Logics of Intuitionistic Kripke-Platek Set Theory
aa r X i v : . [ m a t h . L O ] J u l Logics of Intuitionistic Kripke-Platek Set Theory
Rosalie Iemhoff a,1 , Robert Passmann b,2, ∗ a Department of Philosophy, Utrecht University, Janskerkhof 13, 3512 BL Utrecht, TheNetherlands b Institute for Logic, Language and Computation, Faculty of Science, University ofAmsterdam, P.O. Box 94242, 1090 GE Amsterdam, The Netherlands
Abstract
We investigate the logical structure of intuitionistic Kripke-Platek set theory
IKP , and show that the first-order logic of
IKP is intuitionistic first-order logic
IQC .
1. Introduction
The definition of any formal system has two crucial steps: First, choosing alogic, and second, adding some axioms for mathematical content. For example,Peano Arithmetic PA is defined by some arithmetical axioms on the basis ofclassical first-order logic. Heyting Arithmetic HA uses the same arithmeticalaxioms but is based on intuitionistic first-order logic. Similar situations arisein the context of set theories: Zermelo-Fraenkel Set Theory ZF is based onclassical logic while its intuitionistic and constructive counterparts, IZF and
CZF , are based on intuitionistic logic.A feature of non-classical systems is that their logical strength can increasewith adding mathematical axioms. For example, Diaconescu [8] proved thatadding the Axiom of Choice AC implies the Law of Excluded Middle in thecontext of intuitionistic IZF set theory. In other words, the system
IZF + AC is defined on the basis of intuitionistic logic but its logic is classical. Thisillustrates the importance of determining the logic of any non-classical system ofinterest: By showing that an intuitionistic system indeed has intuitionistic logic, ∗ Corresponding author
Email addresses: [email protected] (Rosalie Iemhoff), [email protected] (RobertPassmann) The first author is supported by the Netherlands Organisation for Scientific Researchunder grant 639.073.807. The second author is supported by a doctoral scholarship of the
Studienstiftung desdeutschen Volkes . The results in Section 3.3.2 and Section 4.3 of this paper are based onthe third chapter of the second author’s master’s thesis [24], supervised by Benedikt Löweat the University of Amsterdam. During the research for his master’s thesis, the second au-thor was partially supported by the Marie Skłodowska-Curie fellowship REGPROP (706219)funded by the European Commission at the Universität Hamburg. ne verifies the conceptual requirement that the theory should be intuitionistic.The first result in this area was of de Jongh [5, 6] who showed that the logicof Heyting Arithmetic HA is intuitionistic logic. This fact is now known as DeJongh’s Theorem (see Definition 33 for more details).Even though there is a rich literature on constructive set theories, there hasnot been much focus on the logics of these theories: Passmann [25] recentlyproved that the propositional logic of
IZF is intuitionistic propositional logic
IPC . On the other hand, a result of H. Friedman and Ščedrov [10] (see Theo-rem 34) implies that the first-order logic of intuitionistic set theories includingfull separation, such as
IZF , must be strictly stronger than intuitionistic first-order logic
IQC . These results show that
IZF is logically well-behaved on thepropositional level but less so on the level of predicate logic.What about other constructive set theories? Determining the first-orderlogic of
CZF , one of the most studied constructive set theories, is still an openproblem. Another natural constructive set theory, that has been studied in theliterature, is intuitionistic Kripke-Platek set theory
IKP , which was introducedby Lubarsky [18] to investigate intuitionistic admissibility theory in the traditionof Barwise [3]. In this article, we show that
IKP is a logically very well-behavedtheory as the following consequences of our more general results illustrate:(i) the propositional logic of
IKP is intuitionistic propositional logic
IPC (seeCorollary 43),(ii) the relative first-order logic of
IKP is intuitionistic first-order logic
IQC (see Corollary 46),(iii) the first-order logic of
IKP is intuitionistic first-order logic
IQC (see Corol-lary 59), and,(iv) the first-order logic with equality of
IKP is strictly stronger than intuition-istic first-order logic with equality
IQC = (see Corollary 62).An important byproduct of our work is a study of the possibilities and limitsof Kripke models whose domains are classical models of set theory: The commonKripke model constructions for intuitionistic or constructive set theories, suchas CZF or IZF , that are stronger than
IKP , usually involve complex recursiveconstructions (see, for example, [20]). We will expose a failure of the exponen-tiation axiom showing that these more complex constructions are necessary toobtain models of many stronger theories (see Section 3.3.2).This article is organised as follows. In Section 2, we will lay out the basicsneeded in terms of Kripke semantics for propositional and first-order logics.Section 3 provides an analysis of a certain Kripke model construction for
IKP .In Section 4 we will analyse the logical structure of
IKP and prove several
DeJongh Theorems for propositional, relative first-order and first-order logics. Weclose with some questions and directions for further research.2 . Logics and Their Kripke Semantics
As usual, we denote intuitionistic propositional logic by
IPC and intuitionis-tic first-order logic by
IQC . The classical counterparts of these logics are called
CPC and
CQC , respectively. We will generally identify each logic with the setof its consequences. A logic J is called intermediate if IPC ⊆ J ⊆ CPC in case J is propositional logic, or IQC ⊆ J ⊆ CQC in case J is a first-order logic.We assume intuitionistic first-order logic IQC to be formulated in a languagewithout equality. Intuitionistic first-order logic with equality will be denoted by
IQC = .A Kripke frame ( K, ≤ ) is a set K equipped with a partial order ≤ . A Kripke model for
IPC is a triple ( K, ≤ , V ) such that ( K, ≤ ) is a Kripke frameand V : Prop → P ( K ) a valuation that is persistent, i.e., if w ∈ V ( p ) and w ≤ v ,then v ∈ V ( p ) . We can then define, by induction on propositional formulas, theforcing relation for propositional logic at a node v ∈ K in the following way fora Kripke model M for IPC :(1)
M, v (cid:13) p if and only if v ∈ V ( p ) ,(2) M, v (cid:13) ϕ ∧ ψ if and only if K, V, v (cid:13) ϕ and K, V, v (cid:13) ψ ,(3) M, v (cid:13) ϕ ∨ ψ if and only if K, V, v (cid:13) ϕ or K, V, v (cid:13) ψ ,(4) M, v (cid:13) ϕ → ψ if and only if for all w ≥ v , K, V, w (cid:13) ϕ implies K, V, w (cid:13) ψ ,(5) M, v (cid:13) ⊥ holds never.We write v (cid:13) ϕ instead of K, V, v (cid:13) ϕ if the Kripke frame or the valuationare clear from the context. We will write K, V (cid:13) ϕ if K, V, v (cid:13) ϕ holds for all v ∈ K . A formula ϕ is valid in K if K, V, v (cid:13) ϕ holds for all valuations V on K and v ∈ K , and ϕ is valid if it is valid in every Kripke frame K .We can now define the propositional logic of a Kripke frame and of a classof Kripke frames. Definition 1. If ( K, ≤ ) is a Kripke frame, we define the propositional logic L ( K, ≤ ) to be the set of all propositional formulas that are valid in K . For aclass K of Kripke frames, we define the propositional logic L ( K ) to be the setof all propositional formulas that are valid in all Kripke frames ( K, ≤ ) in K .Given an intermediate propositional logic J , we say that K characterises J if L ( K ) = J .A Kripke model for
IQC is a triple ( K, ≤ , D, V ) where ( K, ≤ ) is a Kripkeframe, D v a set for each v ∈ K such that D v ⊆ D w for v ≤ w , and V a functionsuch that:(i) if p is a propositional letter, then V ( p ) ⊆ K such that if v ∈ V ( p ) and v ≤ w , then w ∈ V ( p ) ,(ii) if R is an n -ary relation symbol of the language of IQC , then V ( R ) = { R v | v ∈ K } such that R v ⊆ D nv and R v ⊆ R w for v ≤ w , and,3iii) if f is an n -ary function symbol of the language of IQC , then V ( f ) = { f v | v ∈ K } such that f v is a function D nv → D v such that Graph ( f v ) ⊆ Graph ( f w ) for v ≤ w .We now extend the conditions of the forcing relation for IPC to Kripkemodels M for IQC in the following way:(6)
M, v (cid:13) R ( x , . . . , x n − ) if and only if ( x , . . . , x n − ) ∈ R v ,(7) M, v (cid:13) f ( x , . . . , x n − ) = y if and only if f v ( x , . . . , x n − ) = y ,(8) M, v (cid:13) ∃ x ϕ ( x ) if and only if there is some x ∈ D v such that K, V, v (cid:13) ϕ ( x ) , and,(9) M, v (cid:13) ∀ x ϕ ( x ) if and only if for all w ≥ v and x ∈ D w it holds that K, V, w (cid:13) ϕ ( x ) .We can further extend these definitions to Kripke models for IQC = by in-terpreting equality as an equivalence relation ∼ v at every node v ∈ K , andstipulate that:(10) M, v (cid:13) x = y if and only if x ∼ v y .We define the validity of formulas in frames and classes of frames just asin the case of propositional logic. Now, we can define the first-order logic of aKripke frame and of a class of Kripke frames. Definition 2. If ( K, ≤ ) is a Kripke frame, then the first-order logic QL ( K, ≤ ) is defined to be the set of all first-order formulas that are valid in K . For a class K of Kripke frames, we define the first-order logic QL ( K ) to be the set of allfirst-order formulas that are valid in all Kripke frames ( K, ≤ ) in K . Given anintermediate first-order logic J , we say that K characterises J if QL ( K ) = J .Similarly, we define QL = (( K, ≤ )) and QL = ( K ) as the set of all first-orderformulas in the language of equality that are valid in the respective frame orclass of frames.We will sometimes write L ( K ) for L ( K, ≤ ) , QL ( K ) for QL ( K, ≤ ) , and QL = ( K ) for QL = ( K, ≤ ) . The next result is proved by induction on the com-plexity of formulas; it shows that persistence of the propositional variables trans-fers to all formulas. Proposition 3.
Let M be a Kripke model for IPC , IQC or IQC = , v ∈ K and ϕ be a propositional formula such that M, v (cid:13) ϕ holds. Then M, w (cid:13) ϕ holdsfor all w ≥ v . Theorem 4.
A propositional formula ϕ is derivable in IPC if and only if it isvalid in all Kripke models for
IPC . In particular, a propositional formula ϕ isderivable in IPC if and only if it is valid in all finite Kripke models for
IPC . Aformula ϕ of first-order logic is derivable in IQC if and only if it is valid in allKripke models for
IQC . Finally, a formula ϕ of first-order logic with equalityis derivable in IQC = if and only if it is valid in all Kripke models for IQC = . IQC to denote first-order intuitionistic logic in a language without equality, and
IQC = to denote first-order intuitionistic logic with equality (see [28, Chapter2] for a discussion of various versions of intuitionistic first-order logic with and without equality).We will later need the following result on Kripke frames for IQC ( without equality). Definition 5.
We say that a Kripke model M = ( K, ≤ , D, V ) is countable if K is countable and D v is countable for every v ∈ K . A Kripke model M = ( K, ≤ , D, V ) has countably increasing domains if for every v, w ∈ K such that v < w ,we have that D w \ D v is a countably infinite set. Lemma 6.
Let M = ( K, ≤ , D, V ) be a countable Kripke model for intuitionisticfirst-order logic. Then there is a model M ′ = ( K, ≤ , D ′ , V ′ ) with countablyincreasing domains and a family of maps f v : D v → D ′ v such that M, v (cid:13) ϕ (¯ x ) ifand only if M ′ , v (cid:13) ϕ ( f v (¯ x )) holds for every v ∈ K . Further, if M is countable,then so is M ′ .Proof. As M is countable, the Kripke frame ( K, ≤ ) will be countable. So let h v i | i < ω i be a bijective enumeration of all nodes of K . Let M = M . Given M n = ( K, ≤ , D n , V n ) , define M n +1 as follows: Take a countable set X n suchthat X n ∩ S v ∈ K D nv = ∅ . Now let D nw = D nw if w v n , and D nw = D nw ∪ X n if w ≥ v n . Extend the valuation V n of M n to the extended domains as follows:Pick an arbitrary element y n ∈ D nv n and copy the valuation of y n for every x ∈ X n at every w ≥ v n (i.e. such that v (cid:13) P ( x, ¯ z ) if and only if v (cid:13) P ( y n , ¯ z ) ).Finally, take M ′ = ( K, ≤ , D ′ , V ′ ) where D ′ v = S n<ω D nv and V ′ v = S n<ω V nv .Clearly M ′ is countable. Further define f : S v ∈ K D ′ v → S v ∈ K D v by stipulatingthat f ( x ) = x if x ∈ D v , and f ( x ) = y n if x ∈ X n . An easy induction nowshows that the desired statement holds (note that the language of IQC doesnot contain equality).
3. IKP and Its Kripke Semantics
In this section, we will introduce the constructive set theory that we aregoing to analyse in this article, and some Kripke semantics for set theory.
We will list the relevant axioms and axiom schemes. The language L ∈ ofset theory extends the logical language with binary relation symbols ∈ and = denoting set membership and equality, respectively. As usual, the boundedquantifiers ∀ x ∈ a ϕ ( x ) and ∃ x ∈ a ϕ ( x ) are abbreviations for ∀ x ( x ∈ a → ϕ ( x )) ∃ x ( x ∈ a ∧ ϕ ( x )) , respectively. ∃ a ∀ x ∈ a ⊥ (Empty Set) ∀ a ∀ b ∃ y ∀ x ( x ∈ y ↔ ( x = a ∨ x = b )) (Pairing) ∀ a ∃ y ∀ x ( x ∈ y ↔ ∃ u ( u ∈ a ∧ x ∈ u )) (Union) ∀ a ∀ b ( ∀ x ( x ∈ a ↔ x ∈ b ) → a = b ) (Extensionality) ∃ x ( ∅ ∈ x ∧ ( ∀ y y ∈ x → y ∪ { y } ∈ x ) ∧ (Infinity) ( ∀ y y ∈ x → ( y = ∅ ∨ ∃ z ∈ y y = z ∪ { z } )))( ∀ a ( ∀ x ∈ a ϕ ( x ) → ϕ ( a ))) → ∀ aϕ ( a ) (Set Induction)Moreover, we have the axiom schemes of ∆ -separation and ∆ -collection, where ϕ ranges over the bounded formulas: ∀ a ∃ y ∀ x ( x ∈ y ↔ x ∈ a ∧ ϕ ( x )) ( ϕ is a ∆ -formula) ( ∆ -Separation) ∀ a ( ∀ x ∈ a ∃ yϕ ( x, y ) → ∃ b ∀ x ∈ a ∃ y ∈ bϕ ( x, y )) ( ϕ is a ∆ -formula)( ∆ -Collection)Sometimes, these schemes are also referred to as bounded separation and boundedcollection , respectively. Removing the restriction to ∆ -formulas, we obtain theusual schemes of separation and collection . Definition 7.
The theory
IKP of intuitionistic Kripke-Platek set theory IKP consists of the axioms and rules of intuitionistic first-order logic for the language L ∈ extended by the axioms and axiom schemes of empty set, pairing, union,extensionality, infinity, set induction, ∆ -separation, and ∆ -collection. IKP was first introduced and studied by Lubarsky [18]. Denote by
IKP + the theory obtained by adding the schemes of bounded strong collection andset-bounded subset collection to IKP .For reference, we also introduce the well-known theories of
CZF and
IZF . Inthe following strong infinity axiom,
Ind( a ) is the formula denoting that a is aninductive set: Ind( a ) abbreviates ∅ ∈ a ∧ ∀ x ∈ a ∃ y ∈ a y = { x } . ∃ a (Ind( a ) ∧ ∀ b (Ind( b ) → ∀ x ∈ a ( x ∈ b ))) (Strong Infinity)Finally, we have the schemes of strong collection and subset collection for allformulas ϕ ( x, y ) and ψ ( x, y, u ) , respectively. ∀ a ( ∀ x ∈ a ∃ y ϕ ( x, y ) → (Strong Collection) ∃ b ( ∀ x ∈ a ∃ y ∈ b ϕ ( x, y ) ∧ ∀ y ∈ b ∃ x ∈ a ϕ ( x, y ))) ∀ a ∀ b ∃ c ∀ u ( ∀ x ∈ a ∃ y ∈ b ψ ( x, y, u ) → (Subset Collection) ∃ d ∈ c ( ∀ x ∈ a ∃ y ∈ d ψ ( x, y, u ) ∧ ∀ y ∈ d ∃ x ∈ a ψ ( x, y, u ))) The axiom scheme obtained from strong collection when restricting ϕ torange over ∆ -formulas only will be called Bounded Strong Collection . Similarly,we obtain the axiom scheme of
Set-bounded Subset Collection from the axiom6cheme of subset collection when restricting ψ to ∆ -formulas such that z isset-bounded in ψ (i.e., it is possible to intuitionistically derive z ∈ t for someterm t that appears in ψ from ψ ( x, y, z ) ).We also need the power set axiom. ∀ a ∃ y ∀ z ( z ∈ y ↔ z ⊆ a ) (Power Set) Definition 8.
The theory
CZF of constructive Zermelo-Fraenkel set theory con-sists of the axioms and rules of intuitionistic first-order logic for the language L ∈ extended by the axioms of extensionality, empty set, pairing, union and stronginfinity as well as the axiom schemes of set induction, bounded separation,strong collection and subset collection.In the statement of the following axiom of exponentiation, f : x → y is anabbreviation for the ∆ -formula ϕ ( f, x, y ) stating that f is a function from x to y . ∀ x ∀ y ∃ z ∀ f ( f ∈ z ↔ f : x → y ) (Exponentiation, Exp )The axiom of exponentiation is a constructive consequence of the axiom of subsetcollection over
CZF (cf. [1, Theorem 5.1.2]). Hence, a failure of exponentiationimplies a failure of subset collection. We will see in Section 3.3.2 that the Kripkemodels with classical domains do not satisfy the axiom of exponentiation ingeneral, and therefore, they cannot satisfy full
CZF . Definition 9.
The theory
IZF of intuitionistic Zermelo-Fraenkel set theory con-sists of the axioms and rules of intuitionistic first-order logic for the language L ∈ extended by the axioms and axiom schemes of extensionality, pairing, union,empty set, strong infinity, separation, collection, set induction, and powerset. By extending the Kripke models introduced above, we can obtain models forintuitionistic first-order logic. Instead of developing this theory in full generality,we will focus on the subcase of
Kripke models for set theory . Definition 10. A Kripke model ( K, ≤ , D, e ) for set theory is a Kripke frame ( K, ≤ ) for IPC with a collection of domains D = { D v | v ∈ K } and a collectionof set-membership relations e = { e v | v ∈ K } , such that the following hold:(i) e v is a binary relation on D v for every v ∈ K , and,(ii) D v ⊆ D w and e v ⊆ e w for all w ≥ v ∈ K .Examples of Kripke models for set theory are not only the Kripke modelswith classical domains that we will introduce in Section 3.3, but also the Kripkemodels introduced by Lubarsky [19, 20], by Diener and Lubarsky [21] and byLubarsky and Rathjen [22]; recently Passmann [25] introduced the so-called blended Kripke models for set theory to prove de Jongh’s theorem for IZF and
CZF . 7e can now extend the forcing relation to Kripke models for set theory tointerpret the language of set theory L ∈ . For the following definition, we tacitlyenrich the language of set theory with constant symbols for every element of thedomains of the Kripke model at hand. Definition 11.
Let ( K, ≤ , D, e ) be a Kripke model for set theory. We define,by induction on L ∈ -formulas, the forcing relation at every node of a Kripkeframe in the following way, where ϕ and ψ are formulas with all free variablesshown, and ¯ y = y , . . . , y n − are elements of D v for the node v considered onthe left side:(i) ( K, ≤ , D, e ) , v (cid:13) a ∈ b if and only if ( a, b ) ∈ e v ,(ii) ( K, ≤ , D, e ) , v (cid:13) a = b if and only if a = b ,(iii) ( K, ≤ , D, e ) , v (cid:13) ∃ xϕ ( x, ¯ y ) if and only if there is some a ∈ D v with ( K, ≤ , D, e ) , v (cid:13) ϕ ( a, ¯ y ) ,(iv) ( K, ≤ , D, e ) , v (cid:13) ∀ xϕ ( x, ¯ y ) if and only if for all w ≥ v and a ∈ D w we have ( K, ≤ , D, e ) , w (cid:13) ϕ ( a, ¯ y ) .The cases for → , ∧ , ∨ and ⊥ are analogous to the ones in the above definitionof the forcing relation for Kripke models for IPC . We will write v (cid:13) ϕ (or K, v (cid:13) ϕ ) instead of ( K, ≤ , D, e ) , v (cid:13) ϕ if the Kripke model is clear from thecontext. An L ∈ -formula ϕ is valid in K if v (cid:13) ϕ holds for all v ∈ K , and ϕ is valid if it is valid in every Kripke frame K . Finally, we will call ( K, ≤ ) the underlying Kripke frame of ( K, ≤ , D, e ) .Persistence also holds in Kripke models for set theory. Proposition 12.
Let ( K, ≤ , V ) be a Kripke model for set theory, v ∈ K and ϕ be a formula in the language of set theory such that K, v (cid:13) ϕ holds. Then K, w (cid:13) ϕ holds for all w ≥ v . Remark 13.
We have now introduced four kinds of Kripke models: for
IPC ,for
IQC , for
IQC = , and for set theory. The reader might have noticed thatKripke models for set theory are just a special instance of the Kripke models for IQC = where equality is interpreted as actual equality on the domains. Kripkemodels for IQC = do in general not interpret equality this way and only requirean equivalence relation, and Kripke models for IQC do not have equality at all.Using this distinction, we are making explicit when we talk about Kripke modelsfor certain logics and when we are talking about Kripke models for certain settheories . The idea is to obtain models of set theory by assigning classical models of ZF set theory to every node of a Kripke frame. We will first introduce Kripke modelswith classical domains and explain some of their basic properties. Afterwards,we will indicate their limitations in modelling strong set theories by exhibitinga failure of the exponentiation axiom. 8 .3.1. Definitions and Basic Properties We will closely follow the presentation of Iemhoff [15] but give up on somegenerality that is not needed here. We will start by giving a condition for whenan assignment of models to nodes is suitable for our purposes.
Definition 14.
Let ( K, ≤ ) be a Kripke frame. An assignment M : K → V oftransitive models of ZF set theory to nodes of K is called sound for K if for allnodes v, w ∈ K with v ≤ w we have that M ( v ) ⊆ M ( w ) , M ( v ) (cid:15) x ∈ y implies M ( w ) (cid:15) x ∈ y , and M ( v ) (cid:15) x = y implies M ( w ) (cid:15) x = y .For convenience, we will write M v for M ( v ) . Of course, this could be readilygeneralised to homomorphisms of models of set theory that are not necessarilyinclusions, but we will not need this level of generality here. Definition 15.
Given a Kripke frame ( K, ≤ ) and a sound assignment M : K → V , we define the Kripke model with classical domains K ( M ) to be the Kripkemodel for set theory ( K, ≤ , M , e ) where e v = ∈ ↾ ( M v × M v ) .Persistence for Kripke models with classical domains is a special case ofpersistence for Kripke models for set theory. Proposition 16. If K ( M ) is a Kripke model with classical domains with nodes v, w ∈ K such that v ≤ w , then for all formulas ϕ , K ( M ) , v (cid:13) ϕ implies K ( M ) , w (cid:13) ϕ . We will now analyse the set theory satisfied by these models.
Definition 17.
We say that a set-theoretic formula ϕ ( x , . . . , x n − ) is evaluatedlocally if for all Kripke models with classical domains K ( M ) , where M is asound assignment, we have K ( M ) , v (cid:13) ϕ ( a , . . . , a n − ) if and only if M v (cid:15) ϕ ( a , . . . , a n − ) for all a , . . . , a n − ∈ M v . Proposition 18. If ϕ is a ∆ -formula, then ϕ is evaluated locally.Proof. This statement can be shown by actually proving a stronger statementby induction on ∆ -formulas, simultaneously for all v ∈ K . Namely, we canshow that for all w ≥ v it holds that w (cid:13) ϕ ( a , . . . , a n ) if and only if M v (cid:15) ϕ ( a , . . . , a n ) . To prove the case of the bounded universal quantifier and thecase of implication, we need that the quantifier is outside in the sense that ourinduction hypothesis will be: ∀ w ≥ v ( w (cid:13) ϕ ( a , . . . , a n ) ⇐⇒ M v (cid:15) ϕ ( a , . . . , a n )) . With this setup, the induction follows straightforwardly.
Theorem 19 (Iemhoff, [15, Corollary 4]) . Let K ( M ) be a Kripke model withclassical domains. Then K ( M ) (cid:13) IKP + . Recall that Markov’s principle MP is formulated in the context of set theoryas follows: ∀ α : N → ¬∀ n ∈ N α ( n ) = 0 → ∃ n ∈ N α ( n ) = 1) roposition 20. Let K ( M ) be a Kripke model with classical domains. Then K ( M ) (cid:13) MP .Proof. Let v ∈ K and α ∈ M v be given such that v (cid:13) “ α is a function α → ”.By Proposition 18, we know that α is such a function also in the classical model M v . Further observe that ¬∀ n ∈ N α ( n ) = 0 → ∃ n ∈ N α ( n ) = 1 is a ∆ -formula and therefore evaluated locally by Proposition 18. Now this statementis clearly true of α because M v is a classical model of ZF .Extended Church’s Thesis ECT does not hold. Let us conclude this sectionwith the following curious observation.
Proposition 21. If K ( M ) is a Kripke model with classical domains such thatevery M v is a model of the axiom of choice, then the axiom of choice holds in K ( M ) .Proof. Recall that the axiom of choice is the following statement: ∀ a (( ∀ x ∈ a ∀ y ∈ a ( x = y → x ∩ y = ∅ )) → ∃ b ∀ x ∈ a ∃ ! z ∈ b z ∈ x ) . ( AC )Let v ∈ K and a ∈ M v such that v (cid:13) ∀ x ∈ a ∀ y ∈ a ( x = y → x ∩ y = ∅ ) .This is a ∆ -formula, so we can apply Proposition 18 to derive that M v (cid:15) ∀ x ∈ a ∀ y ∈ a ( x = y → x ∩ y = ∅ ) . As M v (cid:15) AC , there is some b ∈ M v suchthat M v (cid:15) ∀ x ∈ a ∃ ! z ∈ b z ∈ x . Again, this is a ∆ -formula, so it holds that v (cid:13) ∀ x ∈ a ∃ ! z ∈ b z ∈ x . As b ∈ M v , we have v (cid:13) ∃ b ∀ x ∈ a ∃ ! z ∈ b z ∈ x . Butthis shows that v (cid:13) AC .As IKP + contains the bounded separation axiom, it follows that AC impliesthe law of excluded middle for bounded formulas in the models of the proposition(see [1, Chapter 10.1]). We summarise the results of this section in the followingcorollary. Corollary 22. If K ( M ) is a Kripke model with classical domains such thatevery M v is a model of the axiom of choice, then K ( M ) (cid:13) IKP + + MP + AC .3.3.2. A Failure of Exponentiation In this section, we will exhibit a failure of the axiom of exponentiation inparticular Kripke models with classical domains.
Proposition 23.
Let K ( M ) be a Kripke model with classical domains suchthat there are v, w ∈ K with v < w . If a, b ∈ M v and g : a → b is a functioncontained in M w but not in M v , then K ( M ) (cid:13) Exp .Proof.
Assume, for a contradiction, that K ( M ) (cid:13) Exp . Further, assume that a, b ∈ M v and g : a → b is a function contained in M w but not in M v . Then, K ( M ) , v (cid:13) ∀ x ∀ y ∃ z ∀ f ( f ∈ z ↔ f : x → y ) , This follows because under MP and ECT all functions f : R → R are continuous (see [1,Theorem 16.0.23]) but that is in general not the case here. c ∈ M v such that K ( M ) , v (cid:13) ∀ f ( f ∈ c ↔ f : a → b ) . By the semantics of universalquantification, this means that K ( M ) , w (cid:13) g ∈ c ↔ g : a → b. Since g is indeeda function from a → b , it follows that K ( M ) , w (cid:13) g ∈ c. As c is a member of M v by assumption, we have g ∈ c ∈ M v . Hence, by transitivity, g ∈ M v . Butthis is a contradiction to our assumption that g is not contained in M v .Of course, when adding a generic filter for a non-trivial forcing notion, wealways add such a function, namely the characteristic function of the genericfilter. Therefore, Proposition 23 yields: Corollary 24.
Let K ( M ) be a Kripke model with classical domains. If thereare nodes v < w ∈ K such that M w is a non-trivial generic extension of M v (i.e., M w = M v [ G ] for some generic G / ∈ M v ), then it is not a model of CZF . In Kripke semantics for intuitionistic logic, K ( M ) (cid:13) ¬ ϕ is strictly strongerthan K ( M ) (cid:13) ϕ . The above results give an instance of the latter (a so-calledweak counterexample), now we will provide an example of the former (a strongcounterexample). Proposition 25.
There is a Kripke model with classical domains K ( M ) thatforces the negation of the exponentiation axiom, i.e., K ( M ) (cid:13) ¬ Exp .Proof.
Consider the Kripke frame K = ( ω, < ) where < is the standard orderingof the natural numbers. Construct the assignment M as follows: Choose M to be any countable and transitive model of ZFC . If M i is constructed, let M i +1 = M i [ G i ] where G i is generic for Cohen forcing over M i (actually, everynon-trivial forcing notion does the job). Clearly, M is a sound assignment ofmodels of set theory. Now, we want to show that for every i ∈ ω we have that i (cid:13) ¬ Exp , i.e., for all j ≥ i we need to show that j (cid:13) Exp implies j (cid:13) ⊥ . This,however, is done exactly as in the proof of Proposition 23, where the witnessesare the characteristic functions χ G i of the generic filters G i . We define the relativisation ϕ ϕ L of a formula of set theory to the con-structible universe L in the usual way. Note, however, that in our setting theevaluation of universal quantifiers and implications is in general not local (incontrast to classical models of set theory). Nevertheless, we will now show that—under mild assumptions—statements about the constructible universe can beevaluated locally. The following is a well-known fact. Fact 26 ([16, Lemma 13.14]) . There is a Σ -formula ϕ ( x ) such that in anymodel M (cid:15) ZFC , we have M (cid:15) ϕ ( x ) ↔ x ∈ L . From now on, let x ∈ L be an abbreviation for ϕ ( x ) , where ϕ is the Σ -formula from Fact 26. 11 roposition 27. Let K be a Kripke frame and M a sound assignment of nodesto transitive models of ZFC . Then K ( M ) , v (cid:13) x ∈ L if and only if M v (cid:15) x ∈ L ,i.e., the formula x ∈ L is evaluated locally.Proof. Recall that the existential quantifier is defined locally, i.e., the witnessfor the quantification must be found within the domain associated to the currentnode in the Kripke model. Then, the statement of the proposition follows fromthe fact that ∆ -formulas are evaluated locally by Proposition 18.The crucial detail of the following technical Lemma 29 is the fact that theconstructible universe is absolute between inner models of set theory. We willtherefore need to strengthen the notion of a sound assignment. If N and M are transitive models of set theory, we say that N is an inner model of M if N ⊆ M , N is a model of ZFC , N is a transitive class of M , and N contains allthe ordinals of M (see [16, p. 182]). Definition 28.
Let K be a Kripke frame. We say that a sound assignment M : K → V agrees on L if there is a transitive model N (cid:15) ZFC + V = L suchthat N is an inner model of M v for every v ∈ K .In particular, if K is a Kripke frame and M : K → V agrees on L , then weare justified in referring to the constructible universe L from the point of viewof all models in M . Lemma 29.
Let K be a Kripke frame and M be a sound assignment that agreeson L . Then the following are equivalent for any formula ϕ ( x ) in the languageof set theory, and all parameters a , . . . , a n − ∈ L :(i) for all v ∈ K , we have K ( M ) , v (cid:13) ( ϕ ( a , . . . , a n − )) L ,(ii) for all v ∈ K , we have M v (cid:15) ( ϕ ( a , . . . , a n − )) L ,(iii) there is a v ∈ K such that M v (cid:15) ( ϕ ( a , . . . , a n − )) L , and,(iv) L (cid:15) ϕ ( a , . . . , a n − ) .Proof. By our assumption, a , . . . , a n − ∈ M v for all v ∈ K as L ⊆ M v for all v ∈ K . The equivalence of (ii), (iii) and (iv) follows directly from the fact that L is absolute between inner models of ZFC .The equivalence of (i) and (ii) can be proved by an induction on set-theoreticformulas simultaneously for all nodes in K with the induction hypothesis as inthe proof of Proposition 18. For the case of the universal quantifier, we makeuse of the fact that M agrees on L (hence, that L is absolute between all models M v for v ∈ K ), and apply Proposition 27.
4. The Logical Structure of IKP
The aim of this section is to analyse the propositional and first-order logicsof
IKP . First, we will introduce the logics of interest in a general way, and thenproceed to introduce a Kripke model construction that we will use to determinecertain logics of
IKP + . 12 .1. Logics and the De Jongh Property We will be concerned with both propositional and first-order logics.
Definition 30. A propositional translation σ : Prop → L sent T is a map frompropositional letters to sentences in the appropriate language that is extendedto formulas in the obvious way.A first-order translation σ : L J → L form T is a map from the collection ofrelation symbols of L J to L ∈ -formulas such that n -ary relation symbols aremapped to formulas with n -free variables. Then σ is extended to all predicateformulas in L J in the obvious way.If J is a first-order logic with equality and T a theory with equality, thena first-order equality translation σ is a first-order translation with the extracondition that equality of J is mapped to equality T .Following Visser [29, Section 2.2], we only consider the case of predicatelanguages that contain only relation symbols by eliminating any function sym-bol f by replacing it with a relation R f ( x , . . . , x n , y ) defined by the equality f ( x , . . . , x n ) = y . If we eliminate a function symbol in such a way, we demandthat the interpreting theory T proves that σ ( R f ) is the graph of a function(i.e., σ being a translation is then dependent on the theory T ). Nested functionsymbols can be eliminated with the usual procedure of introducing variables forthe intermediate values.Further, given a first-order logic J , we will make use of the relative transla-tion ( · ) E (where we shall always tacitly assume that E is a fresh unary predicatesymbol) that acts non-trivially only on quantifiers: ( ∃ x ϕ ( x )) E = ( ∃ x ( Ex ∧ ϕ E ( x ))) , and, ( ∀ x ϕ ( x )) E = ( ∀ x ( Ex → ϕ E ( x ))) . Definition 31.
Given a theory T , formulated in a language L T , we define thefollowing logics:(i) The propositional logic L ( T ) of T consists of the propositional formulas ϕ such that T ⊢ ϕ σ for all propositional translations σ .(ii) The first-order logic QL ( T ) of T consists of the first-order formulas ϕ suchthat T ⊢ ϕ σ for all first-order translations σ in the language L T .(iii) The relative first-order logic QL E ( T ) of T consists of the first-order formu-las ϕ such that T ⊢ ( ϕ E ) σ for all first-order translations σ in the language L T ∪ { E } , where E is the fresh unary predicate symbol introduced for therelative translation.(iv) The first-order logic with equality QL = ( T ) of T consists of the first-orderformulas ϕ such that T ⊢ ϕ σ for all first-order equality translations σ inthe language L T . Definition 32.
Let T be a theory and J a logic. We define the theory T ( J ) asfollows: 13i) If J is a propositional logic, we define T ( J ) to be the theory obtainedfrom T by adding all sentences of the form A σ for formulas A ∈ J andpropositional translations σ .(ii) If J is a first-order logic, we define T ( J ) to be the theory obtained from T by adding all sentences of the form A σ for formulas A ∈ J and first-ordertranslations σ . Definition 33.
We say that a theory T satisfies the de Jongh property for alogic J if L ( T ( J )) = J . A theory T based on intuitionistic logic satisfies deJongh’s theorem if L ( T ) = IPC .Before embarking on determining some logics of
IKP + , let us survey a fewknown results. De Jongh [5, 6] started the investigations of logics of arith-metical theories, establishing that L ( HA ) = IPC and QL E ( HA ) = IQC . DeJongh, Verbrugge and Visser [7] introduced the de Jongh property and showed—among other results—that L ( HA ( J )) = J for logics J that are characterised byclasses of finite frames. Considering a logic that is weaker than intuitionisticlogic, Ardeshir and Mojtahedi [2] proved that the propositional logic of basicarithmetic is the basic propositional calculus.For the sake of a counterexample to the de Jongh property, consider the the-ory HA + MP + ECT , i.e., Heyting arithmetic extended with Markov’s Principle ( MP ) and Extended Church’s Thesis ( ECT ). Even though these principles areconsidered constructive, one can show that the propositional logic of this theoryis an intermediate logic, i.e., IPC ( L ( HA + MP + ECT ) ( CPC (this followsfrom results of Rose [26] and McCarty [23]; for details see the discussion at theend of [7, Section 2]). In conclusion, HA + MP + ECT does not satisfy de Jongh’stheorem.Turning now towards set theory, Passmann [25] used a Kripke-model con-struction to show that L ( IZF ) = L ( CZF ) =
IPC , and, in fact, that L ( T ( J )) = J for every set theory T ⊆ IZF and every logic J characterised by a class of finiteframes. H. Friedman and Ščedrov [10] conclude from their earlier conservativityresults [9] that L ( ZFI ) =
IPC holds for the two-sorted theory
ZFI .If C is a class of formulas, we write L C ( T ) for the propositional logic of T where we restrict to the class of translations to maps σ with ran( σ ) ⊆ C . Wedefine QL C ( T ) and QL CE ( T ) in the same way.An important observation of H. Friedman and Ščedrov is the following. Theorem 34 (H. Friedman and Ščedrov, [10, Theorem 1.1]) . Let T be a settheory based on intuitionistic logic. Suppose that T includes the axioms of Ex-tensionality, Separation, Pairing and (finite) Union. Then IQC ( QL ( T ) , i.e.,the first-order logic of T is stronger than intuitionistic first-order logic. This implies, in particular, that
IZF does not satisfy the de Jongh propertyfor
IQC . As
CZF only contains ∆ -separation but full separation is used in theproof of the above theorem, the theorem does not apply to CZF . However, witha slight adaption of the proof of H. Friedman and Ščedrov we can observe the14ollowing theorem. If C is a class of formulas, we denote the separation schemerestricted to formulas from C by C -Separation . Theorem 35.
Let T be a set theory based on intuitionistic logic and C be aclass of formulas. Suppose that T includes the axioms of Extensionality, C -Separation, Pairing and (finite) Union. Then IQC ( QL C ( T ) , i.e., the C -first-order logic of T , QL C ( T ) , is stronger than intuitionistic first-order logic. So, in particular,
IQC ( QL ∆ ( CZF ) , i.e., the ∆ -first-order logic of CZF is strictly stronger than intuitionistic logic. On the other hand, A ∨ ¬ A / ∈ QL ∆ ( CZF ) , so IQC ( QL ∆ ( CZF ) ( CQC . We will now introduce a class of Kripke models with classical domains thatarise from certain classical models of set theory. These models will later be usedto prove our results on logics of
IKP .S. Friedman, Fuchino and Sakai [11] presented family of sentences that we aregoing to use to imitate the logical behaviour of a given Kripke frame. Considerthe following statements ψ i :There is an injection from ℵ L i +2 to P ( ℵ L i ) . There are different ways of formalising these statements that are classicallyequivalent, but (possibly) differ in the way they are evaluated in a Kripke model.For our purposes, we choose to define the sentence ψ i like this: ∃ x ∃ y ∃ g (( x = ℵ i +2 ) L ∧ ( y = ℵ i ) L ∧ g “is an injective function” ∧ dom( g ) = x ∧ ∀ α ∈ x ∀ z ∈ g ( α ) z ∈ y ) Note that this is a Σ -formula. The main reason for this choice of formalisationis that the semantics of the existential quantifier is local, which will allow us toprove the following crucial observation. Proposition 36.
Let K be a Kripke frame and M a sound assignment thatagrees on L . Then K ( M ) , v (cid:13) ψ i if and only if M v (cid:15) ψ i , i.e., the sentences ψ i are evaluated locally.Proof. This follows from Lemma 29, Proposition 18 and the fact that the se-mantics of the existential quantifier is local, i.e., the sets x , y and g of the abovestatement must (or may not) be found within M v . In this situation, it sufficesto argue that the following conjunction is evaluated locally: ( x = ℵ i +2 ) L ∧ ( y = ℵ i ) L ∧ g “is an injective function” ∧ dom( g ) = x ∧ ∀ α ∈ x ∀ z ∈ g ( α ) z ∈ y.
15t suffices to argue that every conjunct is evaluated locally. For the first twoconjuncts of the form ϕ L this holds by Lemma 29. The final three conjuncts are ∆ -formulas. So we can apply Proposition 18 and the desired result follows.We will now obtain a collection of models of set theory using the forcingnotions from S. Friedman, Fuchino and Sakai in [11]. From this collection, wedefine models with classical domains by constructing sound assignments thatagree on L . Construction 37.
We begin by setting up the forcing construction. By ourassumption that there is a countable transitive model of set theory, we canchoose a minimal countable ordinal α such that L α is a model of ZFC + V =L . We fix this α for the rest of the article. Let Q β,n be the forcing notion Fn( ℵ L β + n +2 , , ℵ L β + n ) , defined within L α . Given A ⊆ ω , we define the followingforcings: P Aβ,n = ( Q β,n , if n ∈ A, , otherwise.Then let P Aβ = Q n<ω P Aβ,n be the full support product of the forcing notions P Aβ,n . Recall that the ordering < on P Aβ is defined by ( a i ) i ∈ ω < ( b i ) i ∈ ω if andonly if a i < i b i for all i ∈ ω . Now, let G β be P ωβ -generic over L , and let G β,n = π n [ G ] be the n -th projection of G β . Let H be the trivial generic filteron the trivial forcing . Now, for A ⊆ ω and n ∈ ω define the collection offilters: G Aβ,n = ( G β,n , if n ∈ A,H, otherwise,and let G Aβ = Q n<ω G Aβ,n . Proposition 38.
The filter G Aβ is P Aβ -generic over L α . Proposition 39. If A ⊆ B ⊆ ω and A ∈ L[ G Bβ ] , then L[ G Aβ ] ⊆ L[ G Bβ ] . Indeed, L[ G Aβ ] is an inner model of L[ G Bβ ] . The additional assumption A ∈ L[ G B ] is necessary because there are forcingextensions that cannot be amalgamated (see [12, Observation 35] for a discussionof this). The following generalised proposition of S. Friedman, Fuchino and Sakaiis crucial for our purposes. Proposition 40 (S. Friedman, Fuchino and Sakai, [11, Proposition 5.1]) . Let β be an ordinal, i ∈ ω and A ⊆ ω . Then L α [ G Aβ ] (cid:15) ψ β + i if and only if i ∈ A . The notation
Fn(
I, J, λ ) is introduced by Kunen in [17, Definition 6.1] and denotes theset of all partial functions p : I → J of cardinality less than λ ordered by reversed inclusion. In different terminology, the statement of the following proposition is that the sentences ψ i constitute a family of so-called independent buttons for set-theoretical forcing. This termi-nology originates from the modal logic of forcing, see the article [14] of Hamkins and Löwe. roof. S. Friedman, Fuchino and Sakai prove this proposition for the case β = 0 .The generalised version can be proved in exactly the same way.This concludes our preparatory work, and we can state our main technicaltool of this section as the following theorem. Theorem 41.
Let β < α be an ordinal, ( K, ≤ ) be a Kripke frame and f : K →P ( ω ) be a monotone function such that f ( v ) ∈ L α for all v ∈ K . Then there isa sound assignment M that agrees on L α such that K ( M ) , v (cid:13) ψ i if and onlyif there is j ∈ f ( v ) such that i = β + j .Proof. Let ( K, ≤ ) be a Kripke frame and f : K → P ( ω ) be a function such that f ( v ) ∈ L α for all v ∈ K . Let M v = L α [ G f ( v ) β ] . This is a well-defined soundassignment that agrees on L by Proposition 39. By Proposition 40, it holdsthat i ∈ f ( v ) if and only if M v (cid:15) ψ i . Proposition 36 implies that the latter isequivalent to K ( M ) , v (cid:13) ψ i . The result follows. We are now ready to prove a rather general result on the logics for which
IKP satisfies the de Jongh property. The essential idea is to transform a Kripkemodel for propositional logic into a Kripke model for set theory in such a waythat the models exhibit very similar logical properties. In particular, if thelogical model does not force a certain formula ϕ , then we will construct a set-theoretic model and a translation τ such that the set-theoretic model will notforce ϕ τ .Recall that an intermediate logic J is called Kripke-complete if there is aclass of Kripke frames C such that J = L ( C ) . Theorem 42.
Let T ⊆ IKP + + MP + AC be a set theory. If J is a Kripke-complete intermediate propositional logic, then L Σ ( T ( J )) = J .Proof. The inclusion from right to left follows directly from the definition of T ( J ) . We show the converse inclusion by contraposition. So assume that thereis a formula ϕ in the language of propositional logic such that J ϕ . By ourassumption that J is Kripke-complete, there is a Kripke model ( K, ≤ , V ) suchthat ( K, ≤ , V ) (cid:13) J but ( K, ≤ , V ) (cid:13) ϕ . Without loss of generality, we canassume that the propositional letters appearing in ϕ are p , . . . , p n . We definea function f : K → P ( N ) by stipulating that: i ∈ f ( v ) if and only if i ≤ n and ( K, ≤ , V ) , v (cid:13) p i . In particular, f ( v ) is finite and thus f ( v ) ∈ L for every v ∈ K . Apply Theo-rem 41 to get a sound assignment M that agrees on L such that K ( M ) , v (cid:13) ψ i if and only if i ∈ f ( v ) .Let σ : Prop → L sent ∈ be the map p i ψ i . It follows via an easy inductionon propositional formulas that K ( M ) , v (cid:13) χ σ if and only if ( K, ≤ , V ) , v (cid:13) χ . Inparticular, K ( M ) (cid:13) ϕ σ but K ( M ) (cid:13) T ( J ) . Hence, ϕ / ∈ L ( T ( J )) .17 orollary 43. Every set theory T ⊆ IKP + + MP + AC has the de Jongh propertywith respect to every Kripke-complete intermediate propositional logic J , i.e., L ( T ( J )) = J . De Jongh, Verbrugge and Visser [7] proved a similar result for Heyting arith-metic HA , namely, that L ( HA ( J )) = J holds for every intermediate propositionallogic J which possesses the finite frame property. Passmann [25] showed that L ( IZF ( J )) = J holds for every intermediate logic J that is complete with respectto a class of finite trees. Our present Corollary 43, however, applies to a muchbroader class of logics: all intermediate logics that are complete with respect toa class of Kripke frames. When it comes to first-order logics, several intricacies arise that concern theinterplay of the logics and the surrounding set theory. We were able to ignorethese intricacies in the previous section when we were dealing with propositionallogics because we effectively reduced the problem to finitely many propositionalletters. In the case of first-order logic, however, we need to deal with infinitedomains and predication.The basic idea remains the same: We will construct a set-theoretical modelbased on a Kripke model for first-order logic. This time, however, we also needto deal with domains and predication. We will see that working with relativeinterpretations allows us to easily adapt the proof of the previous section forour purposes here: We will use the statements ψ i to code domains of Kripkemodels for IQC as subsets of ω as well as coding which predications hold true. Theorem 44.
Let T ⊆ IKP + + MP + AC be a set theory. If J ∈ L α is anintermediate first-order logic such that L α (cid:15) “ J is a Kripke-complete logic in acountable language”, then QL Σ E ( T ( J )) = J .Proof. Again, the inclusion from right to left is trivial and we prove the otherdirection by contraposition.Let J ∈ L α be a first-order logic such that “ J is Kripke-complete” holds in L α . Let J ϕ for some first-order sentence ϕ . We have to find a map σ suchthat T ( J ) ( ϕ E ) σ .Work in L α . By the fact that J ϕ and that J is Kripke-complete, we knowthat there is first-order Kripke model M = ( K, ≤ , D, I ) ∈ L α such that M (cid:13) ϕ .As we work in a classical meta-theory, we apply the downward Löwenheim-Skolem-Theorem by coding M as first-order structure and assume without lossof generality that M is countable. Fix enumerations d : ω → S D of the unionof all domains of the model M , C : ω → L J of all constant symbols appearingin L J , R : ω → L J of all relation symbols appearing in L J , and F : ω → L J of all function symbols appearing in L J (in each case, if there are only finitelymany symbols, restrict the domain to some n ∈ ω ).Still working in L α , we will now code all information about M in sets ofnatural numbers. Without loss of generality, we can assume that D v ⊆ ω forall v ∈ K , and that the transition functions are inclusions. Fix now a map18 ·i : ω <ω → ω . For v ∈ K , we let k ∈ f ( v ) if and only if one of the followingcases holds true:(i) k = h , j i and j ∈ D v ,(ii) k = h , i, j , . . . , j n − i , R i is an n -ary relation symbol, j , . . . , j n − ∈ D v and v (cid:13) R i ( j , . . . , j n − ) . Observe that we have defined a function f : K → P ( ω ) . This f is monotonedue to the persistence property of Kripke models.Now work in V , and apply Theorem 41 to obtain a sounds assignment M that agrees on L such that K ( M ) , v (cid:13) ψ k if and only if k ∈ f ( v ) . We define atranslation σ :(i) if χ = Et , where E is the predicate of the relative translation, then ( Et ) σ = ψ h ,t σ i , and,(ii) if R i ( t , . . . , t n − ) is an n -ary relation symbol different from the existentialpredicate E , then R i ( t , . . . , t n − ) σ = ψ h ,i,t σ ,...,t σn − i .Note that the sentences ψ i are uniformly defined for i ∈ ω , and thereforethe translation σ is well-defined. With an easy induction on formulas χ in thelanguage of J we show that K ( M ) , v (cid:13) χ σ if and only if M, v (cid:13) χ . We can thenconclude that K ( M ) (cid:13) ϕ σ but K ( M ) (cid:13) J σ , i.e., ϕ / ∈ QL E ( T ) .The following corollary shows that the theorem covers many important cases.Recall that a logic is axiomatisable if it has a recursively enumberable axioma-tisation. Corollary 45.
Let T ⊆ IKP + + MP + AC be a set theory. If J is an axioma-tisable intermediate first-order logic that is ZFC -provably Kripke-complete, then QL Σ E ( T ( J )) = J .Proof. By Craig’s Lemma [4], we know that axiomatisable logics are recursivelyaxiomatisable, i.e., we can assume without loss of generality that J is a recursiveset. As recursive sets are ∆ -definable with parameter ω (as a coding of Turingmachines in arithmetic), it follows that J ∈ L ω +2 ⊆ L α . Hence, we can applyTheorem 44 and derive the desired result. Corollary 46.
Let T ⊆ IKP + + MP + AC be a set theory. The relative first-order logic of T is intuitionistic first-order logic IQC , i.e., QL E ( T ) = IQC . Inparticular, QL E ( IKP ) =
IQC . We give a few more examples of logics to which Corollary 45 applies. Tothis end, note that KF is the following scheme: ¬¬∀ x ( P ( x ) ∨ ¬ P ( x )) . Moreover,
QHP k is the first-order logic of frames of depth at most k , and QLC is the first-order logic of linear frames. For more on these logics, we refer thereader to the book of Gabbay, Shehtman and Skvortsov [13].19 orollary 47.
Let T ⊆ IKP + + MP + AC be a set theory. It holds that QL Σ E ( T ( J )) = J in case that J is one of IQC + KF , QHP k or QLC .Proof.
This follows from Corollary 45 and the respective completeness theoremsfrom [13] (see [13, Theorem 6.3.5] for the completeness of
IQC + KF , [13, Theo-rem 6.3.8] for completeness of
QHP k , and [13, Theorem 6.7.1] for completenessof QLC ). The most important result of this section is that the first-order logic of
IKP is intuitionistic first-order logic, i.e., QL ( IKP ) =
IQC . We will show this bygeneralising the argument of the previous sections. Our first step will be toconstruct the necessary Kripke models for set theory.Our approach in this section will be somewhat different from what we did inthe previous two sections. As we now have to deal with unrestricted quantifi-cation, we have to give up the idea of coding directly into the classical models M v which propositions or predications must be true at a certain node. Rather,the idea is we will now encode enough information such that the models knowsinternally which predication must hold at which node. We remind the readerthat we consider IQC to be intuitionistic first-order logic without equality.
Construction 48.
Recall that we take L α to be the least transitive model of ZFC + V = L . Let ( K, ≤ , D, V ) ∈ L α be a well-founded rooted Kripke modelfor IQC . Work in L α . By Lemma 6 we can assume that there is a rootedwell-founded countable Kripke model ( K, ≤ , D, V ) with countably increasingdomains. Without loss of generality, we may assume that D v ⊆ ω for all v ∈ K ,and D v ⊆ D w for v ≤ w . Let D ∗ v = D v \ S w We call K ( M ) a mimic model of M = ( K, ≤ , D, V ) . Further,we say that γ is the essential ordinal of the mimic model K ( M ) .We will sometimes refer to ( K, ≤ , f, t ) as the coded model of K ( M ) . In thefollowing series of lemmas, we will spell out the way in which the mimic modelscan recover the information about the coded model.20 emma 50. There is a Σ -formula ϕ ess ( x ) in the language of set theory suchthat K ( M ) , v (cid:13) ϕ ess ( x ) if and only if x is the essential ordinal γ of K ( M ) .Proof. We define the formula ϕ ess ( x ) as follows: ϕ ess ( x ) ≡ x ∈ Ord ∧ ψ x ∧ ∀ β ∈ x ¬ ψ β By the definition of the mimic model K ( M ) , we know that K ( M ) (cid:13) ψ γ and K ( M ) (cid:13) ψ i for i < γ , i.e., K ( M ) (cid:13) ¬ ψ i for all i < γ . As being an ordinal can beexpressed by a ∆ -formula, it follows that K ( M ) (cid:13) γ ∈ Ord ∧ ψ γ ∧ ∀ β ∈ γ ¬ ψ β .Conversely, if K ( M ) , v (cid:13) ϕ ess ( x ) , then it follows that x ∈ M v is an ordinalsuch that M v (cid:15) ψ x and for all β < x and w ≥ v we have M v (cid:15) ¬ ψ β . By thedefinition of K ( M ) it must hold that x = γ . Lemma 51. There is a Σ -formula ϕ orig ( x ) in the language of set theory, usingthe essential ordinal γ as a parameter, such that K ( M ) , v (cid:13) ϕ orig ( x ) if and onlyif x is the coded model of K ( M ) (i.e., x = ( K, ≤ , f, t ) ).Proof. Consider the following formula: ϕ orig ( x, y ) ≡ “ x is the y -th element in the canonical well-ordering of L ” L Now, by Lemma 29, K ( M ) , v (cid:13) ϕ orig ( x, γ ) is equivalent to L α (cid:15) “ x is the γ -th element int he canonical well-rodering of L ” . The definition of the essential ordinal ensures that this is the case if and onlyif x = ( K, ≤ , f, t ) . To observe that ϕ orig ( x, y ) is a Σ -formula use the fact thatthe canonical well-ordering of L is Σ -definable (see [16, Lemma 13.19]). Lemma 52. There is a Σ -formula ϕ exists ( x, y ) in the language of set theory,using the coded model ( K, ≤ , f, t ) and the essential ordinal γ as parameters, suchthat K ( M ) , v (cid:13) ϕ exists ( x, y ) if and only if y ∈ K such that y ≤ v and x ∈ M y .Proof. Recall from Section 4.2 that M v is the model L α [ G A v γ ] where G A v γ is L α -generic for P A v γ and A v = { } ∪ { f ( w ) | w ≤ v } . Consider the followingformula: ϕ exists ( x, y ) ≡ ∃ P ∈ L( “ P = P Aγ where A = { } ∪ { f ( w ) | w ∈ K ∧ w ≤ y } ” L ∧ ∃ τ ∈ L( “ τ is a P -name” ∧ ∃ G ( G is generic for P and τ G = x ))) . Note the use of the parameters ( K, ≤ , f, t ) and γ , and observe that this formulais evaluated locally as it is constructed from ∆ -formulas, formulas relativisedto L and existential quantification.Let w ∈ K such that w ≤ v . By general facts about set-theoretical forcing, x ∈ M w = L α [ G A w γ ] if and only if there exists a P A w γ -name τ ∈ L α such that τ G Awγ = x . Equivalently, M v (cid:15) ϕ exists ( x, w ) , and in turn holds if and only if K ( M ) , v (cid:13) ϕ exists ( x, w ) , by our observation on local evaluation.21or the next lemma, we introduce some handy notation. Let M ∗ v = M v \ S w There is a Σ -formula ϕ birth ( x, y ) in the language of set theory,using the coded model ( K, ≤ , f, t ) and the essential ordinal γ as parameters, suchthat K ( M ) , v (cid:13) ϕ birth ( x, y ) if and only if y ∈ K such that y ≤ v and x ∈ M ∗ y .Proof. Let ϕ birth ( x, y ) be defined as follows: ϕ birth ( x, y ) ≡ y ∈ K ∧ ϕ exists ( x, y ) ∧ ∀ u ∈ K ( u < y → ¬ ϕ exists ( x, u )) . If w ≤ v and x ∈ M ∗ w , then it follows from the previous lemma that for all u < w , K ( M ) (cid:13) ϕ exists ( x, u ) , i.e., K ( M ) (cid:13) ¬ ϕ exists ( x, u ) . On the other hand,we clearly have v (cid:13) ϕ exists ( x, w ) and hence v (cid:13) ϕ birth ( x, w ) .Conversely, if v (cid:13) ϕ birth ( x, w ) for w ≤ v , it follows that x ∈ M w but x / ∈ M u for u < w . Hence, x ∈ M ∗ w . Lemma 54. There is a Σ -formula ϕ passed ( x ) in the language of set theory, us-ing the coded model ( K, ≤ , f, t ) as a parameter, such that K ( M ) , v (cid:13) ϕ passed ( x ) if and only if x ∈ K such that x ≤ v .Proof. Consider the following formula: ϕ passed ( x ) ≡ ψ f ( x ) The lemma now follows directly from the definition of the mimic model K ( M ) .We have now finished our preparations and can prove the following lemmawhich will show that the mimic model can imitate the predication of the codedmodel. This is a crucial step for connecting truth in the mimic model with truthin the coded model.Given x ∈ M v , let v x ∈ K be the unique node with x ∈ D ∗ v x and r x ∈ ω suchthat rank( x ) = λ + r x for some limit ordinal λ . Define a map g v : M v → D v by g v ( x ) = f v x ( r x ) . Further let p · q : L IQC → ω be a fixed Gödel coding function. Lemma 55. Let P be an n -ary predicate. There is a Σ -formula ϕ P (¯ x ) withparameters only x , . . . , x n − in the language of set theory such that K ( M ) , v (cid:13) ϕ P ( x , . . . , x n − ) if and only if ( K, ≤ , D, V ) , v (cid:13) P ( g v ( x ) , . . . , g v ( x n − )) .Proof. Let ϕ P ( x , . . . , x n ) be the following formula: ∃ K, ≤ , f, t, ¯ r, ¯ u, w, γ ( ϕ ess ( γ ) ∧ ϕ orig ( K, ≤ , f, t ) ∧ ^ i Let K ( M ) be a mimic model of a well-founded rooted Kripke model ( K, ≤ , D, V ) for IQC . For every formula ϕ in the language of first-order logic,we have that K ( M ) , v (cid:13) ϕ (¯ x ) τ if and only if ( K, ≤ , D, V ) , v (cid:13) ϕ ( g v (¯ x )) .Proof. This is proved by an induction on the complexity of ϕ for all v ∈ K . Theatomic cases has been taken care of in Lemma 55 and the cases for the logicalconnectives ∨ , ∧ and → follow trivially. We will now prove the cases for thequantifiers.First observe that the maps defined by g v are surjective. This is due to thefact that M ∗ v contains elements of rank λ + n for any n < ω . For the existential quantifier, assume that K ( M ) , v (cid:13) ( ∃ xϕ ( x, ¯ z )) τ . Thisis equivalent to the existence of some x ∈ M v such that K ( M ) , v (cid:13) ϕ τ ( x, ¯ z ) .By induction hypothesis, this is equivalent to the existence of some x ∈ M v such that ( K, ≤ , D, V ) , v (cid:13) ϕ ( g v ( x ) , g v (¯ z ))) . By the fact that g v is surjective,we know that the latter is equivalent to ( K, ≤ , D, V ) , v (cid:13) ∃ xϕ ( x, g v (¯ z )) .For the universal quantifier, observe that K ( M ) , v (cid:13) ( ∀ xϕ ( x, ¯ z ))) τ is equiv-alent to the fact that for all x ∈ M v it holds that K ( M ) , v (cid:13) ϕ τ ( x, g v (¯ z )) .By induction hypothesis this holds if and only if for all x ∈ M v we have ( K, ≤ , D, V ) , v (cid:13) ϕ ( g v ( x ) , g v (¯ z )) . Again, by using the surjectivity of g v , thisis equivalent to ( K, ≤ , D, V ) , v (cid:13) ∀ xϕ ( x, g v (¯ z )) .We are now ready to derive the final result. This can be shown via a construction starting with the generic x := G ∈ M ∗ v anditerating the operation x n := { x n } . Then take y = S n<ω x n and y n +1 := { y n } . It followsthat y n has rank λ + n for some limit ordinal λ . heorem 57. Let T ⊆ IKP + + MP + AC be a set theory. If J ∈ L α is anintermediate first-order logic that is ZFC -provably Kripke-complete with respectto a class of well-founded frames, then QL Σ1 ( T ( J )) = J .Proof. Let J ∈ L α be ZFC -provably Kripke-complete first-order logic. It isclear that J ⊆ QL ( T ( J )) . For the other direction, assume that J ϕ . Byour assumptions, there is a Kripke model ( K, ≤ , D, V ) ∈ L α such that ( K, ≤ , D, V ) (cid:13) ϕ . Due to Lemma 6 we can assume without loss of generality that ( K, ≤ , D, V ) has countably increasing domains. Let K ( M ) be a mimic modelobtained from ( K, ≤ , D, V ) . By Lemma 56 it follows that K ( M ) , v (cid:13) ϕ τ . As K ( M ) is a model of IKP + and T ⊆ IKP + , it follows that IKP + ϕ τ so that ϕ / ∈ QL ( IKP + ) . This finishes the proof of the theorem.We conclude this section by stating some important corollaries. Corollary 58. Let T ⊆ IKP + + MP + AC be a set theory. If J ∈ L α is anintermediate first-order logic that is ZFC -provably Kripke-complete with respectto a class of well-founded frames, then QL ( T ( J )) = J .Proof. As in the proof of Corollary 45, we use the fact that every axiomatisablefirst-order logic is contained in L α . The result then follows with Theorem 57. Corollary 59. Let T ⊆ IKP + + MP + AC be a set theory. The first-order logicof T is IQC , i.e., QL ( T ) = IQC . In particular, QL ( IKP ) = IQC .Proof. This follows from the fact that IQC is ZFC -provably Kripke-completewith respect to a class of well-founded frames (see the proof of [27, Theorem8.17]), and applying the previous corollary. Corollary 60. Let T ⊆ IKP + + MP + AC be a set theory. Then QL ( T ( QHP k )) = QHP k for k < ω .Proof. This follows from the fact that QHP k is complete with respect to theclass of frames of depth at most k (see [13, Theorem 6.3.8]). Given the results in the previous section, a natural question would be whetherthese results extend to logic with equality. In this section we show that that isnot the case. Theorem 61. Let T be a set theory based on intuitionistic logic containing theaxioms of extensionality, empty set and pairing. Then the first-order logic withequality of T , QL = ( T ) , is strictly stronger than IQC = , i.e., IQC = ( QL = ( T ) .Proof. Let ϕ denote the following formula in the language of IQC = : [ ∃ x ∃ y ∀ z ( z = x ∨ z = y )] → [ ∃ x ∀ z ( z = x )] . Intuitively, ϕ formalises the statement “if there are at most two objects, thenthere is at most one object.” Note that ϕ σ = ϕ holds for any first-order equal-ity translation σ into the language of set theory. By the principle of ex falso uodlibet it therefore suffices to show that the antecedent of ϕ is false in T . Letus call this antecedent ψ. We give an informal argument that can be easily transferred into a formalproof in the theory T . By pairing and emptyset, we can obtain the sets ∅ , {∅} , and {∅ , {∅}} . Suppose ψ . Then, by transitivity of equality, weknow that ∨ ∨ must hold. In each case, we can derivefalsum, ⊥ , using extensionality and the empty set axiom. With ∨ -eliminationand → -introduction, we conclude that ¬ ψ holds.This argument shows that ϕ ∈ QL = ( T ) . To finish the proof of the theorem,it is enough to show that ϕ / ∈ IQC = . This follows by completeness as follows.Consider the Kripke model for IQC = that consists of one node with a domainof two distinct points: the antecedent of ψ will be true in this model but theconsequent fails. Corollary 62. The first-order logic with equality of any set theory T consideredin this paper, such as IKP , IKP + + MP + AC , CZF and IZF , is stronger than IQC = . We close this section with the following question. Question 63. What is the first-order logic with equality QL = ( IKP ) of IKP ? 5. Conclusions and Open Questions We have seen that IKP is a very well-behaved theory from the logical pointof view—in fact, this applies to every subtheory of IKP + + MP + AC . This resultis also conceptually important: Constructive set theories are usually formulatedon the basis of IQC in such a way that the set-theoretic axioms should notstrengthen the logic, ensuring intuitionistic reasoning . We have shown that IKP indeed satisfies this requirement. Determining the first-order logic of CZF , andthus ensuring that CZF is logically and conceptually well-behaved, is an openproblem. Question 64. What is the first-order logic of CZF ? Is it the case that QL ( CZF ) = IQC ?Due to the failure of exponentiation (see Section 3.3.2) it is clear that ourtechniques above cannot directly be used to obtain the results of this article for CZF . With the semantics for CZF that the authors are currently aware of, itseems not possible to obtain mimic models for CZF .The situation for IZF is slightly different as Friedman and Ščedrov (see The-orem 34) showed that IQC ( QL ( IZF ) ( CQC . A challenging open problemis to give a better description of the first-order logic of IZF . Question 65. What is the first-order logic of IZF ? For example, is it possibleto give an axiomatisation of QL ( IZF ) or a concrete class of Kripke models thatcharacterise QL ( IZF ) ? 25 first step in this direction might be to determine the relative first-orderlogic of IZF .Moreover, our study also contributes to the analysis of the admissible rulesof the theory IKP : Knowing the logic of a theory is the first step in analysing itsadmissible rules. For example, as we have shown that L ( IKP ) = IPC , it followsthat any propositional rule that is admissible in IKP must be admissible in IPC as well. It remains to determine the lower bound. Question 66. What are the admissible rules of IKP ? References [1] Peter Aczel and Michael Rathjen. Notes on Constructive Set Theory . 2010.Draft.[2] Mohammad Ardeshir and S. Mojtaba Mojtahedi. The de Jongh propertyfor basic arithmetic. Archive for Mathematical Logic , 53(7):881–895, Nov2014.[3] Jon Barwise. Admissible sets and structures . Springer-Verlag, Berlin-NewYork, 1975. 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