aa r X i v : . [ m a t h . L O ] A p r Impredicative consistency and reflection
David Fern´andez-Duque
Centre International de Math´ematiques et d’Informatique, University of Toulouse, FranceDepartment of Mathematics, Instituto Tecnol´ogico Aut´onomo de M´exico, Mexico [email protected]
Abstract
Given a set X of natural numbers, we may formalize “The formula φ is atheorem of ω -logic over the theory T using an oracle for X ” by an expres-sion [ I | X ] T φ , defined using a least fixed point in the language of second-orderarithmetic. We will prove that the consistency and reflection principles aris-ing from this notion of provability lead to axiomatizations of Π -CA and Π -CA with bar induction. We compare this to well-known results thatreflection for ω -derivable formulas and ω -model reflection are equivalent tobar induction. Keywords: reflection principles, second-order arithmetic, proof theory
1. Introduction
Reflection principles in formal arithmetic are statements of the form “If φ is a theorem of T , then φ ” [12]. Using notation from provability logic [4],for a computably enumerable theory T we may use (cid:3) T φ to denote a naturalformalization of “ φ is a theorem of T ”. Then, the above statement may bewritten succinctly as (cid:3) T φ → φ . If φ is a sentence, this gives us an instanceof local reflection. Although such principles merely state the soundness of T , they can almost never be proven within T itself. For example, setting φ ≡ = , we see that (cid:3) T φ → φ is equivalent to ∼ (cid:3) T = , which assertsthe consistency of T and hence is unprovable within T itself (if T satisfiesthe assumptions of G¨odels second incompleteness theorem). More generally,by L¨ob’s theorem we have that T ⊢ (cid:3) T φ → φ only if φ is already a theoremof T [13]. Preprint submitted to Annals of Pure and Applied Logic October 13, 2018 e can extend reflection to formulas φ ( x ), obtaining uniform reflectionprinciples, denoted RFN [ T ]. These are given by the scheme ∀ x (cid:0) (cid:3) T φ (¯ x ) → φ ( x ) (cid:1) , where ¯ x denotes the numeral of x .Uniform reflection principles are particularly appealing because they some-times give rise to familiar theories. If we use PRA to denote primitive recur-sive arithmetic, Kreisel and L´evy proved in [12] thatPA ≡ PRA +
RFN [PRA];in fact, we may replace PRA by the weaker elementary arithmetic (EA), ob-tained by restricting the induction shema in Peano arithmetic to ∆ formulasand adding an axiom asserting that the exponential function is total [2].Recall that the ω -rule is an infinitary deduction rule that has the followingform: φ (¯0) , Γ φ (¯1) , Γ φ (¯2) , Γ . . . ∀ xφ ( x ) , Γ , and ω -logic is the logic generated by the ω -rule together with the standardfinitary rules of the Tait calculus. More generally, ω -logic over T allows forsequents derivable in T to be used as axioms.In this article, we will study formalizations of ω -reflection in second-orderarithmetic; that is, statements of the form “If φ is a theorem of ω -logic, then φ ”. The question readily arises as to what it means for φ to be a theorem of ω -logic. There are at least three ways to model this. Informally, they are:( i ) There is a well-founded derivation tree formalizing an ω -proof of φ , inwhich case we will write [ P ] φ .( ii ) There is a well-order Λ such that φ belongs to the set of theorems of ω -logic defined by transfinite recursion on Λ, in which case we will write[ R ] φ .( iii ) The formula φ belongs to the least set closed under the rules and axiomsof ω -logic. If this is the case, we will write [ I ] φ .Although we will discuss these in greater detail later, the ideas behind [ P ] φ and [ I ] φ should be clear; [ P ] φ gives a ‘local’ view of φ being a theorem of ω -logic by considering (infinite) ω -proofs of φ , while [ I ] φ gives a more global2erspective, describing the set of theorems of ω -logic as a whole via an induc-tive definition. Meanwhile, [ R ] φ describes the approximations to the fixedpoint used in [ I ] φ via transfinite recursion.Over a strong enough formal theory, one can show that all of these notionsof provability are equivalent. However, from the point of view of a weaktheory, they may vary in strength. For X ∈ { P , R , I } and A ⊆ N , let us write[ X | A ] φ if φ is provable in the sense of X from the atomic diagram of A . Then,we define a schema ω X -RFN ≡ ∀ A ∀ n (cid:0) [ X | A ] φ (¯ n, ¯ A ) → φ ( n, A ) (cid:1) ;the notation ¯ A indicates a second-order constant added to represent A . IfΓ is a set of formulas, ω X -RFN Γ is the restriction of this scheme to φ ∈ Γ.Then, over RCA we have that: ω P -RFN ≡ Π ω -BI ; (1) ω R -RFN Π ≡ ATR . (2)(We will review the theories Π ω -BI of full bar induction and ATR of arith-metical transfinite recursion in § ω X -RFN Γ [ T ] to denote a variant of thescheme where ω -logic is extended by theorems of T , (2) generalizes to ω R -RFN Σ n +1 [ACA ] ≡ ATR + Π n - BI (3)(which is just Π n -BI if n > ω -model re-flection, the scheme asserting that any formula true in every ω -model mustbe true [11]. This begs the question: is ω I -RFN also equivalent to a naturaltheory? In this article, we answer the question affirmatively, and prove that: ω I -RFN Π ≡ Π -CA ; (4) ω I -RFN Σ n +1 [ACA ] ≡ Π -CA + Π n -BI . (5)Both equivalences are proven over the theory ECA of elementary compre-hension, which is strictly weaker than RCA or even RCA ∗ . Layout of the article In § § ω -logic in the literature, and in § ω -models,which give rise to another family of reflection principles, also equivalent tobar induction. In § § ω -logic and prove (3), and § § β -models.
2. Second-order arithmetical theories
In this section we review some basic notions of second-order arithmeticand mention some important theories that will appear throughout the article.
It will be convenient to work within a Tait-style calculus, so we willconsider a language without negation, except on primitive predicates. Thusterms and formulas will be built from the symbols , , + , × , exp , = , = , ∈ , ,representing the standard constants, operations and relations on the naturalnumbers, along with the Booleans ∧ , ∨ and the quantifires ∀ , ∃ . The rank of a formula is the number of logical symbols (Booleans and quantifiers)that appear in it. We assume a countably infinite set of first-order variables n, m, x, y, z . . . , which will always be denoted by lower-case letters, as wellas a countably infinite set of second-order variables. It will be convenientto assume that the second-order variables are enumerated by V = h V i i i ∈ N ,although we may also use X, Y, Z, . . . to denote set-variables. Tuples of first-order terms or second-order variables will be denoted with a boldface font,e.g. t , X . In general, if S = h S i i i ∈ N is a sequence we will write S The theory Π - CA is equivalent to Q + + Ind + (Π / Σ ) - CA . Proof sketch. In [17, Lemma V.1.4], it is proven that any Π formula isequivalent to one of the form ∀ f : N → N φ ( f ) , where φ ∈ Σ . If fun ( F ) ∈ Π ( F ) is a formula stating that F is the graph of a function, this is in turnequivalent to some formula ∀ F (cid:0) ∼ fun ( F ) ∨ φ ′ ( F ) (cid:1) ∈ Π / Σ , where φ ′ isobtained by modifying φ in the obvious way. We mention two further theories that will appear later and require a moreelaborate setup. We may represent well-orders in second-order arithmetic aspairs of sets Λ = h| Λ | , < Λ i , and define Prog φ (Λ) = ∀ λ (cid:16)(cid:0) ∀ ξ< Λ λ φ ( ξ ) (cid:1) → φ ( λ ) (cid:17) TI φ (Λ) = ∀ λ ∈ | Λ | (cid:0) Prog φ (Λ) → ∀ λ ∈ | Λ | φ ( λ ) (cid:1) WF (Λ) = ∀ X TI λ ∈ X (Λ) WO (Λ) = LO (Λ) ∧ WF (Λ) , where LO (Λ) is a formula expressing that Λ is a linear order.8iven a set X whose elements we will regard as ordered pairs h λ, n i , let X λ be the set of all n with h λ, n i ∈ X , and X < Λ λ be the set of all h η, n i with η < Λ λ . With this, we define the transfinite recursion scheme by TR φ ( X, Λ) = ∀ λ ∈ | Λ | ∀ n (cid:0) n ∈ X λ ↔ φ ( n, X < Λ λ ) (cid:1) . Finally, we defineATR ≡ ACA + n ∀ Λ (cid:0) WO (Λ) → ∃ X TR φ ( X, Λ) (cid:1) : φ ∈ Π ω o ;Γ-BI ≡ ACA + n ∀ Λ (cid:0) WO (Λ) → TI φ (Λ) (cid:1) : φ ∈ Γ o .These theories are rather powerful, yet as we will see, Π -CA proves verystrong reflection principles for both of them; this is particularly remarkablein the case of Π ω -BI , which is not a subtheory of Π -CA . The following isproven in [16]: Lemma 2.2. Π - BI ATR ( Σ - BI . To be precise, Π -BI ≡ Σ -DC , a theory known to be incomparablewith ATR . 3. Formalized ω -logic In this section we will give the necessary definitions in order to reasonabout ω -logic within second-order arithmetic, and introduce the provabilityoperator [ P ] based on ω -proofs. For our purposes, a theory is a set of sequents defined by an arithmeticalformula (cid:3) T γ , where γ is a first-order variable. For ρ ≤ ω , fix Rule ρ ( x, y ) ∈ ∆ such that it is provable in ECA that if Rule ρ ( x, y ) holds, then x codes asequence of sequents h δ i i i 9e can formalize this using the following expression: ω -Rule ( P, γ ) ≡ ∃ φ ∈ γ ∃ x, ψ < φ (cid:16) φ = ∀ xψ ( x ) ∧ ∀ z (cid:0) γ , ψ ( ˙ z ) ∈ P (cid:1)(cid:17) . Here, P is a set-variable. The formula ω -Rule ( P, γ ) states that γ follows byapplying one ω -rule to elements of P , and will be used in our formalizationsof ω -logic. In order to deal with free second-order variables, we will enrich theorieswith oracles. As we have mentioned previously, we will use countably manyconstants O = h O i i i ∈ N in order to ‘feed’ information about any tuple of setsof numbers into T . The O i ’s are assumed to be disjoint from the second-ordervariables.To be precise, we first encode finite sequences of sets in a natural way:for example, we may enconde h A i i i When workingin T | A , . . . , A n we may write x ∈ ¯ A i instead of x ∈ O i to increase legibility;for example, instead of (cid:3) T | A,B φ ( O , O ), we may write (cid:3) T | A,B φ ( ¯ A, ¯ B ).10 .3. Formalizing ω -logic using proof trees In [1, 9], derivability in ω -logic is formalized by the existence of an (infi-nite) derivation tree. It will be convenient to use a standardized representa-tion of such trees. Let N <ω denote the set of all finite sequences of naturalnumbers. We will represent ω -trees as subsets of N <ω . If s , t ∈ N <ω , define s t if s is an initial segment of t , and ↓ s = { t ∈ S : t s } . Then, saythat an ω -tree is a set S ⊆ N <ω such that ↓ S = S . A labeled ω -tree is a pair h S, L i such that S is an ω -tree and L : S → N . Definition 3.1. A preproof (for T ) of cut-rank at most ρ ≤ ω is a labeled ω -tree h S, L i such that for every s ∈ S , L ( s ) is a sequent, and there is aninstance h δ i i i<ξ γ of a rule of ω - Tait ρ with ξ ≤ ω such that L ( s ) = γ and forall i ∈ N , s ⌢ i ∈ S if and only if i < ξ , in which case L ( s ⌢ i ) = δ i , orelse s is a leaf and T ⊢ L ( s ) . Let PreProof ρT ( S, L ) be a Π ( S, L ) formulastating that h S, L i is a preproof for T of cut-rank at most ρ .If S is (upwards) well-founded, we will say that h S, L i is an ω -proof . The formula PreProof ρT ( S, L ) would make use of the formulas Rule and(a mild variant of) ω -Rule defined in § Definition 3.2. Given ρ ≤ ω , define a formula [ P ] ρT γ by ∃ S ∃ L (cid:16) WF ( h S, < i ) ∧ PreProof ρT ( S, L ) ∧ L ( hi ) = γ (cid:17) . We write [ P | X ] ρT γ instead of [ P ] ρT | X γ . The following is immediate from the definition: Lemma 3.3. Given ρ ≤ σ ≤ ω , it is provable in ECA that [ P | X ] ρT γ implies [ P | X ] σT γ . The notion of provability [ P ] gives rise to a natural reflection scheme. Definition 3.4. Given a theory T , ρ ≤ ω , and a set of formulas Γ , we definea schema ω P -RFN ρ Γ [ T ] ≡ ∀ A ∀ n (cid:16) [ P | A ] ρT φ ( ˙ n , O ) → φ ( n , A ) (cid:17) , where φ ( z , X ) ∈ Γ with all free variables shown. 11e may omit the parameter ρ when ρ = ω , as well as the parameter T when T is just the Tait calculus. This form of reflection gives an alternativeaxiomatization for bar induction, as shown by Arai [1]. Theorem 3.5. RCA + ω P -RFN Π ω ≡ RCA + Π ω - BI . Note the analogy with Kreisel and L´evy’s result; just as reflection isequivalent to induction, ω -reflection is equivalent to transfinite induction.As we will see, different formulations of ω -logic can also give rise to certainforms of comprehension. 4. Countable ω -models and reflection Another notion of reflection can be defined using ω -models. An ω -model isa second-order model whose first-order part consists of the standard naturalnumbers with the usual arithmetical operations. Because this part of ourmodel is fixed, we only need to specify the second-order part, which consistsof a family of sets over which we interpret second-order quantifiers. Moreover,if this family is countable, we can represent it using a single set.In order to have names for all the sets appearing in our ω -model, we in-troduce countably many set-constants C = h C i i i<ω and let Π ω ( C ) be thesecond-order language enriched with these constants. With this, a satisfac-tion notion can be associated to each countable coded ω –model in a naturalway. If M codes a sequence of sets, a satisfaction class on M is a set whichobeys the usual recursive clauses of Tarski’s truth definition, where eachconstant C n is interpreted as M n . Let us give a precise definition: Definition 4.1. Let M ⊆ N . A satisfaction class on M is a set S ⊆ Π ω ( C ) such that, for any terms t, s , n ∈ N , and sentences φ, ψ, ( t ◦ s ) ∈ S ⇒ J t K ◦ J s K ( ◦ ∈ { = , = } );( t ◦ C n ) ∈ S ⇒ h n, J t K i ◦ M ( ◦ ∈ {∈ , );( φ ∧ ψ ) ∈ S ⇒ φ ∈ S and ψ ∈ S ;( φ ∨ ψ ) ∈ S ⇒ φ ∈ S or ψ ∈ S ;( ∃ u φ ( u )) ∈ S ⇒ for some n ∈ N , φ (¯ n ) ∈ S ;( ∀ u φ ( u )) ∈ S ⇒ for all n ∈ N , φ (¯ n ) ∈ S ;( ∃ X φ ( X )) ∈ S ⇒ for some n ∈ N , φ ( C n ) ∈ S ;( ∀ X φ ( X )) ∈ S ⇒ for all n ∈ N , φ ( C n ) ∈ S. Given a set of sentences Γ ⊆ Π ω ( C ) closed under subformulas and substi-tution by closed terms (including set-constants), if for every φ ∈ Γ we have hat either φ ∈ S or ∼ φ ∈ S , we will say that S is a Γ-satisfaction class. If Γ contains all formulas of rank ρ ≤ ω , we say that S is a satisfaction class ofrank ρ . A pair M = h| M | , S M i , where | M | is a set and S M is a Γ -satisfactionclass on | M | of rank ρ is a Γ-valued ω -model of rank ρ . If Γ is the set of allsentences of Π ω ( C ) , we say that M is a full ω -model.Satisfaction classes are used to define truth in a model: Definition 4.2. Given an ω –model M , we write M | = φ if φ ∈ S M . If T is a theory, we say that M is a (partial) ω -model of T if, whenever φ is atheorem of T , it follows that M = ∼ φ . If A is an a -tuple of sets, we write [ M | A ] ρT φ for the formula stating that, for every Γ -valued ω -model M of rankat least ρ of T with φ ∈ Γ and | M |
Let T be any theory and ρ ≤ ω . Then, if φ ( z , X ) ∈ Π withall free variables shown, ECA ⊢ ∀ A ∀ n (cid:0) φ ( n , A ) → [ M | A ] ρT φ ( ˙ n , C ) (cid:1) . Proof. First assume that φ is arithmetical, and let M be a model of T of rank ρ . Then, an external induction using the definition of a satisfaction classshows that, if φ holds, then M | = φ . Otherwise, assume that φ = ∀ X ψ ( X )and M = ∀ Xψ ( X ), so that M = ψ ( C k ) for some k . But then, by thearithmetical case, ψ ( C k ) fails, so that ∀ X ψ ( X ) fails.The following claim is immediate from observing that every model of rank σ is already a model of any rank ρ ≤ σ : Lemma 4.4. Let φ be an arbitrary formula and ρ ≤ σ ≤ ω . Then, ECA ⊢ ∀ A (cid:0) [ M | A ] ρT φ ( C ) → [ M | A ] σT φ ( C ) (cid:1) . We may use ω -models to define a notion of reflection ω M -RFN ρ Γ [ T ], anal-ogously to Definition 3.4. The following is proven by J¨ager and Strahm [11],and is a refinement of results of Friedman [8] and Simpson [16]:13 heorem 4.5. Let < n ≤ ω , and fix a finite axiomatization of ACA ofrank ρ . Then, ACA + ω M -RFN ρ Σ n [ACA ] ≡ Π n - BI . In fact, [ P ] γ and [ M ] γ are equivalent [9]. In the next section we will useinductive definitions to define two further notions of provability, which arealso equivalent over a strong enough base theory. Remark 4.6. In the literature, ω -model reflection is often presented as ‘If φ is true, then φ is satisfiable in an ω -model’. We have presented it duallyas ‘If φ holds in every ω -model, then φ is true’. The two schemes are clearlyequivalent, but we prefer the latter for its symmetry with the other notions ofreflection we consider. Note, however, that we must replace φ by ∼ φ to passfrom one to the other, and thus Theorem 4.5 is stated with Σ n +1 in place of Π n +1 as in [11]. 5. Inductive definitions of ω -logic We may also formalize ‘provable in ω -logic’ in second-order arithmeticusing a least fixed point construction. To this end, let us review how suchfixed points may be treated in this framework. Let us quickly review inductive definitions in the context of second-orderarithmetic. Below, recall that we are working in a language without negationfor non-atomic formulas. Definition 5.1. Let φ be any formula and X a set-variable. We say φ ispositive on X if φ contains no occurrences of t X . A positive formula φ induces a map F = F φ : 2 N → N , which is monotonein the sense that X ⊆ Y implies that F ( X ) ⊆ F ( Y ). It is well-known thatany such operator has a least fixed point. Definition 5.2. Given a formula φ ( n, X ) , we define the abbreviations Closed φ ( X ) ≡ ∀ n (cid:0) φ ( n, X ) → n ∈ X (cid:1)(cid:0) X = µX.φ (cid:1) ≡ Closed φ ( X ) ∧ ∀ Y (cid:0) Closed φ ( Y ) → X ⊆ Y (cid:1) . 14t is readily checked that n ∈ µX.φ if and only if φ ( n, µX.φ ) holds. Suchfixed points can be constructed ‘from below’ using transfinite iterations of F :if we define F ( X ) = X , F ξ +1 ( X ) = F ( F ξ ( X )) and F ξ ( X ) = S ζ<ξ F ζ ( X ),then by cardinality considerations one can see that µX.φ = F ω ( ∅ ) . (6)On the other hand, we may define µX.φ ‘from above’ as the intersection ofall sets Y such that Closed ( Y ) holds. The latter definition is available in Π -CA , as is well-known (see e.g. [5]), and thus we see that: Lemma 5.3. Given φ ( X ) ∈ Π ω which is positive on X , it is provable in Π - CA that ∃ Y (cid:0) Y = µX.φ (cid:1) . In particular, the rules of ω -logic give rise to a positive operator, and atheorem of ω -logic is any element of its least fixed point. Below, we developthis idea to give alternative formalizations of ω -logic. ω -logic We may use (6) to formalize ‘ φ is a theorem of ω -logic’, as in [6, 7].There, provability along a countable well-order Λ is modeled using an ‘it-erated provability class’ P , defined by arithmetical transfinite recursion asfollows: Definition 5.4. Let Λ be a second-order variable that will be used to denotea well-order and T be a formal theory. Define Iter T ( φ, P ) to be the formula (cid:3) T φ ∨ ∃ ψ (cid:0) ω -Rule ( P, ψ ) ∧ (cid:3) T ( ψ → φ ) (cid:1) . Then, define [Λ] T φ ≡ ∀ P (cid:0) TR Iter T ( P, Λ) → ∃ λ ∈ | Λ | ( φ ∈ P λ ) (cid:1) ;[ R ] T φ ≡ ∃ Λ (cid:0) WO (Λ) ∧ [Λ] T φ (cid:1) . As before, write [ R | A ] T φ instead of [ R ] T | A φ , and for a set of formulas Γ and ρ ≤ ω , define ω R -RFN ρ Γ [ T ] analogously to Definition 3.4. Recall that, by our convention, the parameter ρ will be omitted when ρ = ω . This form of reflection gives rise to an axiomatization of ATR [6]:15 heorem 5.5. Let U, T be c.e. theories such that ECA ⊆ U ⊆ ATR , ECA ⊆ T and such that ATR proves that any set X can be included in afull ω -model for T . Let Γ be any set of formulas such that { = } ⊆ Γ ⊆ Π .Then, ATR ≡ U + ω R -RFN Γ [ T ] . In Theorem 6.6, we will extend this result to reflection over higher com-plexity classes, and show that it also gives rise to an axiomatization of barinduction. ω -logic via a least fixed point We obtain strictly more powerful reflection principles if we model ω -logicby an inductively defined fixed point, rather than its transfinite approxima-tions. Definition 5.6. Fix a theory T , possibly with oracles, and ρ ≤ ω . Then,define a formula SPC ρT ( Q ) ≡ Q = µP. (cid:16) (cid:3) T γ ∨ ∃ x ⊆ Q Rule ρ ( x , γ ) ∨ ω -Rule ( Q, γ ) (cid:17) . If SPC ρT ( Q ) holds we will say that Q is a saturated provability class of rank ρ ( ρ -SPC) for T . With this, we may define our fixed point provability operator. Definition 5.7. We define a formula [ I ] ρT γ ≡ ∀ P (cid:0) SPC ρT ( P ) → γ ∈ P (cid:1) . We will write [ I | X ] ρT γ instead of [ I ] ρT | X γ . We will often want to apply this operator to formulas rather than se-quents; when this is the case, we will identify a formula φ with the singletonsequent h φ i , and write [ I | X ] ρT φ instead of [ I | X ] ρT h φ i . Since SPC’s are definedvia an inductive definition, their existence can be readily proven in Π -CA . Lemma 5.8. Let T be any theory and ρ ≤ ω . Then, it is provable in Π - CA that for every tuple of sets A there exists a set P such that SPC ρT | A ( P ) holds.Proof. Immediate from Lemma 5.3. 16t is important to note that we have defined [ I | X ] ρT γ by quantifying uni-versally over all SPCs, so that ∼ [ I | X ] ρT γ quantifies existentially over them.This means that such consistency statements automatically give us a bit ofcomprehension: Lemma 5.9. If T is any theory and γ any sequent, then ECA ⊢ ∀ X (cid:0) ∼ [ I | X ] ρT γ → ∃ P SPC ρT | X ( P ) (cid:1) . However, this instance of comprehension by itself does not necessarilycarry additional consistency strength, in the following sense: Lemma 5.10. If T is a Tait theory extending ECA , T ≡ Π T + ∀ X ∃ P SPC ρT | X ( P ); that is, the two theories prove the same Π sentences. This is proven in [7] for a weaker notion of provability, but the argumentcarries through in our setting. Roughly, we observe that T + (cid:3) T ⊥ ≡ Π T ,but T + (cid:3) T ⊥ ⊢ T + ∀ X ∃ P SPC ρT | X ( P ), since in this case an SPC wouldsimply consist of the set of all formulas.Unlike the existence of SPCs, their uniqueness is immediate from theirdefinition. Lemma 5.11. If T is any theory and ρ ≤ ω , we have that ECA ⊢ ∀ X ∃ ≤ P SPC ρT | X ( P ) , where ∃ ≤ P φ ( P ) is an abbreviation of ∀ P ∀ Q (cid:0) φ ( P ) ∧ φ ( Q ) → P = Q (cid:1) . As one might expect, adding new sets to our oracle gives us a strongertheory: Lemma 5.12. Let T be any theory and ρ ≤ ω . It is provable in ECA thatif A is a tuple of sets and there exists an SPC for T | A , then for any sequent γ and any set B , [ I | A ] ρT γ → [ I | A , B ] ρT γ . Proof. Suppose that [ I | A ] ρT γ . Using our assumption, we may choose an SPC P for T | A , so that γ ∈ P . Let Q be an arbitrary SPC for T | A , B . Observethat Q contains all axioms of T | A and is closed under all of its rules, so thatby the minimality of P , we have that P ⊆ Q and thus γ ∈ Q . Since Q wasarbitrary, it follows that [ I | A , B ] ρT γ , as needed.17bseve also that our least-fixed-point formalization of ω -provability is atleast as strong as the formalization using ω -proofs: Lemma 5.13. Given any formula φ and ρ ≤ σ ≤ ω , it is provable in ACA that [ P | A ] ρT γ → [ I | A ] σT γ .Proof. Assume that [ P | A ] ρT γ holds, and let h S, L i be an ω -proof of γ . Now,consider any SPC P , and consider the set S ′ = { s ∈ S : L ( s ) P } , which isavailable in ACA . By the closure conditions of P , one readily checks that S ′ cannot have a minimal element, and thus must be empty. In particular, γ = L ( hi ) ∈ P .Our goal now is to prove impredicative reflection within Π -CA . Thefollowing is a first approximation: Π -CA proves that any formula provenin ω -logic with oracles is true in any ω -model. Lemma 5.14 ( ω –model soundness) . Given any theory T , a -tuple A , and ρ ≤ ω ,1. ACA ⊢ ∀ P ∀ A ∀ n (cid:16) SPC ρT ( P ) ∧ p φ ( ˙ n , O
For the first claim, reason in ACA . Let M be any model of T of rank ρ and let P be a saturated provability class for T | A of rank ρ . Let S ′ be obtained from S M by replacing each C i with by O i if i < a and by V a + i otherwise. Then, S ′ is closed under all the rules and axioms defining P , sothat, by minimality, P ⊆ S ′ . It follows that if φ ( O
6. Completeness and strong predicative reflection In this section we will recall some completeness results for formalized ω -logic. It is well-known that ω -logic is Π -complete [15], but it will beconvenient to keep track of the second-order axioms needed to prove this.From these results, we will obtain a more general form of Theorem 5.5.18 .1. Completeness results for ω -logic We begin with a weak completeness result available in ECA . Lemma 6.1. Fix a theory T and ρ ≤ ω . Let γ ( z , X ) ⊆ Π ω with all freevariables shown. Then, it is provable in ECA that ∀ A ∀ n (cid:16) _ γ ( n , A ) → [ I | A ] ρT γ ( ˙ n , O ) (cid:17) . (7) Proof. Reasoning within ECA , fix a tuple n of natural numbers and A ofsets and assume that W γ ( n , A ) holds, and write γ = ( δ , φ ) so that φ ∈ γ holds. We proceed by an external induciton on φ . Assume that P is anarbitrary SPC for T | X ; we must prove that (cid:0) δ , φ ( ¯ n , O ) (cid:1) ∈ P . If φ does notcontain quantifiers we proceed as in a standard Σ -completeness proof, asin e.g. [10, pp. 175–176]; we omit the details, but remark that the case foratomic formulas requires a secondary external induction on the complexityof the terms that may appear.Now assume that φ contains quantifiers. Let us consider the case where φ = ∀ x θ . By the external induction hypothesis we have, for every k , that (cid:0) δ , θ (¯ k, ¯ n , O ) (cid:1) ∈ P. But, P is closed under the ω -rule, so we also have that (cid:0) δ , ∀ x θ ( x, ¯ n , O ) (cid:1) ∈ P. The remaining cases follow a similar structure; the case where φ is aBoolean combination of its subformulas is straightforward using the rules ofthe Tait calculus, and if φ = ∃ x θ ( x ), then for some k we have that θ ( k ) is trueand we may use the induction hypothesis plus existential introduction.So, ECA already proves the completeness of ω -logic for arithmeticalformulas, but we need to turn to ACA to prove that it is also completefor Π formulas. The following is a mild modification of the Henkin-Orey ω -completeness theorem [9, 14]: Theorem 6.2. For any formula φ ( X ) ∈ Π ω and ρ ≤ ω , ACA ⊢ ∀ A ∀ n (cid:16) [ M | A ] ρT φ ( ˙ n , O ) → [ P | A ] ρT φ ( ˙ n , C ) (cid:17) . The following is then immediate from Lemma 5.13:19 orollary 6.3. For any formula φ ( X ) ∈ Π ω and any ρ ≤ ω , ACA ⊢ ∀ A ∀ n (cid:16) [ M | A ] ρT φ ( ˙ n , O ) → [ I | A ] ρT φ ( ˙ n , C ) (cid:17) . For formulas of relatively low complexity, we can replace [ M | A ] ρT φ by φ : Corollary 6.4. Let ρ ≤ ω .1. Given φ ( z , X ) ∈ Π with all free variables shown, ACA ⊢ ∀ A ∀ n (cid:16) φ ( n , A ) → [ I | A ] ρT φ ( ˙ n , O ) (cid:17) . 2. Given φ ( z , X ) ∈ Σ with all free variables shown, ACA ⊢ ∀ A ∀ n (cid:16) φ ( n , A ) → ∃ B [ I | A , B ] ρT φ ( ˙ n , O ) (cid:17) . Proof. The first clam is immediate from Lemma 4.3 and Corollary 6.3. Forthe second, suppose that φ ( z , X ) = ∃ Y ψ ( z , X , Y ), with ψ ∈ Π ( X , Y ).Then, if φ ( n , A ) holds we can fix B so that ψ ( n , A , B ) is the case, andwe may use the first claim to conclude that [ I | A , B ] ρT ψ ( ¯ n , O , ¯ B ), so that byexistential introduction we have [ I | A , B ] ρT φ ( ¯ n , O ). Using the results we have discussed on completeness of ω -logic and The-orem 4.5, we may extend Theorem 5.5 to consider reflection for higher com-plexity classes. Below, recall that the parameter ρ may be omitted when ρ = ω . Lemma 6.5. Let T be any theory. Then, over ATR , the following areprovably equivalent: 1. [ P | A ] T φ , 2. [ M | A ] T φ , 3. [ R | A ] T φ .Proof. That 3 implies 2 is proven in [6], and that 2 implies 1 follows fromTheorem 6.2. Thus it remains to show that 1 implies 3.Reasoning in ATR , suppose that h S, L i is an ω -proof of φ . We use awell-known technique of ‘linearizing’ , as in e.g. [1]. Consider the ordering E on S given by s E t if one of the following occurs: (a) t s , or (b) s , t areincomparable under , and for the least i such that s i = t i , we have that( s ) i ≤ ( t ) i . Then, it is readily verified that E is a well-order on S . Usingarithmetical transfinite recursion, let P be an IPC for T | A along h S, E i .20hen, a straightforward transfinite induction along shows that, for all s ∈ S , W L ( s ) ∈ P s ; in particular, φ ∈ P hi . Since P was arbitrary, weconclude that [ R | A ] T φ . Theorem 6.6. Let U be a theory such that ECA ⊆ U ⊆ ATR . Then, forany n ≤ ω , ATR + Π n - BI ≡ U + ω R -RFN Σ n [ACA ] . (8) Proof. The case for n = 0 follows from Theorem 5.5, in view of the fact thatATR ⊢ Π - BI , so we assume n > 0. Let R ≡ U + ω R -RFN Σ n [ACA ]. Let ρ be the rank of an axiomatization of ACA . Note that by Theorem 5.5,ATR ⊆ R , and hence R ≡ ATR + ω R -RFN Σ n [ACA ]. But, in view ofLemma 6.5, R ≡ ATR + ω M -RFN Σ n [ACA ] ≡ ATR + ω M -RFN ρ Σ n [ACA ] , where the second equivalence is due to the fact that ATR proves that anysatisfaction class extends to a full satisfaction class. But, by Theorem 4.5,ATR + ω M -RFN ρ Σ n [ACA ] ≡ ATR + Π n - BI , as needed.In view of Lemma 2.2, it follows that Theorem 5.5 is sharp: Corollary 6.7. ATR ω R -RFN Σ [ACA ] . Remark 6.8. We could instead use Theorem 3.5 to obtain a variant of Theo-rem 6.6 with the pure Tait calculus in place of ACA . For greater generality,it may be of interest to analyze the proof in [11] to identify the minimalrequirements on a theory T which would allow us to replace ACA by T . 7. Consistency and reflection using inductive definitions In this section we will define the notions of reflection and consistencythat naturally correspond to [ I | A ] ρT . Moreover, we will link the two notionsto each other and see how they relate to comprehension. Below, recall that ⊥ denotes the empty sequent. 21 efinition 7.1. Given a theory T , ρ ≤ ω , and a class of formulas Γ , wedefine the schemas ω I -RFN ρ Γ [ T ] = ∀ A ∀ n (cid:16) [ I | A ] ρT φ ( ˙ n , O ) → φ ( n , A ) (cid:17) ,ω I -CONS ρ Γ [ T ] = ∀ A ∀ n ∼ (cid:16) [ I | A ] ρT φ ( ˙ n , O ) ∧ [ I | A ] ρT ∼ φ ( ˙ n , O ) (cid:17) ,ω I -Cons ρ [ T ] = ∀ A ∼ [ I | A ] ρT ⊥ , for φ ( z , X ) ∈ Γ with all free variables shown. Lemma 7.2. Given any theory T ,1. if ρ ≤ ω , ACA + ω I -RFN ρ Γ [ T ] ⊢ ω M -RFN ρ Γ [ T ] ;2. if ρ ≤ ω, Π - CA + ω I -RFN ρ Γ ≡ Π - CA + ω M -RFN ρ Γ [ T ] . Proof. For the first claim, reason in ACA + ω I -RFN ρ Γ [ T ]. Suppose that φ ∈ Γand [ M | A ] ρT φ ( ¯ n , C ). Then, by Corollary 6.3, [ I | A ] ρT φ ( ¯ n , O ), and thus φ ( n , A )holds by ω I -RFN ρ Γ . For the second claim, the remaining inclusion follows fromLemma 5.14.Of course, the schema ω I -CONS ρ Γ [ T ] is only interesting when ρ < ω , sinceotherwise it is just equivalent to consistency. Lemma 7.3. If T is any theory and ρ ≤ ω , then ECA + ω I -CONS ρ Π ω [ T ] ⊆ ECA + ω I -Cons ω [ T ] . Proof. Reasoning by contrapositive, if ω I -CONS ρ Π ω [ T ] fails, then for some for-mula φ ( z , X ), some tuple of sets A and some tuple of natural numbers n ,we have that [ I | A ] ρT φ ( ¯ n , O ) ∧ [ I | A ] ρT ∼ φ ( ¯ n , O ) , which applying one cut gives us [ I | A ] ωT ⊥ .Let us now see that with just a little amount of reflection we get arithmeti-cal comprehension. The fist step is to build new sets out of our provabilityoperators. Lemma 7.4. Let T be any Tait theory, φ ( z, X ) be any formula and ρ ≤ ω .Then, ECA ⊢ ∀ A ∃ W ∀ n (cid:16) n ∈ W ↔ [ I | A ] ρT φ ( ˙ n, O ) (cid:17) . roof. Reason within ECA and pick a tuple of sets A . Consider two cases;if there does not exist a ρ -SPC for T | A , then we may set W = N and observethat ∀ n (cid:0) n ∈ W ↔ [ I | A ] ρT φ ( ˙ n, O ) (cid:1) holds trivially by vacuity.If such an SPC does exist, by Lemma 5.11 it is unique; call it P . WithinECA we may form the set W = { n : φ (¯ n, O ) ∈ P } . Then, if n ∈ W is arbitrary we have by the uniqueness of P that [ I | A ] ρT φ (¯ n, O )holds. Conversely, if [ I | A ] ρT φ (¯ n, O ) holds, then in particular φ (¯ n, O ) ∈ P holds and n ∈ W by definition, so W has all desired properties.Since A was arbitrary, the claim follows. Lemma 7.5. Let T be any theory and ρ ≤ ω . Then, ACA ⊆ ECA + ω I -RFN ρ Σ [ T ] . Proof. Work in ECA + ω I -RFN ρ Σ [ T ]. We only need to prove Σ - CA , that is, ∀ X ∃ Y ∀ n (cid:0) n ∈ Y ↔ φ ( n, X ) (cid:1) , where φ ( n, X ) can be any formula in Σ ( X ).Fix some tuple of sets A . By Lemma 7.4, we can form the set Z = { n : [ I | A ] ρT φ (¯ n, O ) } . We claim that ∀ n (cid:0) n ∈ Z ↔ φ ( n, A ) (cid:1) which finishes the proof. If n ∈ Z , then, by reflection, φ ( n, A ). On the other hand, if φ ( n, A ) we get byarithmetical completeness (Lemma 6.1) that [ I | A ] ρT φ (¯ n, O ), so that n ∈ Z .The above result along with the completeness theorems mentioned ear-lier may be used to prove that many theories defined using reflection andconsistency are equivalent. Below, ∼ Γ = {∼ φ : φ ∈ Γ } . Lemma 7.6. Let T be a theory extending Q + , and ρ ≤ ω . Then:1. if Σ ⊆ Γ ⊆ Π , ECA + ω I -CONS ρ Γ [ T ] ≡ ECA + ω I -RFN ρ Γ ∪∼ Γ [ T ];23 . ECA + ω I -Cons ω [ T ] ≡ ECA + ω I -RFN ω Π [ T ] . Proof. For the first claim, let us begin by proving thatECA + ω I -CONS ρ Γ [ T ] ⊆ ECA + ω I -RFN ρ Γ ∪∼ Γ [ T ] . Assume ω I -RFN ρ Γ ∪∼ Γ [ T ] and let φ ∈ Γ. Towards a contradiction, suppose thatfor some tuple of natural numbers n and some tuple of sets A ,[ I | A ] ρT φ ( ¯ n , O ) ∧ [ I | A ] ρT ∼ φ ( ¯ n , O ) . By reflection, this gives us φ ( n , A ) ∧ ∼ φ ( n , A ) , which is impossible. Since φ was arbitrary, the claim follows.Next we prove thatECA + ω I -CONS ρ Γ [ T ] ⊇ ECA + ω I -RFN ρ Γ ∪∼ Γ [ T ] . For this, fix φ ∈ Γ ∪ ∼ Γ and reason in ECA + ω I -CONS ρ Γ [ T ]. We first considerthe case where φ = φ ( z , X ) is arithmetical.Let n be a tuple of natural numbers and A a tuple of sets such that[ I | A ] ρT φ ( ¯ n , O ). If φ ( n , A ) were false, by Lemma 6.4.1, we would also havethat [ I | A ] ρT ∼ φ ( ¯ n , O ); but this contradicts ω I -CONS ρ Γ [ T ]. We conclude that φ ( n , A ) holds, as desired.Before considering the case where φ is not arithmetical, observe that since Σ ⊆ Γ, it follows thatECA + ω I -CONS ρ Γ [ T ] ⊇ ECA + ω I -RFN ρ Σ [ T ] , and by Lemma 7.5, we have thatACA ⊆ ECA + ω I -CONS ρ Γ [ T ] , so we may now use arithmetical comprehension.With this observation in mind, the argument will be very similar tothe one before. Once again, suppose that [ I | A ] ρT φ ( ¯ n , O ) for some tuples n , A . If φ ( n , A ) were false, by Corollary 6.4.2, there would be B such that[ I | A , B ] ρT ∼ φ ( ¯ n , O ). By Lemma 5.9, ECA + ω I -CONS ρ Γ [ T ] implies that thereexists a ρ -SPC for T | A , and hence we may use Lemma 5.12 to see that[ I | A , B ] ρT φ ( ¯ n , O ) ∧ [ I | A , B ] ρT ∼ φ ( ¯ n , O ) . 24s before, this contradicts ω I -CONS ρ Γ [ T ]. We conclude that φ ( n , A ) holds, asdesired.Now we prove the second claim. The right-to-left implication is obvious,so we focus on the other. Reason in ECA + ω I -Cons ω [ T ]. By Lemma 7.3,this implies ω I -CONS ω Π ω [ T ], so that using Lemma 7.5, we may reason in ACA .Fix φ ( z , X ) ∈ Π and assume that [ I | A ] ωT φ ( ¯ n , O ). If φ ( n , A ) were false,then by Corollary 6.4, we would also have [ I | A , B ] ωT ∼ φ ( ¯ n , O ) for some set B ,and using Lemma 5.12 as above,[ I | A , B ] ωT φ ( ¯ n , O ) ∧ [ I | A , B ] ωT ∼ φ ( ¯ n , O ) . But this contradicts ω I -CONS ω Π ω [ T ], and we conclude that φ ( X ) holds.Next, we turn our attention to proving that reflection implies Π -CA .This fact will be an easy consequence of the following: Lemma 7.7. Let T be any theory, ρ ≤ ω , Γ ⊆ Π ω ( X ) , and φ ( z , X ) ∈ Π / Γ .Then, it is provable in ACA + ω I -RFN ρ Π / Γ [ T ] that ∀ A ∀ n (cid:0) φ ( n , A ) ↔ [ I | A ] ρT φ ( ˙ n , O ) (cid:1) . Proof. Reason in ACA + ω I -RFN ρ Π / Γ [ T ] and let A and n be arbitrary. Forthe left-to-right direction we see that if φ ( n , A ) holds, then by provable Π -completeness (Corollary 6.4), [ I | A ] ρT φ ( ¯ n , O ) holds as well. For the right-to-left direction, if [ I | A ] ρT φ ( ¯ n , O ), by ω I -RFN ρ Π / Γ [ T ], φ ( n , A ) holds.We can now finally combine all our previous results and formulate themain theorem of this section. Theorem 7.8. Given any theory T , ACA + ω I -RFN ρ Π / Σ [ T ] ⊢ Π - CA . Proof. Work in ACA + ω I -RFN ρ Π / Σ [ T ] . By Theorem 2.1, we need only provecomprehension for arbitrary φ ( n, X ) ∈ Π / Σ ( X ).Fix a tuple of sets A . By Lemma 7.4, there is a set W satisfying ∀ n (cid:0) n ∈ W ↔ [ I | A ] ρT φ ( ˙ n, O ) (cid:1) . But by Lemma 7.7, this is equivalent to ∀ n (cid:0) n ∈ W ↔ φ ( n, A ) (cid:1) . Since φ and A were arbitrary, we obtain Π -CA , as desired.25hus impredicative reflection implies impredicative comprehension, asclaimed. Next we will prove the opposite implication, but for this we willfirst need to take a detour through β -models. 8. Countable β -models and impredicative reflection Our goal in this section is to derive a converse of Theorem 7.8. The maintool for this task will be the notion of a countable coded β –model . In whatfollows we shall discuss the definition and basic existence results for suchmodels.Note that the converse of Lemma 4.3 is not always true for Π -sentences,as we are not truly quantifying over all subsets of N . Nevertheless, for specialkinds of models it may actually be the case that M | = ∀ Xφ ( X ) implies that ∀ Xφ ( X ) when φ is arithmetical; such models are called β -models. Below, recall that V = h V i i i ∈ N is assumed to be a sequence listing allsecond-order variables, and that S
A countable coded ω -model M is a β -model if for every φ ( z , V
Fix a formula φ ( z , V
It is provable in ATR that, for every countable coded β -model M , M | = Π ω - BI . 26e remark that Theorem 8.3 obviously holds if we replace Π ω -BI by aweaker theory, such as ACA , ATR , or others we have mentioned earlier.However, Π -CA is required to construct β -models: Theorem 8.4. It is provable in Π - CA that for every a -tuple of sets A there is a full β -model M such that | M |
Lemma 8.5. Let