some d ∈ N such that for any y ∈ P N → B (and so in particular y ∈ T N → B ) we have ψy = ψx ∈ T θ whenever[ x ] d = [ x (0) , . . . , x ( d − y (0) , . . . , y ( d − y ] d .Next define < on T N → B by x < y iff x ( i ) = 1 ⇒ y ( i ) = 1 for all i ∈ N andthere exists at least one j ∈ N with x ( j ) = 0 and y ( j ) = 1. Then < is compatiblewith ( ⊕ , ≺ ), and moreover, for any nonempty chain γ ⊆ T N → B define ˜ γ ∈ L by˜ γ ( n ) := ( x ( n ) = 1 for some x ∈ γ γ ( ⊥ ) = ⊥ . Then clearly x ≤ ˜ γ for all x ∈ γ , and moreover for any d ∈ N = T N , by a variant of the argument in Example 2.6 we have [˜ γ ] d = [ x ] d for some x ∈ γ . Thus < is chain bounded w.r.t. [ · ] and L .Now, let Φ denote the least fixed point in P ω ofΦ f x = θ f x ( λ ( n, y ) . Φ f ( x ∪ y ) if x ( n ) · y ( n ) = 1 else 0 θ ) . (2)By Theorem 4.4, taking any total f , since Φ f ∈ P ( N → B ) → θ is automaticallypiecewise continuous, we have that Φ f is total, and therefore Φ is a total objectin P ω . This implies that C ω | = PA ω + Φ for the extension of PA ω with somenew constant satisfying the defining axiom (2). In Example 4.6 we have essentially shown that a simple variant of ‘updateinduction’ in the sense of [4] is total. In fact, with a slight modification ofthe above proof we would be able to reprove totality of update induction inits general form. However, in this paper we are primarily interested in forms ofrecursion on chain bounded partial orders which do not correspond to the simplerecursive scheme (1). The reason for this is that in order to establish totality ofΦ f for any total f , we are typically required to restrict the complexity of theoutput type θ to being discrete, so that something along the lines of Lemma 4.5applies. As we will see, this is a problem for the functional interpretation.Before we go on, we illustrate why extending PA ω / HA ω with (2) for non-discrete types does not even result in a consistent theory! Let us set θ := N → N and define f by f xp := λn . p ( n, { n } )( n + 1)where we identify the set { n } with its characteristic function of type N → B .Then defining k := Φ f ( ∅ )0 : N we have k = 1 + Φ f ( { } )(1) = 2 + Φ f ( { , } )(2)= . . . = k + 1 + Φ f ( { , . . . , k } )( k + 1) > k ω / HA ω . The key point at which the argument fromExample 4.6 fails is that Lemma 4.5 is no longer valid for θ := N → N : if ψx isa function then it can in general query an infinite part of x . To overcome thiswe could restrict our attention to those f such that Φ f is piecewise continuousand thus total for non-discrete output type: For example, let f xp := λn < N . p ( n, { n } )( n + 1)for some numeral N : N , so that p is only queried finitely many times. Thenworking in P ω , for total x we would haveΦ f x = λn . ( Φ f ( x ∪ { n } )( n + 1) if x ( n ) = 1 ∧ n < N f x could be taken to be the maximum of allpoints of continuity of the functions λy. Φ f ( y ∪ { n } )( n + 1) for n < N and atpoint y := x .We now propose cleaner way of extending (2) to non discrete output types.Instead of restricting f , we add a new parameter ω which controls the recursiondirectly. ( ⊕ , ≺ )We modify the scheme (1), resulting in a slightly more elaborate mode of recur-sion in which the continuity behaviour is controlled by some auxiliary functional ω . Define the constant Ψ θ ⊕ , ≺ (from now on omitting the parameters) byΨ ωf x = θ f { x } Ψ ω,f ( λa . Ψ ωf ( { x } Ψ ω,f ⊕ a ) if a ≻ { x } Ψ ω,f else 0 θ ) (3)where f : σ → ( ρ → θ ) → θ and ω : σ → ( ρ → θ ) → σ and { x } Ψ ω,f is defined by { x } Ψ ω,f := σ ωx ( λa . Ψ ωf ( x ⊕ a ) if a ≻ x else 0 θ ) . Observe that Ψ is still defined as the fixed point of a simple closed term ofPA ω , and as it will turn out, this modified scheme will allow us to admit outputof arbitrary type level. Moreover, we will show later that by instantiating ω by a suitable closed term of PA ω , we can use this recursive scheme to define arealizer for the functional interpretation of our axiomatic form of Zorn’s lemma.As before, we use the abbreviationΨ ω,f,x := λa . Ψ ωf ( x ⊕ a ) if a ≻ x else 0 θ so that the defining equation (3) now becomesΨ ωf x = f { x } Ψ ω,f Ψ ω,f, { x } Ψ ω,f for { x } Ψ ω,f := ωx Ψ ω,f,x . We now give a totality theorem analogous to Theorem4.4, but with the notion of piecewise continuity replaced by a slightly moresubtle property. 13 efinition P ω ) . We say that a pair of functionals ψ ∈ P σ → σ and φ ∈ P σ → θ form a truncation with respect to [ · ], L ⊆ T σ and some partial order < on T σ ifthe following two conditions are satisfied:(a) For any x, y ∈ T σ , if ψx ∈ T σ and ψx < y then x < y .(b) For any x ∈ L such that ψx ∈ T σ and φ ( ψx ) ∈ T θ there exists some d ∈ T δ such that ∀ y ∈ T σ ([ x ] d = [ y ] d ⇒ ψx = ψy ) . Example . Continuing from Example 4.6 with σ := N → B but θ now arbi-trary, for any N ∈ N the continuous functional ψ N ∈ T σ → σ defined by ψ N x ( n ) := ( x ( n ) if n < N ψ N x ( ⊥ ) = ⊥ forms a truncation with any other functional φ ∈ P σ → θ w.r.t [ · ], L and < . To see this, observe that for any strict x , if y ∈ T σ satisfies[ x (0) , . . . , x ( N − y (0) , . . . , y ( N − ψ N x = ψ N y and so ψ N satisfiescondition (b) of being a truncation for d := N . For condition (a), if ψ N x < y this means that x ( i ) = 1 ⇒ y ( i ) = 1 for all i < N , y ( i ) = 1 for all i ≥ N , and x ( j ) = 0 and y ( j ) = 1 for some j < N , from which it follows easily that x < y .On the other hand, the continuous functional φ N ∈ T σ → σ defined by φ N x ( n ) := ( x ( n ) if n < N φ N x ( ⊥ ) = ⊥ satisfies condition (b) of being a truncation, but not condition(a), since for x representing the characteristic function of the singleton set { N } we have φ N x < x but not x < x . Finally, the identity functional ι ∈ T σ → σ clearly satisfies condition (a) of being a truncation, but condition (b) fails fore.g. φ also the identity function, since for some arbitrary x ∈ L there is no d ∈ N such that for any total y , [ x (0) , . . . , x ( d − y (0) , . . . , y ( d − x ( n ) = y ( n ) for all n ∈ N . Theorem 4.9 ( P ω ) . Let Ψ denote the least fixed point of the primitive recur-sive defining equation (3), and suppose that there exist < on T σ and L ⊆ T σ such that < is compatible with ( ⊕ , ≺ ) and chain bounded w.r.t. [ · ] and L . Let f ∈ T σ → ( ρ → θ ) → θ and ω ∈ T σ → ( ρ → θ ) → σ . Then whenever {·} Ψ ω,f ∈ P σ → σ and λx . f x Ψ ω,f,x ∈ P σ → θ form a truncation w.r.t. [ · ] , L and < , it follows that Ψ ωf ∈ T σ → θ .Proof. We again appeal to Zorn’s lemma, but this time on the set S ⊆ T σ givenby S := { x ∈ T σ | either { x } Ψ ω,f / ∈ T σ or Ψ ωf x / ∈ T θ } . To show that ( S, < ) is chain bounded in the usual sense, take some nonemptychain γ ⊆ S and consider its upper bound ˜ γ ∈ L in the sense of Definition 4.3.14s before, we want to show that ˜ γ ∈ S , so we assume for contradiction that thisis not the case, which means that both { ˜ γ } Ψ ω,g ∈ T σ and Ψ ωf ˜ γ ∈ T θ . But thenby Definition 4.7 (b) - observing that { ˜ γ } Ψ ω,g and f { ˜ γ } Ψ ω,g Ψ ω,f, { ˜ γ } Ψ ω,g = Ψ ωf ˜ γ are both total - there exists some d ∈ T δ such that { ˜ γ } Ψ ω,f = { y } Ψ ω,f for any y ∈ T σ satisfying [˜ γ ] d = [ y ] d . But since by Definition 4.3 there exists some x ∈ γ such that [˜ γ ] d = [ x ] d it therefore follows that { ˜ γ } Ψ ω,f = { x } Ψ ω,f and thusΨ ωf ˜ γ = f { ˜ γ } Ψ ω,f Ψ ω,f, { ˜ γ } Ψ ω,f = f { x } Ψ ω,f Ψ ω,f, { x } Ψ ω,f = Ψ ωf x which imply that { x } Ψ ω,f ∈ T σ and Ψ ωf x ∈ T θ and thus x / ∈ S , a contradiction.Thus ˜ γ ∈ S and S is chain bounded.We now suppose that the conclusion of the main result is false, which meansthat there exists some x ∈ T σ such that Ψ ωf x / ∈ T θ , and so in particular x ∈ S and thus S is nonempty. By Zorn’s lemma, S contains a maximal element x .We now show that x / ∈ S , a contradiction. Since x is maximal, for any x < y we must have { y } Ψ ω,f ∈ T σ and Ψ ωf y ∈ T θ .We first show that Ψ ω,f,x is total: For any a ∈ T ρ , either x ≻ a and soΨ ω,f,x a = 0 θ ∈ T θ , or x ≻ a and thus by compatibility we have x < x ⊕ a andtherefore Ψ ω,f,x a = Ψ ωf ( x ⊕ a ) ∈ T θ . But then since ω , x and Ψ ω,f,x are alltotal, it follows that { x } Ψ ω,f = ωx Ψ ω,f,x ∈ T σ .We now show that Ψ ω,f, { x } Ψ ω,f is total: For a ∈ T ρ , either { x } Ψ ω,f ≻ a andso Ψ ω,g, { x } Ψ ω,f a = 0 θ ∈ T θ , or { x } Ψ ω,f ≻ a and thus by compatibility we have { x } Ψ ω,f < { x } Ψ ω,f ⊕ a . But now using condition (a) of {·} Ψ ω,f forming a truncation,we have x < { x } Ψ ω,f ⊕ a and thus Ψ ωf ( { x } Ψ ω,f ⊕ a ) ∈ T θ . Now, since f , { x } Ψ ω,f and Ψ ω,f, { x } Ψ ω,f are all total, it follows that Ψ ωf x = f { x } Ψ ω,f Ψ ω,f, { x } Ψ ω,f ∈ T θ .We have therefore proven that if x is maximal, then both { x } Ψ ω,f and Ψ ωf x are total and so x / ∈ S , contradicting that S has a maximal element. Therefore S = ∅ and so Ψ ωf x ∈ T θ for any x ∈ T σ and we have shown totality of Ψ ωf . Example . We now consider Example 4.6 from the perspective of controlledrecursion, using a truncation similar to that given in Example 4.8 above. Letus extend the language of PA ω with a new constant Ω with defining equationΩ nf x = f h x i n ( λa . Ω nf ( h x i n ⊕ a ) if a ≻ h x i n else 0) (4)where n : N , f : σ → ( ρ → θ ) → θ and h x i n is defined by h x i n := N → B λi . x ( i ) if i < n else 1 . Then Ω is definable as Ω nf x := Ψ( cn ) f x where Ψ satisfies (3) and c : N → σ → ( ρ → θ ) → σ is the primitive recursive functional defined by cnxp := λi . x ( i ) if i < n else 1 . Working from now on in P ω , for each n ∈ N we can interpret Ω n as being aleast fixed point of the equation (3) for ω instantiated as the total representation15n P ω of cn as above. We apply Theorem 4.9 to show that Ω n is total. Thecompatibility and chain boundedness requirements are the same as in Theorem4.4, and so in our setting have already been dealt with in Example 4.6. Take f ∈ T σ → ( ρ → θ ) → θ with ω := cn ∈ T σ → ( ρ → θ ) → σ . To see that {·} Ψ ω,f and λx . f x Ψ ω,f,x form a truncation, we use a similar argument to Example 4.8. We can assumethat ω = cn is interpreted in P ω as ωxp ( i ) = ( x ( i ) if i < n ωxp ( ⊥ ) = ⊥ , and so for any strict x , if y ∈ T σ satisfies [ x (0) , . . . , x ( n − y (0) , . . . , y ( n − ωxp = ωyp = ωyq for any functionals p, q ∈ P ρ → θ , andso in particular { x } Ψ ω,f = ωx Ψ ω,f,x = ωy Ψ ω,f,y = { y } Ψ ω,f . This establishesproperty (b), and property (a) follows analogously to Example 4.8. ThereforeΨ ωf = Ψ( cn ) f ∈ T σ → θ for arbitrary n ∈ N and f ∈ T σ → ( ρ → θ ) → θ , which impliesthat the object Ω defined by Ω nf x := Ψ( cn ) f x is total and satisfies the equation(4) in P ω . Therefore Ω also has an interpretation in C ω , i.e. C ω | = PA ω + Ω.Note that no conditions were imposed on θ , and so totality of Ω also holds fornon-discrete θ .The functional defined in (4) is rather strongly controlled by cn (we claimin fact that Ω is definable as a term of System T). This deliberately simplisticexample was chosen simply to illustrate Theorem 4.9. Nevertheless, later we willrequire a much more subtle truncation for realizing the functional interpretationof lexicographic induction, which certainly does lead us beyond the realm ofprimitive recursion. In the last section we introduced two general variants of recursion over chainbounded partial orders. We will now show that our controlled variant is wellsuited for solving the functional interpretation of our axiomatic formulation ofZorn’s lemma from Section 3. We begin by recalling some essential facts aboutthe functional interpretation. Full details can be found in [1] or [14, Chapters8 & 10]. For those readers not familiar with the functional interpretation, weaim to at least present, in a self contained manner, the concrete computationalproblem we need to solve. This alone should suffice in order to understand latersections. Such a reader is advised to skip directly ahead to Section 5.3 (perhapsskimming through Section 5.2 on the way). So that we can formally state a higher-type variant of G¨odel’s soundness the-orem for the functional interpretation, we need to recall the so-called weaklyextensional variant WE-PA ω of PA ω , which is obtained from the latter by sim-ply replacing the axiom of extensionality with a quantifier-free rule form (see[14, Definition 3.12] for details, though this is not necessary to understand what16 A | : ≡ A for A atomic | A ∧ B | x,uy,v : ≡ | A | xy ∧ | B | uv | A ∨ B | i,x,uy,v : ≡ | A | xy ∨ i | B | uv | A → B | U,Yx,v : ≡ | A | xY xv → | B | Uxv |∃ zA ( z ) | x,uv : ≡ | A ( x ) | uv |∀ zA ( z ) | Ux,v : ≡ | A ( x ) | Uxv Figure 1: The functional interpretationfollows). This is because the interpretation is unable to deal with the axiom ofextensionality and thus cannot be applied directly to PA ω (see [1, pp. 15] or[14, pp. 126–127]).The functional interpretation assigns to each formula A of WE-PA ω a newformula | A | xy where now x and y are (possibly empty) tuples of variables of somefinite type. The precise definition is by induction over the logical structure of A ,and is given in Figure 1, where in the interpretation of disjunction, i is an objectof natural number type and P ∨ i Q denotes ( i = 0 → P ) ∧ ( i = 0 → Q ). Thebasic functional interpretation applies only to intuitionistic theories. In orderto deal with classical logic, we need to combine the interpretation with somevariant of the negative translation A A N as an initial step. We do not giveany further details, but simply state the main soundness theorem for classicalarithmetic. In the following, QF-AC denotes the axiom of quantifier-free choicei.e. the schema ∀ x ρ ∃ y σ A ( x, y ) → ∃ f ρ → σ ∀ x A ( x, f x )where ρ and σ are arbitrary types and A ( x, y ) ranges over quantifier-free for-mulas. Theorem 5.1 (cf. Theorem 10.7 of [14], but essentially due to G¨odel [11]) . Let A be a formula in the language of WE - PA ω . Then whenever WE - PA ω + QF - AC ⊢ A we can extract a term t of WE - HA ω whose free variables are the same as thoseof A , and such that HA ω ⊢ ∀ y | A N | ty . Remark. Note that ∀ y | A N | ty is provable even in a quantifier-free fragment ofWE-HA ω , the intuitionistic variant of WE-PA ω .The main result in the remainder of this section is to extend Theorem 5.1above to include our formulation of Zorn’s lemma. Generally speaking, in orderto expand the soundness theorem to incorporate extensions of WE-PA ω +QF-ACwith new axioms X , it suffices to provide a new recursive scheme Ω such thatthe functional interpretation of X N has a solution in HA ω + Ω. A classicalexample of this is with X as the axiom of countable choice, and Ω the scheme ofbar recursion in all finite types (cf. [1, Chapter 6] or [14, Chapter 11]). Here on17he other hand, we set X to be our syntactic formulation of Zorn’s lemma, andΩ a functional definable from our scheme of controlled recursion from Section4.3. ZL [] , ⊕ , ≺ We now outline how the combination of the functional interpretation with thenegative interpretation acts on the axiom ZL [] , ⊕ , ≺ as given in Definition 3.2,subject to the additional restriction that Q ( u ) ranges over quantifier-free for-mulas of WE-PA ω (similar restrictions can be found in [4] and [7] in the contextof open induction). This restriction still allows us to deal with most concreteexamples we are interested in (including the existence of maximal ideals incountable commutative rings in Example 3.3 and also Higman’s lemma, whichwe will discuss later), but simplifies the interpretation considerably (though weconjecture that in many cases, and in particular for concrete example consideredin Section 6, a solution for general Q ( u ) can be reduced to that of quantifier-free Q ( u ), subject to modification of the parameters [] , ⊕ , ≺ ).In what follows, we make use of the fact that the quantifier-free formulas ofWE-PA ω are decidable, in the sense that whenever A ( x , . . . , x n ) is quantifier-free with free variables x , . . . , x n there is a closed term t A of System T sothat HA ω ⊢ ∀ x , . . . , x n ( t A x . . . x n = 1 ↔ A ( x , . . . , x n )). This also meansthat the functional interpretation essentially interprets quantifier-free formulasas themselves.Let us now fix closed terms [] , ⊕ , ≺ and consider ZL [] , ⊕ , ≺ as given in Defini-tion 3.2, but where now Q ( u ) is assumed to be quantifier-free. There are severalvariants of the negative translation which can be applied. Applying standardvariant due to Kuroda, as used in [14, Chapter 10], and using a few standardintuitionistic laws together with Markov’s principle (all of which can be inter-preted by the intuitionistic functional interpretation), it suffices to solve thefunctional interpretation of ∃ x σ ∀ d δ Q ([ x ] d ) →¬¬∃ y σ ( ∀ d Q ([ y ] d ) ∧ ∀ a ≻ y ∃ d ¬ Q ([ y ⊕ a ] d )) . (5)We must now apply the rules of Figure 1 to (5), which we do step by step.We first observe that the inner part of the conclusion of (5) within the doublenegations is translated to ∃ y σ , h ρ → δ ∀ d, a ( Q ([ y ] d ) ∧ ( a ≻ y → ¬ Q ([ y ⊕ a ] ha )) . Therefore the double negated conclusion is partially interpreted (i.e. before thefinal instance of the ∀ -rule) as ∀ F, G ∃ y, h ( Q ([ y ] F yh ) ∧ ( Gyh ≻ y → ¬ Q ([ y ⊕ Gyh ] h ( Gyh ) ))where here F : σ → ( ρ → δ ) → δ and G : σ → ( ρ → δ ) → ρ . Therefore,interpreting the main implication, in order to solve the functional interpretation18f (5) we must produce three terms r, s, t which take as input x, F, G and haveoutput types δ, σ and ρ → σ respectively, and satisfy Q ([ x ] r ) → Q ([ s ] F st ) ∧ ( Gst ≻ s → ¬ Q ([ s ⊕ Gst ] t ( Gst ) )) (6)where for readability we suppress the input parameters, so that r should actuallyread rxF G throughout, and similarly for s and t . Though (6) looks complicated,it can be given a fairly intuitive characterisation as follows.The original axiomatic formulation of Zorn’s lemma is equivalent (usingQF-AC) to the statement that given some x σ satisfying ∀ dQ ([ x ] d ) we can findsome y also satisfying ∀ dQ ([ y ] d ) together with an h : ρ → δ witnessing maximal-ity of y in the sense that ¬ Q ([ y ⊕ a ] ha ) for any a ≻ x . On the other hand, thecomputational interpretation of Zorn’s lemma given as (6) says that for any x σ together with ‘counterexample functionals’ F, G we can produce elements s and t (in terms of x, F, G ), where s approximates our maximal element y in the sensethat it satisfies Q ([ s ] d ) not for all d but just for d := F st , while t approximates h in the sense that it satisfies ¬ Q ([ s ⊕ a ] t ( a ) not for all a ≻ s but just for a := Gst whenever Gst ≻ s . Indeed, this can be seen as a slightly more intricate ver-sion of Kreisel’s no-counterexample interpretation, and the relationship betweenZL [] , ⊕ , ≺ and (6) is similar to the relationship between Cauchy convergence and‘metastability’ (see [14, Section 2.3]). ZL [] , ⊕ , ≺ From this point onwards, we no longer need to deal directly with the functionalinterpretation. Rather, our focus is on solving the functional interpretationof ZL [] , ⊕ , ≺ as given in (6). To be more precise, we will construct realizingterms r , s and t which each take as input x : σ , F : σ → ( ρ → δ ) → δ and G : σ → ( ρ → δ ) → ρ and satisfy, Q ([ x ] r ) → Q ([ s ] F st ) ∧ C ( G, s, t )for any input, where C ( G, y, h ) abbreviates the formula C ( G, y, h ) : ≡ Gyh ≻ y → ¬ Q ([ y ⊕ Gyh ] h ( Gyh ) ) . Interestingly, we do not require ZL [] , ⊕ , ≺ in order to verify our realizing terms.Instead, we work in HA ω extended with two recursively defined constants to-gether with a simple universal axiom which we label ‘relevant part’. That thisformal theory has a model is a separate question, which we discuss after pre-senting our main result (Theorem 5.6). Definition ω ) . Let t C denote the term of System T satisfying t C Gyh =1 ↔ C ( G, y, h ), which exists since ≺ is decidable and Q ( u ) is quantifier-free.For the remainder of this section, we fix some closed term e : ( σ → ( ρ → δ ) → δ ) → ( σ → ( ρ → δ ) → σ ) of System T, so that all definitions and resultsthat follows are parametrised by e . 19 efinition ω ) . Define the new constant Ω e : ( σ → ( ρ → δ ) → δ ) → σ → δ by Ω e F x = F h x i Ω e F ( λa . Ω e F ( h x i Ω e F ⊕ a ) if a ≻ h x i Ω e F else 0 δ ) (7)where h x i Ω e F is shorthand for h x i Ω e F := eF x ( λa . Ω e F ( x ⊕ a ) if a ≻ x else 0 δ )Furthermore, we use the abbreviationΩ e,F,x := λa . Ω e F ( x ⊕ a ) if a ≻ x else 0 δ so that (7) can be expressed as Ω e F x = F h x i Ω e F Ω e,F, h x i Ω eF for h x i Ω e F = eF x Ω e,F,x . Definition ω + Ω e ) . We define in the language of HA ω + Ω e the ‘relevantpart’ axiom for Ω e asRP e : ∀ x, F ([ x ] Ω e F x = [ h x i Ω e F ] Ω e F x ) . Intuitively, the relevant part axiom says that if we take the approximationof x of size Ω e F x , then actually this approximation has no more informationthan that of the truncated version h x i Ω e F of x , and so the latter already containsthe ‘relevant part’ of this approximation. We will see a natural example of an e which satisfies this axiom in Section 6. Definition ω + Ω e ) . Define the constant Γ e : ( σ → ( ρ → δ ) → δ ) → ( σ → ( ρ → δ ) → ρ ) → σ → σ ∗ in the language of HA ω + Ω e byΓ e F Gx := y :: ( [] if t C Gy Ω e,F,y = 1Γ e F G ( y ⊕ Gy Ω e,F,y ) otherwise (8)for y := h x i Ω e F , where here y :: l denotes the appending of y to the front of thelist l . i.e. y :: [ l , . . . , l j − ] := [ y, l , . . . , l j − ]. Theorem 5.6 (HA ω + Ω e + Γ e + RP e ) . Define terms r , s and t as follows: rxF G := δ Ω e F xsxF G := σ tail(Γ e F Gx ) txF G := ρ → δ Ω e,F, tail(Γ e F Gx ) where tail( l ) denotes the last element of the list l (and tail([]) = 0 σ ). Thenprovably in HA ω + Ω e + Γ e + RP e we have ∀ x, F, G ( Q ([ x ] r ) → Q ([ s ] F st ) ∧ C ( G, s, t )) (9) where in the above formula we write just r instead of rxF G , and similarly for s and t . roof. Fixing F and G , we prove by induction on n that ∀ x ( | Γ e F Gx | = n ∧ Q ([ x ] rx ) → Q ([ sx ] F ( sx )( tx ) ) ∧ C ( G, sx, tx )) (10)where here rx is shorthand for rxF G (i.e. the parameter x is now explicitlywritten since it varies in the induction). Since | Γ e F Gx | ≥ 1, our base case is n = 1 which means that t C Gy Ω e,F,y = 1 and Γ e F Gx = [ y ] for y := h x i Ω e F . Butthis implies that sx = y and tx = Ω e,F,y , and thus in particular C ( G, sx, tx )holds. Next, we observe that[ x ] rx = [ x ] Ω e F x ( a ) = [ sx ] Ω e F x ( b ) = [ sx ] F ( sx )( tx ) where (a) follows from RP e and the definitions of rx and sx , while for (b) weuse that Ω e F x = F y Ω e,F,y = F ( sx )( tx ) . Thus from Q ([ x ] rx ) we can infer Q ([ sx ] F ( sx )( tx ) ), which establishes (10) for n = 1.For the induction step, suppose that | Γ e F Gx | = n + 1, which implies that t C Gy Ω e,F,y = 0. Setting y := h x i Ω e F as before, and in addition a := Gy Ω e,F,y ,by unwinding definitions it follows from ¬ C ( G, y, Ω e,F,y ) that(i) a ≻ y and thus Ω e,F,y ( a ) = Ω e F ( y ⊕ a ),(ii) Q ([ y ⊕ a ] Ω e,F,y ( a ) ) and thus Q ([ y ⊕ a ] Ω e F ( y ⊕ a ) ) by (i).Now since Γ e F Gx = y :: Γ e F G ( y ⊕ a ) and thus | Γ e F G ( y ⊕ a ) | = n , we canapply the induction hypothesis for x ′ := y ⊕ a . Since rx ′ = Ω e F x ′ = Ω e F ( y ⊕ a )it follows from (ii) that Q ([ x ′ ] rx ′ ) and therefore we have Q ([ sx ′ ] F ( sx ′ )( tx ′ ) ) and C ( G, sx ′ , tx ′ ). But since sx = tail( y :: Γ e F G ( y ⊕ a )) = tail(Γ e F G ( x ′ )) = sx ′ and similarly tx = tx ′ , it follows that Q ([ sx ] F ( sx )( tx ) ) ∧ C ( G, sx, tx ), whichestablishes (10) for n ′ = n + 1. This completes the induction, and (9) followsby taking some arbitrary F, G, x and letting n := | Γ e F Gx | in (10).The above result which solves the functional interpretation of ZL [] , ⊕ , ≺ isvalid for arbitrary e . However, it is only useful if the theory HA ω + Ω e + Γ e +RP e has a reasonable model. The final results of this section establish someconditions by which both Ω e and Γ e give rise to total objects and hence exist in C ω . An example of a setting where RP e is also valid in C ω is given in Section 6. Theorem 5.7 ( P ω ) . Let Ω e denote a fixed point of the primitive recursivedefining equation (7) - where the closed primitive recursive term e is interpretedas some total object in P ω - and suppose that there exist < and L such that < iscompatible with ( ⊕ , ≺ ) and chain bounded w.r.t. [ · ] and L . Suppose in additionthat h·i Ω e F ∈ P σ → σ and λx . F x Ω e,F,x ∈ P σ → δ form a truncation w.r.t. [ · ] , L and < for any total F . Then Ω e is total. roof. This is a simple adaptation of Theorem 4.9, taking Ω e := λF . Ψ( eF ) F .If F ∈ T σ → ( ρ → δ ) → δ then also eF ∈ T σ → ( ρ → δ ) → σ by totality of e , and thuswhenever {·} Ψ eF,F and λx . F x Ψ eF,F,x form a truncation w.r.t. [ · ], L and < thenΩ e F = Ψ( eF ) F ∈ T σ → δ . But the truncation condition is exactly that given asthe statement of this theorem, and if this holds for arbitrary total F then Ω e isalso total. Theorem 5.8 ( P ω ) . Let Γ e denote a fixed point of the defining equation (8).Under the assumptions of Theorem 5.7, Γ e is total.Proof. We can define Γ e := λF, G . Ψ( ωF )( f F G ) where ω and f are totalrepresentations in P ω of the following functionals definable in HA ω + Ω e : ωF xp := σ h x i Ω e F f F Gxp := σ ∗ x :: ( [] if t C Gx Ω e,F,x p ( Gx Ω e,F,x ) otherwisewhere here p : ρ → σ ∗ (note that totality of ω and f follows from totality ofprimitive recursive functionals plus totality of Ω e as established in Theorem 5.7above). To see that Γ e satisfies (8) is just a case of unwinding the definitions.Now, if F and G are total it follows that ωF and f F G are also total, and soby Theorem 4.9, Γ e F G = Ψ( ωF )( f F G ) is total if we can show that {·} Ψ ωF,fF G and λx. ( f F G ) x Ψ ωF,fF G,x form a truncation. But { x } Ψ ωF,fF G = h x i Ω e F , and sothis follows from the assumption that h·i Ω e F and λx.F x Ω e,F,x form a truncation.Formally, if h x i Ω e F is total for x ∈ L (which it always is by totality of Ω e ),then since in addition F y Ω e,F,y is total for y := h x i Ω e F then h·i Ω e F has a pointof continuity d for x . Condition (a) follows trivially. Therefore we have shownthat Γ e is total. Remark. Our use of controlled recursion means that there are no type levelrestrictions on the output types Ω e F x : δ or Γ e F Gx : σ ∗ . This not only permitsa greater degree of generality but is essential even for simple applications: InExample 3.3, σ := N → B and thus σ ∗ is a higher type. We conclude the paper by showing how our parametrised results can now beimplemented in the special case of induction over the lexicographic orderingon sequences. This constitutes a direct counterpart to open induction as pre-sented in [4], and is closely related to the recursive scheme introduced in [22]for extracting a witness from the proof of Higman’s lemma. Definition ω ) . Let θ be some arbitrary type, and suppose that ✁ : θ × θ → B is a decidable relation on θ such that induction over ✁ is provable in22A ω . Setting σ := N → θ , δ := N , ρ := N × ( N → θ ) and ν := θ ∗ , define[ x ] n := [ x (0) , . . . , x ( n − x ⊕ ( n, y ) := [ x ] n @ y ( n, y ) ≻ x := y ( n ) ✁ x ( n )where ([ x (0) , . . . , x ( n − y )( i ) := x ( i ) if i < n and y ( i ) otherwise. Wedefine LEX ✄ to be the principle ZL [ · ] , ⊕ , ≺ for the parameters given above i.e. ∃ x ∀ dQ ([ x ] d ) → ∃ y ( ∀ dQ ([ y ] d ) ∧ ∀ ( n, z )( z ( n ) ✁ y ( n ) → ∃ d ¬ Q ([[ y ] n @ z ] d )) . Our axiom LEX ✄ is essentially the contrapositive of open induction as pre-sented in [4], and as such the theory WE-PA ω + QF-AC + LEX ✄ (for variousinstantiations of ✄ ) is capable not only of formalizing large parts of mathe-matical analysis but also giving direct formalizations of minimal bad sequencearguments common in the theory of well quasi orderings. We now show how itcan be given a direct computational interpretation using the theory developedso far. Lemma 6.2 ( P ω ) . Define L ⊂ T σ to be the set of all strict total objects i.e.those satisfying x ( ⊥ ) = ⊥ θ (recall that σ = N → θ ), and let the partial order < on T σ by defined by y > x : ⇔ ∃ n ∈ N ([ y ] n = N ∗ [ x ] n ∧ y ( n ) ✁ x ( n )) where here ✁ is now interpreted as a total functional T θ × θ → B . In other words, y > x if it is lexicographically smaller than x w.r.t. ✁ . Then < is compatiblewith ( ⊕ , ≺ ) and chain bounded w.r.t. [ · ] and L .Proof. Compatibility is clear, while chain boundedness follows easily using astandard construction for the lexicographic ordering. Take some nonempty chain γ ⊂ T σ and inductively define the sequence of total objects u k ∈ T θ for k ∈ N by taking u k to be the ✁ -minimal element of the set S k := { x ( k ) | x ∈ γ and ( ∀ i < k )( x ( i ) = u i ) } ⊆ T θ . Note that S k are nonempty by induction on k , and u k is well-defined since the ✁ -minimum principle is provable from induction over ✁ , which is provable in HA ω and thus satisfied by the total elements T θ . Now define ˜ γ ( k ) := u k for k ∈ N and ˜ γ ( ⊥ ) = ⊥ , which is clearly an element of L ⊂ T N → θ . It follows by definitionthat for any d ∈ N there exists some x ∈ γ with [ x ] d = [ u , . . . , u d − ] = [˜ γ ] d .To see that ˜ γ is an upper bound, take some x ∈ γ and assume that x = ˜ γ . Let k ∈ N be the least with x ( k ) = ˜ γ ( k ) = u k . Then by definition of u k there issome y ∈ γ with [ y ] k = [ u , . . . , u k − ] = [ x ] k and y ( k ) = u k . Since < is a totalorder on γ we must have either x < y or y < x , and since x ( k ) = y ( k ) thismeans that either x ( k ) ✁ y ( k ) or y ( k ) ✁ x ( k ). But by minimality of u k = y ( k )we must have ˜ γ ( k ) = y ( k ) ✁ x ( k ) and thus ˜ γ > x . This proves that x ≤ ˜ γ forany x ∈ γ . 23ur next step is to define a suitable closed term e of HA ω which not onlyinduces a truncation in the sense of Theorem 5.7 but also satisfies RP e in thetotal continuous functionals. For this, we introduce a powerful idea that isalready implicit in Spector’s fundamental bar recursive interpretation of theaxiom of countable choice [25], and has been studied in more detail in [19].From now on we make the fairly harmless assumption that the canonicalobject 0 θ is minimal w.r.t to ✁ (this could in theory be circumvented but havingit makes what follows slightly simpler). Definition ω ) . For x : σ and n : N let x, n := [ x ] n @ ( λi. θ ) : σ, and define the primitive recursive functional η : ( σ → N ) → σ → σ by ηφxk := θ ( θ if ( ∃ i ≤ k )( φ ( x, i ) < i ) x ( k ) if ( ∀ i ≤ k )( φ ( x, i ) ≥ i )where we note that the bounded quantifiers can be represented as boundedsearch terms in System T. Lemma 6.4 ( P ω ) . Let us represent η in P ω by the total continuous functional ηφxk := θ if ( ∃ i ≤ k )( ∀ j ≤ i ( φ ( x, j ) ∈ N ) ∧ φ ( x, i ) < i ) x ( k ) if ( ∀ i ≤ k )( φ ( x, i ) ∈ N ∧ φ ( x, i ) ≥ i ) ⊥ otherwisewith ηφx ⊥ = ⊥ . Then for any φ ∈ P σ → N , the functionals ηφ ∈ P σ → σ and φ form a truncation w.r.t. [ · ] , L and < .Proof. Part (a) is simple: Suppose that x, y, ηφx ∈ T σ and ηφx < y so thatthere exists some n ∈ N with [ y ] n = [ ηφx ] n and y ( n ) ✁ ηφx ( n ). Since we cannothave y ( n ) ✁ θ by our minimality assumption, we must have ηφx ( n ) = x ( n ).But then by definition of η it follows that ηφx ( k ) = x ( k ) for all k < n , and thus[ y ] n = [ x ] n and so x < y .For part (b), let us now assume that x ∈ L with ηφx ∈ T σ and φ ( ηφx ) ∈ N .We first show that there exists some n ∈ N with φ ( x, n ) < n . Suppose forcontradiction that for all i ∈ N we have either φ ( x, i ) = ⊥ or φ ( x, i ) ≥ i . Thefirst possibility is ruled out since if φ ( x, i ) = ⊥ then ηφxi = ⊥ contradictingtotality of ηφx . But this means that ηφx = x (since also ηφ ⊥ = ⊥ = x ( ⊥ )).But then φ ( ηφx ) = φx ∈ N and so by Lemma 4.5 there exists some d ∈ N suchthat φx = φy whenever x ( i ) = y ( i ) for all i < d . Now set N := max { φx + 1 , d } and consider y := x, N . Then x ( i ) = y ( i ) for all i < N and so also for all i < d ,which implies that φ ( x, N ) = φx < φx + 1 ≤ N This is a standard domain theoretic interpretation of η , where the bounded search termi-nates with 0 for the first i ≤ k it finds with φ ( x, i ) < i , and returns ⊥ if φ ( x, i ) is undefinedfor any i that is queried. 24 contradiction. Therefore we have shown there exists some n ∈ N with φ ( x, n ) < n , from which it follows that ηφx = x, m for the least such m ∈ N with this property (again, φ ( x, j ) ∈ N for all j ≤ m by totality of ηφx ). Letus now suppose that y ∈ P σ satisfies [ x ] m = [ y ] m . Then for k < m , since φ ( y, i ) = φ ( x, i ) ≥ i for all i ≤ k it follows that ηφyk = y ( k ) = x ( k ), andif k ≥ m , since φ ( y, m ) = φ ( x, m ) < m it follows that ηφyk = 0, and thus ηφy = x, m = ηφx . Lemma 6.5 ( P ω ) . Let Ω e be a fixed point of the equation (7) as in Theorem5.7, where now e is defined by eF xp := η ( λy.F y ( p | y )) x for η as in Definition 6.3 (resp. Lemma 6.4) and p | y ( n, z ) := p ( z ) if z ( n ) ✁ y ( n ) else 0 . Then h·i Ω e F ∈ P σ → σ and λx.F x Ω e,F,x ∈ P σ → N form a truncation w.r.t. [ · ] , L and < for any F .Proof. We first observe that h x i Ω e F = eF x Ω e,F,x = η ( λy.F y (Ω e,F,x | y )) x. We now argue that for any i ∈ N we haveΩ e,F,x | x,i = Ω e,F,x,i . For this we only need to check arguments ( n, y ) which satisfy ( n, y ) ≻ x, i i.e. y ( n ) ✁ ( x, i )( n ). But by minimality of 0 θ this is only possible if n < i and y ( n ) ✁ x ( n ), in which caseΩ e,F,x | x,i ( n, y ) = Ω e,F,x ( n, y ) = Ω e F ([ x ] n @ y )= Ω e F ([ x, i ] n @ y ) = Ω e,F,x,i ( n, y ) . Since ηφx only depends on φ for arguments of the form x, i , it follows that h x i Ω e F = ηφ F, Ω x for φ F, Ω := λy.F y Ω e,F,y . But for any F , by Lemma 6.4 applied to φ := φ F, Ω as defined above, we havethat ηφ F, Ω and φ F, Ω form a truncation w.r.t. [ · ], L and < , and the resultfollows. Corollary 6.6 ( P ω ) . Let Ω e and Γ e be fixed points of the equations (7) and (8)respectively, for e be as defined in Lemma 6.5. Then Ω e and Γ e are total, andthus C ω | = HA ω + Ω e + Γ e .Proof. Directly from Lemmas 6.2 and 6.5 together with Theorems 5.7 and The-orem 5.8. 25 emma 6.7. RP e is valid in C ω for e as in Lemma 6.5.Proof. The argument in the proof of Lemma 6.5 that h·i Ω e F = ηφ F, Ω for φ F, Ω := λy.F y Ω e,F,y is also valid in C ω , and a simpler version of the argument in theproof of Lemma 6.4 verifies that there is some n ∈ N such that φ F, Ω ( x, n ) < n ,and moreover h x i Ω e F = ηφ F, Ω x = x, m where m ∈ N is the least satisfying thisproperty. But since φ F, Ω ( x, m ) = φ F, Ω ( h x i Ω e F ) = Ω e F x and thus Ω e F x < m , itfollows that [ x ] Ω e F x = [ x, m ] Ω e F x = [ h x i Ω e F ] Ω e F x and so RP e is satisfied. Theorem 6.8. For any type θ and relation on ✁ such that induction over ✁ isprovable in HA ω , the functional interpretation of (the negative translation of ) LEX ✁ can be solved by a term in HA ω +Ω e +Γ e , provably in HA ω +Ω e +Γ e +RP e ,for any closed term e of System T. Moreover, defining e as in Lemma 6.5, wehave C ω | = HA ω + Ω e + Γ e + RP e .Proof. The first claim follows directly from Theorem 5.6, and the second fromCorollary 6.6 and Lemma 6.7. In this paper, we explored various notions of recursion over chain bounded par-tial orders, and gave a general theorem on solving the functional interpretationof an axiomatic, parametrised form of Zorn’s lemma.We intend this work to be taken as the starting point for a number of muchbroader research questions in both proof theory and computability theory, whichwe hope to pursue in the future. These include the following:1. Can particular instances of Φ and Ψ as in Section 4 be connected toknown forms of strong recursion, particularly variants of bar recursion?We conjecture, for example, that Ω e and Γ e as given in Section 6 aredefinable using Spector’s variant of bar recursion, using ideas from [21].Are more general results along the lines of [6, 10, 19, 21] possible?2. The relationship between our simple and controlled recursors has manyparallels to that between modified bar recursion and Spector’s variant. Itwas shown in [6] that the former in fact defines the latter over System T.Under certain conditions, can we show that our simple recursor actuallydefines the controlled variant? It was also shown in [6] that Spector’s barrecursion is S1-S9 computable in C ω , but modified bar recursion is not.Does an analogous result hold in our setting?3. Can we formulate Theorems 4.4 and 4.9 so that they apply to non-continuous models, such as the majorizable functionals [12]?26. What other applications of our abstract computational interpretation ofZorn’s lemma are possible? Are there cases where a sensible choice ofthe parameters could lead to a more concise formalisation of a well-knownproof, and consequently a more natural and efficient extracted program?In the other direction, can our framework be applied to give a compu-tational interpretation to instances of Zorn’s lemma stronger than evencountable dependent choice?5. 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