aa r X i v : . [ m a t h . L O ] J u l Not all Kripke models of HA are locally PA Erfan Khaniki ∗ Faculty of Mathematics and Physics, Charles University Institute of Mathematics, Czech Academy of SciencesJuly 10, 2020
To Mohammad Ardeshir
Abstract
Let K be an arbitrary Kripke model of Heyting Arithmetic, HA . For every node k in K ,we can view the classical structure of k , M k as a model of some classical theory of arithmetic.Let T be a classical theory in the language of arithmetic. We say K is locally T , iff for every k in K , M k | = T . One of the most important problems in the model theory of HA is thefollowing question: Is every Kripke model of HA locally PA ? We answer this question negatively.We introduce two new Kripke model constructions to this end. The first construction actuallycharacterizes the arithmetical structures that can be the root of a Kripke model K (cid:13) HA + ECT ( ECT stands for Extended Church Thesis). The characterization says that for every arithmeticalstructure M , there exists a rooted Kripke model K (cid:13) HA + ECT with the root r such that M r = M iff M | = Th Π ( PA ). One of the consequences of this characterization is that there isa rooted Kripke model K (cid:13) HA + ECT with the root r such that M r = I ∆ and hence K isnot even locally I ∆ . The second Kripke model construction is an implicit way of doing thefirst construction which works for any reasonable consistent intuitionistic arithmetical theory T with a recursively enumerable set of axioms that has the existence property. We get a sufficientcondition from this construction that describes when for an arithmetical structure M , thereexists a rooted Kripke model K (cid:13) T with the root r such that M r = M . As applications ofthis sufficient condition, we construct two new Kripke models. The first one is a Kripke model K (cid:13) HA + ¬ θ + MP ( θ is an instance of ECT and MP is Markov’s principle) which is not locally I ∆ . The second one is a Kripke model K (cid:13) HA such that K forces exactly the sentencesthat are provable from HA , but it is not locally I ∆ . Also, we will prove that every countableKripke model of intuitionistic first-order logic can be transformed into another Kripke modelwith the full infinite binary tree as the Kripke frame such that both Kripke models force thesame sentences. So with the previous result, there is a binary Kripke model K of HA such that K is not locally I ∆ . Heyting Arithmetic ( HA ) is the intuitionistic counterpart of Peano Arithmetic ( PA ). HA has thesame non-logical axioms as PA with intuitionistic first-order logic as the underlying logic. Thistheory is one of the well-known and most studied theories of constructive mathematics, and it wasinvestigated in many proof-theoretic and model-theoretic aspects in the literature (see [20] for moreinformation). This paper aims to answer a question about the model theory of HA . Let T be aclassical theory in the language of arithmetic. A Kripke model of HA is called locally T , iff forevery node k ∈ K , the classical structure M k associated with k , is a model of T . One of the mostimportant problems in the model theory of HA is the following question: ∗ [email protected] roblem 1.1 Is every Kripke model of HA locally PA ? This problem was first asked and investigated in the seminal paper [21] by van Dalen et al. in 1986.They proved that every finite Kripke model of HA is locally PA . Furthermore, they proved thata Kripke model of HA with the Kripke frame ( ω, ≤ ) as the underlying frame has infinitely manylocally PA nodes. This work initiated a research line into Problem 1.1 and also about the followinggeneral question: Problem 1.2
For a Kripke model K of the theory T in a language σ , and a node k ∈ K , what isthe relationship between the sentences forced in k and the sentences satisfied in M k ? There are several works that deal with these problems. We will review those works in the followingparagraphs. Wehmeier in [22], investigated Problem 1.1 and extended the results of [21] to a largerclass of frames. In particular, he proved that every Kripke model of HA with ( ω, ≤ ) as the Kripkeframe is indeed locally PA . Moniri, in [14], considered these problems and proved that every once-branching Kripke model of HA + MP (Markov’s principle) is locally PA . Ardeshir and Hesaam in [4]generalized the results of [22] to rooted narrow tree Kripke models of HA . Recently, Mojtahedi in[13] considered Problem 1.2 and answered this problem in the case of finite depth Kripke models.As an application, he generalized the result of [4] to rooted semi-narrow tree Kripke models of HA .Regarding Problem 1.1, the strongest positive result about the strength of induction axioms thatare true in a node of a Kripke model of HA was proved by Markovi in [12]. He proved that everynode of a Kripke model of HA satisfies induction for formulas that are provably ∆ in PA . Also,from Π conservativity of PA over HA (see [10]), we know that every Kripke model of HA is locally Th Π ( PA ).Buss studied another question related to these problems in [8]. For every language σ and everyclassical theory T in it, he characterized the sentences that are true in every locally T Kripkemodel. As a result, he proved that HA is complete with respect to the locally PA Kripke models.In a similar direction, Ardeshir et al. in [7] presented a set of axiom systems for the class of end-extension Kripke models. As an application, they proved that HA is strongly complete for its classof end-extension Kripke models. For the case of fragments of HA , Problem 1.1 was investigated andanswered negatively in [15].To best of our knowledge, the above theorems are all results relevant to Problem 1.1 in theliterature. There are some other papers such as [1, 2] that investigated Problem 1.2 in general andpartially answered this question.In this paper, we will present two new model construction to answer Problems 1.1 and 1.2. Themain technical theorem of the first construction says that the theory HA + ECT + Diag ( M ) forevery M | = Th Π ( PA ) has the existence and disjunction properties (Theorem 3.5). This theoremprovides the right tool for constructing rooted Kripke models of HA with control over the structureof the root (Theorem 3.8). This construction theorem moreover characterizes the necessary andsufficient conditions for an arithmetical structure M to be the root of a Kripke model of HA + ECT (Corollary 3.9). Using this characterization we will construct a Kripke model of HA + ECT thatis not even locally I ∆ . This answers Problem 1.1 negatively. Moreover, this is optimal, becauseit is well-known that every node of a Kripke model of HA satisfies induction for formulas that areprovably ∆ in PA . The second construction is an implicit way of doing the first construction and itworks for any reasonable consistent intuitionistic arithmetical theory with a recursively enumerableset of axioms that has the existence property (Theorem 3.16). This construction gives us a sufficientcondition for an arithmetical structure M to be the root of a Kripke model of T . As applications ofthis sufficient condition, we will construct two new Kripke models. The first one is a Kripke modelof HA + ¬ θ + MP where θ is an instance of ECT and MP is Markov’s principle that is not locally I ∆ (Corollary 3.19). The second one is a Kripke model of HA that forces exactly all sentences thatare provable in HA , but it is not locally I ∆ (Corollary 3.20).The second construction is general and also works for HA + ECT , but some Kripke models can beconstructed for HA + ECT with the first construction, but not possible with the second one. We will2iscuss this matter in more detail at the end of Section 3. The new model constructions imply theexistence of a large class of Kripke models of reasonable intuitionistic arithmetical theories including HA , which cannot be constructed by previous methods, so we think that these model constructionsare interesting in their own rights.We will also prove that every countable Kripke model of intuitionistic first-order logic can betransformed into another Kripke model with the full infinite binary tree as the Kripke frame (Lemma4.1). Using this result, we will prove that there exists a Kripke model of HA with the full infinitebinary tree as the Kripke frame that is not locally I ∆ (Corollary 4.3). Let L be the language of Primitive Recursive Arithmetic in which it has a function symbol for everyprimitive recursive function. HA is the intuitionistic theory with the following non-logical axioms:1. Axioms of Robinson Arithmetic Q .2. Axioms defining the primitive recursive functions.3. For each formula φ ( x, ~y ) ∈ L , the axiom ∀ ~y I φ in which I φ := φ (¯0) ∧ ∀ x ( φ ( x ) → φ ( Sx )) → ∀ xφ ( x ) . PA is the classical theory that has the same non-logical axioms as HA . i PRA (intuitionistic PrimitiveRecursive Arithmetic) has axioms of Q and induction for every atomic formula of L . The underlyinglogic of i PRA is intuitionistic logic.
PRA is the classical counter part of i PRA . T ⊢ c φ means thatthere exists a proof of φ from axioms of T using first-order classical logic Hilbert system. ⊢ i denotesthe same thing for intuitionistic proofs. An important set of intuitionistic arithmetical theories forthe purpose of this paper is defined in the following definition. Definition 2.1 I is the set of all intuitionistic arithmetical theories T in L such that:1. T is consistent.2. i PRA ⊆ T .3. The set of axioms of T is recursively enumerable. Note that with the power of primitive recursive functions we can define finite sequences of numbers,so we can code finite objects such as formulas, proofs, and etc. as numbers. This is a standardtechnique and it is called G¨odel numbering (see [19]). With the help of this coding we can talkabout proofs of theories in arithmetical theories (see [19]). For every L sentence φ , p φ q denotes thenumber associated with φ . If φ ( x ) is an L formula, then p φ ( ˙ c ) q denotes the number associated with ψ ( x ) when we substitute the numeral with value c for x . Suppose T ∈ I . Let Axiom ( x, y ) be theprimitive recursive function such that for every L sentence φ , φ is a T -axiom iff ∃ x Axiom ( x, p φ q ) istrue. Then it is possible to define the provability predicate of T , Proof T ( x, y ) as a primitive recursivepredicate as follows. Let h . i be a natural primitive recursive coding function. Then Proof T ( x, y ) istrue iff there exist two sequences L sentences { φ i } i ≤ n and numbers { w i } i ≤ n for some n such that:1. x = hh w , p φ q i , ..., h w n , p φ n q ii .2. For every i ≤ n :(a) If w i >
0, then
Axiom ( w i − , p φ i q ) is true.3b) If w i = 0, then φ i can be derived from { φ j } j
For every T ∈ I the following statements are true:1. For every L sentence φ , if T ⊢ i φ , then PRA ⊢ c Pr T ( p φ q ) .2. PRA ⊢ c ∀ x, y ( Pr T ( x ) ∧ Pr T ( x → y ) → Pr T ( y )) .3. PRA ⊢ c ∀ x, y ( Pr T ( x ) ∧ Pr T ( y ) → Pr T ( x ∧ y )) .4. For every L formula φ ( x ) with x as the only free variable, PRA ⊢ c Pr T ( p ∀ xφ ( x ) q ) → ∀ x Pr T ( p φ ( ˙ x ) q ) .5. For every Σ formula φ ( ~x ) , PRA ⊢ c ∀ ~x ( φ ( ~x ) → Pr T ( p φ ( ˙ x , ..., ˙ x n ) q )) . Proof.
See [19] for a detailed discussion of these statements. ⊣ For proving the first model construction theorem, we need some definitions and theorems aboutKleene’s realizability.
Definition 2.3
Let T ( x, y, z ) be the primitive recursive function called Kleene’s T-predicate and U ( x ) be the primitive recursive function called result-extracting function. Note that HA ⊢ i ∀ x, y, z, z ′ ( T ( x, y, z ) = 0 ∧ T ( x, y, z ′ ) = 0 → z = z ′ ) . We use T ( x, y, z ) instead of T ( x, y, z ) = 0 for simplicity. For more information, see section 7 of thethird chapter of [20]. Let j ( x ) and j ( x ) be the primitive recursive projections of the pairing function j ( x, y ) = 2 x · (2 y +1) ∸
1. Kleene’s realizability is defined as follows.
Definition 2.4 x r φ ( x realizes φ ) is defined by induction on the complexity of φ where x F V ( φ ) .1. x r p := p for atomic p ,2. x r ( ψ ∧ η ) := j ( x ) r ψ ∧ j ( x ) r η ,3. x r ( ψ ∨ η ) := ( j ( x ) = 0 ∧ j ( x ) r ψ ) ∨ ( j ( x ) = 0 ∧ j ( x ) r η ) ,4. x r ( ψ → η ) := ∀ y ( y r ψ → ∃ u ( T ( x, y, u ) ∧ U ( u ) r η ) , u F V ( η ) ,5. x r ∃ yψ ( y ) := j ( x ) r ψ ( j ( x )) ,6. x r ∀ yψ ( y ) := ∀ y ∃ u ( T ( x, y, u ) ∧ U ( u ) r ψ ( y )) , u F V ( ψ ) . Definition 2.5
A formula φ ∈ L is almost negative iff φ does not contain ∨ , and ∃ only immediatelyin front of atomic formulas. Definition 2.6
The extended Church’s thesis is the following schema, where φ is almost negative: ECT := ∀ ~v ( ∀ x ( φ ( x, ~v ) → ∃ yψ ( x, y, ~v )) → ∃ z ∀ x ( φ ( x, ~v ) → ∃ u ( T ( z, x, u ) ∧ ψ ( x, U ( u ) , ~v )))) . HA , ECT and Kleene’s realizability. Theorem 2.7
For every formula φ ∈ L :1. HA + ECT ⊢ i φ ↔ ∃ x ( x r φ ) ,2. HA + ECT ⊢ i φ ⇔ HA ⊢ i ∃ x ( x r φ ) . Proof.
See Theorem 4.10 in the fourth chapter of [20]. ⊣ Another important properties of HA are the existence and disjunction properties. We will usenotation ¯ n as the syntactic term corresponds to natural number n . Theorem 2.8
The following statements are true:1. Disjunction property: For every sentences φ, ψ ∈ L , if HA ⊢ i φ ∨ ψ , then HA ⊢ i φ or HA ⊢ i ψ ,2. Existence property: For every sentence ∃ xφ ( x ) ∈ L , if HA ⊢ i ∃ xφ ( x ) , then there exists anatural number n such that HA ⊢ i φ (¯ n ) . Proof.
See Theorem 5.10 of the third chapter of [20]. ⊣ Although HA is an intuitionistic theory, it can prove decidability of some restricted class offormulas. The next theorem explains this fact. Theorem 2.9
For every quantifier free formula φ ∈ L , HA ⊢ i φ ∨ ¬ φ . Proof.
See [20]. ⊣ A Kripke model for a language σ is a triple K = ( K, ≤ , M ) such that:1. ( K, ≤ ) is a nonempty partial order.2. For every k ∈ K , M k ∈ M is a classical structure in the language σ ( M k ) = σ ∪ { c | c ∈ M k } .3. For every k, k ′ ∈ K , if k ≤ k ′ , then σ ( M k ) ⊆ σ ( M k ′ ) and also M k ′ | = Diag + ( M k ) ( M k is asub-structure of M k ′ ).For every Kripke model K , there is a uniquely inductively defined relation (cid:13) ⊆ K × (cid:0)S k ∈ K σ ( M k ) (cid:1) that is called forcing. Definition 2.10
For every k ∈ K , and every sentence φ ∈ σ ( M k ) , the relation k (cid:13) φ is defined byinduction on complexity of φ :1. k (cid:13) p iff M k | = p , for atomic p ,2. k (cid:13) ψ ∧ η iff k (cid:13) ψ and k (cid:13) η ,3. k (cid:13) ψ ∨ η iff k (cid:13) ψ or k (cid:13) η ,4. k (cid:13) ¬ ψ iff for no k ′ ≥ k , k ′ (cid:13) ψ ,5. k (cid:13) ψ → η iff for every k ′ ≥ k , if k ′ (cid:13) ψ , then k ′ (cid:13) η ,6. k (cid:13) ∃ xψ ( x ) iff there exists c ∈ σ M k such that k (cid:13) ψ ( c ) ,7. k (cid:13) ∀ xψ ( x ) iff for every k ′ ≥ k and every c ∈ σ ( M k ′ ) , k ′ (cid:13) ψ ( c ) . We use the notation K (cid:13) φ ( φ ∈ T k ∈ K σ ( M k ) is a sentence) as an abbreviation that for every k ∈ K , k (cid:13) φ which simply means that the Kripke model K forces φ . The important property ofthe forcing relation is its monotonicity. This means that for every k ′ ≥ k and every φ ∈ σ ( M k ), if k (cid:13) φ , then k ′ (cid:13) φ . Also, note that first-order intuitionistic logic is sound and is strongly completewith respect to the Kripke models. For more details see [20].5 Kripke model constructions for intuitionistic arithmeticaltheories
We will explain the first model construction in this subsection. This construction will be presentedin a sequence of lemmas and theorems.
Lemma 3.1
For every quantifier-free formula φ ∈ L there exists an atomic formula p ∈ L with thesame free variables such that HA ⊢ i φ ↔ p . Proof.
By induction on the complexity of φ and using Theorem 2.9. ⊣ Lemma 3.2
Let h . i and ( . ) x be a primitive recursive coding and decoding functions, then for everyformula Qx , ..., x n φ ( ~x, ~y ) ∈ L where Q ∈ {∀ , ∃} and n > , HA ⊢ i Qx , ..., x n φ ( ~x, ~y ) ↔ Qxφ (( x ) , ..., ( x ) n , ~y ) . Proof.
Straightforward by properties of the coding and decoding functions. ⊣ We will use notation φ ([ x ] , ~y ) instead of φ (( x ) , ..., ( x ) n , ~y ) for simplicity. Theorem 3.3
For every Π sentence φ := ∀ ~x ∃ ~yψ ( ~x, ~y ) , if HA + ECT ⊢ i φ , then PA ⊢ c φ . Proof.
Let φ be a Π sentence and HA + ECT ⊢ i φ . By Lemmas 3.2 and 3.1 there exists an atomicformula p ( x, y ) such that HA ⊢ i φ ↔ ∀ x ∃ yp ( x, y ) and therefore HA + ECT ⊢ i ∀ x ∃ yp ( x, y ). ByTheorem 2.7 HA ⊢ i ∃ n ( n r ∀ x ∃ yp ( x, y )). Because ∃ n ( n r ∀ x ∃ yp ( x, y )) is a sentence, there exists anatural number n such that HA ⊢ i ¯ n r ∀ x ∃ yp ( x, y ). Therefore by definition of the realizability:1. ⇒ HA ⊢ i ∀ x ∃ u ( T (¯ n, x, u ) ∧ U ( u ) r ∃ yp ( x, y )) , ⇒ HA ⊢ i ∀ x ∃ u ( T (¯ n, x, u ) ∧ j ( U ( u )) r p ( x, j ( U ( u )))) , ⇒ HA ⊢ i ∀ x ∃ u ( T (¯ n, x, u ) ∧ p ( x, j ( U ( u )))) , ⇒ HA ⊢ i ∀ x ∃ up ( x, u ) , hence PA ⊢ c φ . ⊣ In the rest of the paper, for every L structure M , T M := HA + ECT + Diag ( M ) . Theorem 3.4 If M | = Th Π ( PA ) , then T M is consistent. Proof.
Suppose T M is inconsistent, so there exists a finite number of L ( M ) sentences { φ i ( ~c i ) } i ≤ n ⊆ Diag ( M ) such that HA + ECT + V ni =1 φ i ( ~c i ) ⊢ i ⊥ , therefore HA + ECT ⊢ i ¬ V ni =1 φ i ( ~c i ). Because ~c i are not used in the axioms of HA + ECT , we have HA + ECT ⊢ i ∀ ~x , ..., ~x n ( ¬ V ni =1 φ i ( ~x i )). Note that ∀ ~x , ..., ~x n ( ¬ V ni =1 φ i ( ~x i )) is a Π sentence and therefore by Theorem 3.3, PA ⊢ c ∀ ~x , ..., ~x n ( ¬ V ni =1 φ i ( ~x i )).This implies that M | = ∀ ~x , ..., ~x n ( ¬ V ni =1 φ i ( ~x i )) and especially M | = ¬ V ni =1 φ i ( ~c i ), but by defini-tion of Diag ( M ) we know M | = V ni =1 φ i ( ~c i ) and this leads to a contradiction, hence T M is consistent. ⊣ If an L structure M satisfies a strong enough theory of arithmetic, then T M has actually theexistence and disjunction properties. 6 heorem 3.5 (Existence and disjunction properties). Suppose M is a model of Th Π ( PA ) , thenthe following statements are true:1. For every L ( M ) sentence ∃ zφ ( z ) such that T M ⊢ i ∃ zφ ( z ) , there exists a constant symbol c ∈ L M such that T M ⊢ i φ ( c ) .2. For every L ( M ) sentence φ ∨ ψ such that T M ⊢ i φ ∨ ψ , T M ⊢ i φ or T M ⊢ i ψ . Proof.
1. Suppose φ ( z ) is ψ ( z,~d ) such that ψ ( z, ~y ) is an L formula. By assumption of the theorem thereexists a finite number of L ( M ) sentences { φ i ( ~c i ) } i ≤ n ⊆ Diag ( M ) such that HA + ECT + n ^ i =1 φ i ( ~c i ) ⊢ i ∃ zψ ( z,~d ) , so HA + ECT ⊢ i V ni =1 φ i ( ~c i ) → ∃ zψ ( z,~d ). Because L ( M ) constants that appear in V ni =1 φ i ( ~c i ) →∃ zψ ( z,~d ) are not used in the axioms of HA + ECT , therefore HA + ECT ⊢ i ∀ ~y, ~x , ..., ~x n ( n ^ i =1 φ i ( ~x i , ~y ) → ∃ zψ ( z, ~y )) . Note that V ni =1 φ i ( ~x i , ~y ) is a quantifier free formula, hence by Lemma 3.1 there exists an atomicformula p such that HA ⊢ i p ( ~x , ..., x n , ~y ) ↔ V ni =1 φ i ( ~x i , ~y ). Also note that by Theorem 2.9 HA ⊢ i p ∨ ¬ p , hence HA + ECT ⊢ i ∀ ~y, ~x , ..., ~x n ∃ z ( p ( ~x , ..., ~x n , ~y ) → ψ ( z, ~y )) . By Lemma 3.2 HA + ECT ⊢ i ∀ x ∃ z ( p ([ x ]) → ψ ( z, [ x ])). Note that ∀ x ∃ z ( p ([ x ]) → ψ ( z, [ x ])) isan L sentence and therefore by Theorems 2.7 and 2.8 there exists a natural number n suchthat HA ⊢ i ¯ n r ∀ x ∃ z ( p ([ x ]) → ψ ( z, [ x ])) . By definition of realizability we get HA ⊢ i ∀ x ∃ u ( T (¯ n, x, u ) ∧ U ( u ) r ∃ z ( p ([ x ]) → ψ ( z, [ x ]))) . Note that HA ⊢ i ∀ x ∃ uT (¯ n, x, u ), hence PA ⊢ c ∀ x ∃ uT (¯ n, x, u ) and therefore M | = ∀ x ∃ uT (¯ n, x, u ).Let M | = e = D ~c , ..., ~c n , ~d E and M | = T (¯ n, e, f ) ∧ U ( f ) = g for some e, f, g ∈ M . This implies T (¯ n, e, f ) , U ( f ) = g ∈ Diag ( M ) and therefore we get T M ⊢ i T (¯ n, e, f ) ∧ g r ∃ z ( p ([ e ]) → ψ ( z, [ e ])) . By applying realizability definition we get T M ⊢ i j ( g ) r ( p ([ e ]) → ψ ( j ( g ) , [ e ])). Note that byTheorem 2.7, HA + ECT ⊢ i v r ( p [ x ] → ψ ( w, [ x ])) → ( p ([ x ]) → ψ ( w, [ x ])) , so T M ⊢ i p ([ e ]) → ψ ( j ( g ) , [ e ]) . Because p ([ e ]) ∈ Diag ( M ), we get T M ⊢ i ψ ( j ( g ) , [ e ]) and this implies T M ⊢ i ψ ( c, [ e ]) for some c ∈ L ( M ) such that M | = j ( g ) = c . 7. Suppose T M proves φ ∨ ψ , therefore T M ⊢ i ∃ x (( x = 0 → φ ) ∧ ( x = 0 → ψ )). By the previouspart there exists a constant symbol c ∈ L ( M ) such that T M ⊢ i ( c = 0 → φ ) ∧ ( c = 0 → ψ ).Note that c = 0 is an atomic formula, hence c = 0 ∈ Diag ( M ) or c = 0 ∈ Diag ( M ) and thisimplies T M ⊢ i φ or T M ⊢ i ψ . ⊣ Definition 3.6
Let M be an L structure and T be an intuitionistic theory in the language L ( M ) .For every L ( M ) sentence φ such that T i φ , fix a Kripke model K φ (cid:13) T such that K φ φ . The following definition is based on Smory´nski collection operation in [18].
Definition 3.7
Let M be an L structure and T be an intuitionistic theory in the language L ( M ) .Define S ( M , T ) = { φ ∈ L ( M ) | T i φ, φ is a sentence } . Define the universal model K ( M , T ) as follows. Take the disjoint union { K φ } φ ∈S ( M , T ) and then adda new root r with domain M r = M . Theorem 3.8 If M is a model of Th Π ( PA ) , then K ( M , T M ) is a well-defined Kripke model andfor every L ( M ) sentence φ , K ( M , T M ) (cid:13) φ ⇔ T M ⊢ i φ. Proof.
First note that by Theorem 3.4 T M ⊥ , hence S ( M , T M ) is not empty and therefore K ( M , T M ) has other nodes except r . To make sure that K ( M , T M ) is well-defined, we shouldcheck the three conditions in the definition of Kripke models. It is easy to see that the first twoconditions hold for K ( M , T M ). For the third condition, we need to show that for every node k = r , L ( M r ) ⊆ L ( M k ) and M k | = Diag + ( M r ). By definition of K ( M , T M ), L ( M r ) ⊆ L ( M k ) holds. Forthe condition M k | = Diag + ( M r ), note that T M ⊢ i Diag ( M ) which implies M k | = Diag ( M r ).( ⇒ ). Let K ( M , T M ) (cid:13) φ . If T M i φ , then K φ exists and K φ ⊆ K ( M , T M ). By the assumptionwe get K φ (cid:13) φ , but this leads to a contradiction by definition of K φ , hence T M ⊢ i φ .( ⇐ ). We prove this part by induction on the complexity of φ :1. φ = p : Note that if T M ⊢ i p , then p ∈ Diag ( M ). Because if p Diag ( M ), then ¬ p ∈ Diag ( M ), hence T M ⊢ i ⊥ which leads to a contradiction by Theorem 3.4. Therefore p ∈ Diag ( M ) and by the fact that M | = p we get K ( M , T M ) (cid:13) p .2. φ = ψ ∧ η : By the assumption we get T M ⊢ i ψ and T M ⊢ i η , therefore by the inductionhypothesis K ( M , T M ) (cid:13) ψ and K ( M , T M ) (cid:13) η , hence K ( M , T M ) (cid:13) ψ ∧ η .3. φ = ψ ∨ η : By Theorem 3.5 T M ⊢ i ψ or T M ⊢ i η , therefore by the induction hypothesis K ( M , T M ) (cid:13) ψ or K ( M , T M ) (cid:13) η , hence K ( M , T M ) (cid:13) ψ ∨ η .4. φ = ψ → η : By the assumption for every θ ∈ S ( M , T M ), K θ (cid:13) ψ → η , so for proving K ( M , T M ) (cid:13) ψ → η we only need to show that if r (cid:13) ψ , then r (cid:13) η . Let r (cid:13) ψ ,therefore we have K ( M , T M ) (cid:13) ψ , hence by the previous part, T M ⊢ i ψ . Note that Bythe assumption T M ⊢ i ψ → η , hence T M ⊢ i η and therefore by the induction hypothesis K ( M , T M ) (cid:13) η which implies r (cid:13) η .5. φ = ∃ xψ ( x ): By Theorem 3.5 there exists a constant symbol c ∈ L ( M ) such that T M ⊢ i ψ ( c ), therefore by the induction hypothesis K ( M , T M ) (cid:13) ψ ( c ), hence K ( M , T M ) (cid:13) ∃ xψ ( x ).6. φ = ∀ xψ ( x ): By the assumption for every θ ∈ S ( M , T M ), K θ (cid:13) ∀ xψ ( x ), so for proving K ( M , T M ) (cid:13) ∀ xψ ( x ) we only need to show that for every c ∈ M , r (cid:13) ψ ( c ). Let c ∈ M .By the assumption T M ⊢ i ∀ xψ ( x ), therefore T M ⊢ i ψ ( c ), hence by induction hypothesis K ( M , T M ) (cid:13) ψ ( c ). This implies that r (cid:13) ψ ( c ). Note that c is interpreted by c ∈ M ,hence k (cid:13) ψ ( c ). 8 From the last theorem, we can get the characterization of the structure of the roots of Kripkemodels of HA + ECT . Corollary 3.9
For every L structure M , there exists a rooted Kripke model K (cid:13) HA + ECT withthe root r such that M r = M iff M | = Th Π ( PA ) . Proof.
As we mentioned before, it is known that every Kripke model of HA is locally Th Π ( PA )which proves the left to the right direction.For the case of right to left direction, note that if M | = Th Π ( PA ), then by Theorem 3.8 K ( M , T M ) (cid:13) HA + ECT and moreover the classical structure attached to the root is M . ⊣ Now we have the right tool for constructing a counter example for Problem 1.1. In general wecan get a lot of new models for every M | = Th Π ( PA ). For our purpose, it is sufficient to know that Th Π ( PA ) c PA to get the result. The next two theorems established the stronger fact which says Th Π ( PA ) c I ∆ . I ∆ is a classical theory in the language L with the following non-logical axioms:1. Axioms of Robinson Arithmetic Q .2. Axioms defining the primitive recursive functions.3. ∆ induction: ∀ ~y [ ∀ x ( φ ( x, ~y ) ↔ ¬ ψ ( x, ~y )) → I φ ]for every Σ formulas φ, ψ ∈ L For stating the theorems we need also another arithmetical theory that is called B Σ with thefollowing non-logical axioms:1. Axioms of Robinson Arithmetic Q .2. Axioms defining the primitive recursive functions.3. Induction for quantifier free formulas.4. Bounded Σ collection: ∀ ~y, x [ ∀ z ( z < x → ∃ wφ ( z, w, ~y )) → ∃ r ∀ z ( z < x → ∃ w ( w < r ∧ φ ( z, w, ~y ))]for every Σ formulas φ, ψ ∈ L It is worth mentioning that these theories usually are defined over the language of Peano Arithmetic,and not over the language of Primitive Recursive Arithmetic, hence our definitions of I ∆ and B Σ are stronger than the usual definition, but for our use this does not cause a problem. Now we knowthe definitions, we will state the theorems. Theorem 3.10 I ∆ c ⊣⊢ c B Σ . Proof.
As we explained before, this version of these theories are stronger that the original ones.Therefore by the result of [17] these two theories are the same. ⊣ Theorem 3.11
There exists a model M | = Th Π ( N ) such that M = I ∆ . Proof.
By the result of [3] there exists a model M | = Th Π ( N ) such that M = B Σ , hence byTheorem 3.10 M = I ∆ too. ⊣ orollary 3.12 There exists a rooted Kripke model of HA + ECT which is not locally I ∆ . Proof.
By Theorem 3.11 there exists a model M | = Th Π ( N ) such that M = I ∆ . Note that byTheorem 3.8, K ( M , T M ) (cid:13) HA + ECT , and also K ( M , T M ) is not locally I ∆ . ⊣ ECT is a very powerful non-classical axiom schema, so a natural question is that: Is it the casethat for every Kripke model K (cid:13) HA + ECT and every node k in K , M k = PA ? This question hasa negative answer, because K ( N , T N ) (cid:13) HA + ECT , but M r | = PA . In this subsection, we will explain the generalized construction which works for any reasonableintuitionistic arithmetical theory. We will also mention an application of it at the end of thissubsection.For every T ∈ I , the existence property of T is the following Π sentence: EP ( T ) := ∀ x ( x = p ∃ yφ ( y ) q for some formula φ ( y ) ∧ x is a sentence ∧ Pr T ( x ) → ∃ y Pr T ( p φ ( ˙ y ) q )) . For an L structure M and a theory T ∈ I , let extension of T with respect to M be the followingtheory: EXT ( M , T ) := { φ ∈ L ( M ) | φ is a sentence , M | = Pr T ( p φ q ) } . The following lemma states that
EXT ( M , T ) is closed under finite conjunctions. Lemma 3.13
Let M | = PRA and T ∈ I . Then for every L ( M ) sentences φ and ψ , if φ, ψ ∈ EXT ( M , T ) , then φ ∧ ψ ∈ EXT ( M , T ) . Proof. If φ, ψ ∈ EXT ( M , T ), then M | = Pr T ( p φ q ) ∧ Pr T ( p ψ q ), so by Theorem 2.2 M | = Pr T ( p φ ∧ ψ q ).Hence φ ∧ ψ ∈ EXT ( M , T ). ⊣ Define C M , T := T + EXT ( M , T ) . The crucial property of C M , T is the following lemma. Lemma 3.14
Suppose M | = PRA . Then for every T ∈ I and every L ( M ) sentence ψ , if C M , T ⊢ i ψ ,then M | = Pr T ( p ψ q ) . Proof.
Let ψ ( ~d ) be an L ( M ) sentence such that C M , T ⊢ i ψ ( ~d ). So there exists a finite number of L ( M ) sentence { φ i ( ~c i ) } i ≤ n ⊆ EXT ( M , T ) such that T ⊢ i n ^ i =1 φ i ( ~c i ) → ψ ( ~d ) . Because L ( M ) constants that appear in V ni =1 φ i ( ~c i ) → ψ ( ~d ) are not used in the axioms of T , therefore T ⊢ i ∀ ~y, ~x , ..., ~x n ( n ^ i =1 φ i ( ~x i , ~y ) → ψ ( ~y )) . So by Theorem 2.2 M | = Pr T ( p ∀ ~y, ~x , ..., ~x n ( n ^ i =1 φ i ( ~x i , ~y ) → ψ ( ~y )) q ) . Hence again by Theorem 2.2 M | = Pr T ( p n ^ i =1 φ i ( ~ ˙ c i ) → ψ ( ~ ˙ d ) q ) .
10n the other hand by Lemma 3.14
EXT ( M , T ) is closed under finite conjunctions, so V ni =1 φ i ( ~c i ) ∈ EXT ( M , T ) which means M | = Pr T ( p V ni =1 φ i ( ~ ˙ c i ) q ). So by Theorem 2.2 M | = Pr T ( p ψ ( ~ ˙ d ) q ). ⊣ Theorem 3.15
For every T ∈ I and every M | = PRA + EP ( T ) + Con ( T ) , the following statementsare true:1. C M , T is consistent.2. C M , T has the existence and disjunction properties. Proof.
1. Suppose C M , T ⊢ i ⊥ . Then by Lemma 3.14 M | = Pr T ( p ⊥ q ), but this is not possible becausewe assumed M | = Con ( T ), hence C M , T is consistent.2. We will prove the existence property of C M , T . The disjunction property will follow from it bythe same argument as in the proof of Theorem 3.5. Let ψ ( x ) be a formula in L ( M ) with only x as the free variable. Suppose C M , T ⊢ i ∃ xψ ( x ). Then by Lemma 3.14 M | = Pr T ( p ∃ xψ ( x ) q ).Note that M | = EP ( T ), hence M | = ∃ x Pr T ( p ψ ( ˙ x ) q ). This means there exists a c ∈ M suchthat M | = Pr T ( p ψ ( ˙ c ) q ). This implies ψ ( c ) ∈ EXT ( M , T ), so C M , T ⊢ i ψ ( c ). ⊣ This is the generalized version of the Theorem 3.8 which gives us the sufficient condition.
Theorem 3.16
Let T ∈ I and M | = PRA + EP ( T ) + Con ( T ) . Then K ( M , C M , T ) is a well-definedKripke model and for every L ( M ) sentence φ , K ( M , C M , T ) (cid:13) φ ⇔ C M , T ⊢ i φ. Proof.
The proof of this theorem is essentially the same as the proof of the Theorem 3.8 by usingthe Theorem 3.15. The only part that needs some extra work is the fact that C M , T ⊢ i Diag ( M ) andmoreover if C M , T ⊢ i p for atomic p , then p ∈ Diag ( M ).Let p ∈ Diag ( M ). We know by Theorem 2.2 M | = p → Pr T ( p p q ). This implies M | = Pr T ( p p q ).So p ∈ EXT ( M , T ) which implies C M , T ⊢ i p .Now if we have C M , T ⊢ i p for some atomic L ( M ) sentence p , then by Lemma 3.14 M | = Pr T ( p p q ).Note that M | = Con ( T ), so in presence of PRA , M | = Π - RFN ( T ) which Π - RFN ( T ) is the followingsentence: ∀ x ( x ∈ Π ∧ Pr T ( x ) → Tr ( x ))where Tr is a natural Π formula which works as the truth predicate for Π sentence. Substituting p p q for x in Π - RFN ( T ), we get M | = Tr ( p p q ), hence M | = p which means p ∈ Diag ( M ). ⊣ As we already see, using the first construction, we provide a Kripke model of HA + ECT whichis not locally I ∆ . A natural conjecture would be that the existence of such a Kripke model waspossible because the base theory has a very powerful non-classical schema ECT . As an applicationof Theorem 3.16 we will show this is not the case. Let H ( x ) be a Σ formula that is a naturalformalization of the statement ”The Turing machine with code x halts on input x ”. Let θ be aninstance of ECT in Definition 2.6 such that φ ( x ) := ⊤ and ψ ( x, y ) := ( y = 0 ∧ H ( x )) ∨ ( y = 0 ∧¬ H ( x )).We also need the definition of Markov’s principle. Definition 3.17
Markov’s principle is the following schema: MP := ∀ ~y ( ∀ x ( φ ( x, ~y ) ∨ ¬ φ ( x, ~y )) ∧ ¬¬∃ xφ ( x, ~y ) → ∃ xφ ( x, ~y )) . emma 3.18 The following statements are true:1. HA + ¬ θ + MP is consistent.2. HA + ¬ θ + MP has the existence and disjunction properties. Proof.
1. It is easy to see that PA ⊢ c ¬ θ and also PA ⊢ c MP . So HA + ¬ θ + MP is a sub-theory of PA and it is consistent.2. We will prove the existence property of HA + ¬ θ + MP here. The disjunction property willfollow from it like before. This part is a standard application of Kripke models (see [18]). Let ∃ xψ ( x ) be an L sentence such that HA + ¬ θ + MP ⊢ i ∃ xψ ( x ), but for every natural number n , HA + ¬ θ + MP i ψ (¯ n ). It is well-know that K ( N , HA + ¬ θ + MP ) is a well-defined Kripke modeland moreover K ( N , HA + ¬ θ + MP ) (cid:13) HA (see Theorem 5.2.4 in [18]). Moreover we can assumethat K ⊥ (Note that ⊥ ∈ S ( N , HA + ¬ θ + MP )) is a Kripke model with just one node withthe classical structure N . Note that r θ , because otherwise by the monotonicity of forcingrelation for every φ ∈ S ( N , HA + ¬ θ + MP ), K φ (cid:13) θ which is not true. Moreover for every node k = r , k (cid:13) ¬ θ , so with the last argument r (cid:13) ¬ θ which implies K ( N , HA + ¬ θ + MP ) (cid:13) ¬ θ . Notethat MP is forced in every node k = r . So we only need to show that r (cid:13) MP . For this mattersuppose r (cid:13) ∀ x ( φ ( x, ~ ¯ a ) ∨ ¬ φ ( x, ~ ¯ a )) ∧ ¬¬∃ xφ ( x, ~ ¯ a ) where ~a ∈ N . If for every n ∈ N , r φ (¯ n, ~ ¯ a ),then because of decidability of φ ( x, ~ ¯ a ) in the point of view of r , for every n ∈ N , r (cid:13) ¬ φ (¯ n, ~ ¯ a ).This implies K ⊥ (cid:13) ∀ x ¬ φ ( x, ~ ¯ a ). But this leads to a contradiction because K ⊥ (cid:13) ¬¬∃ x ¬ φ ( x, ~ ¯ a ).This means that there exists a natural number n such that r (cid:13) φ (¯ n, ~ ¯ a ).By the above arguments, we have K ( N , HA + ¬ θ + MP ) (cid:13) HA + ¬ θ + MP . So K ( N , HA + ¬ θ + MP ) (cid:13) ∃ xψ ( x ). This implies that there exists a natural number n such that r (cid:13) ψ (¯ n ). But this leads to a contradiction because we know K ψ (¯ n ) ψ (¯ n ). This implies thatour assumption was false and there exists a natural number n such that HA + ¬ θ + MP ⊢ i ψ (¯ n ). ⊣ The following corollary is the first application of Theorem 3.16.
Corollary 3.19
There exists a rooted Kripke model of HA + ¬ θ + MP which is not locally I ∆ . Proof.
By Theorem 3.11 there exists a model M | = Th Π ( N ) such that M = B Σ and hence byTheorem 3.10 M = I ∆ . Note that by Lemma 3.18 HA + ¬ θ + MP is consistent and has the existenceproperty. This implies that EP ( HA + ¬ θ + MP ) and Con ( HA + ¬ θ + MP ) are true in N . Note thatthese sentences are Π , so they are also true in M . This implies that M satisfies the conditionsneeded in the Theorem 3.16, hence K ( M , C M , HA + ¬ θ + MP ) (cid:13) HA + ¬ θ + MP and also it is not locally I ∆ . ⊣ It is worth mentioning that HA + ¬ θ + MP does not prove anything contradictory with PA andin some sense, it is close to PA , but still, we were able to construct a Kripke model of it which isnot locally I ∆ . The following corollary is the second application of Theorem 3.16. Corollary 3.20
There exists a rooted Kripke model K (cid:13) HA which is not locally I ∆ , but for every L sentence φ , K (cid:13) φ ⇔ HA ⊢ i φ. roof. Define U = {¬ Pr HA ( p φ q ) | HA i φ, φ is a sentence } . Let T := PRA + EP ( HA ) + U . It is easy to see that T is a Π axiomatized theory and moreover N | = T . By Theorem 3.11 there exists a model M | = Th Π ( N ) such that M = I ∆ . By the factsthat N | = T and also T is a Π axiomatized theory, we get M | = T . So by these explanations, M hasthe required property that is needed in Theorem 3.16, hence K ( M , C M , HA ) (cid:13) HA . This means thatfor every L sentence φ , if HA ⊢ i φ , then K ( M , C M , HA ) (cid:13) φ .For the opposite direction, let φ be an L sentence such that K ( M , C M , HA ) (cid:13) φ . Then by Theorem3.16 C M , HA ⊢ i φ . So by Lemma 3.14 M | = Pr HA ( p φ q ). If HA i φ , then ¬ Pr HA ( p φ q ) ∈ U , hence T ⊢ c ¬ Pr HA ( p φ q ) which implies M | = ¬ Pr HA ( p φ q ), but this leads to a contradiction, hence HA ⊢ i φ . ⊣ As we already mentioned in the Introduction, we can get more Kripke models for HA + ECT from the first construction than by the second construction. We will show this fact in the rest ofthis subsection. For this matter, we need the following theorem. Theorem 3.21
For any constant k , there is no consistent Π k -axiomatized theory T such that T ⊢ c PA . Proof.
See [11]. ⊣ Theorem 3.22
The following statements are true:1. For every L structure M , if K ( M , C M , HA + ECT ) (cid:13) HA + ECT , then K ( M , T M ) (cid:13) HA + ECT .2. There exists an L structure M such that K ( M , T M ) (cid:13) HA + ECT , but K ( M , C M , HA + ECT ) HA . Proof.
1. Suppose K ( M , C M , HA + ECT ) (cid:13) HA + ECT . Let φ := ∀ ~x ∃ ~yψ ( ~x, ~y ) be a Π sentence such that PA ⊢ c φ . Then by Π conservativity of HA over PA , we have HA ⊢ i φ , hence K ( M , C M , HA + ECT ) (cid:13) φ . This implies r (cid:13) ∀ ~x ∃ ~yψ ( ~x, ~y ). So for every ~a ∈ M :(a) ⇒ r (cid:13) ∃ ~yψ ( ~a, ~y ),(b) ⇒ there exist ~b ∈ M such that r (cid:13) ψ ( ~a,~b ),(c) ⇒ M | = ψ ( ~a,~b ).Hence M | = φ . This implies that M | = Th Π ( PA ), so by Theorem 3.8 K ( M , T M ) (cid:13) HA + ECT .2. By G¨odel’s second incompleteness theorem, PA + ¬ Con ( PA ) is consistent. So this impliesthat Th Π ( PA ) + ¬ Con ( HA ) is also consistent. Th Π ( PA ) + ¬ Con ( HA ) is a Π -axiomatizedtheory, hence by Theorem 3.21 there exists a model M | = Th Π ( PA ) + ¬ Con ( HA ) such that M = PA . Note that by Theorem 3.8 K ( M , T M ) (cid:13) HA + ECT . On the other hand M | = ¬ Con ( HA + ECT ), so ⊥ ∈ EXT ( M , HA + ECT ). This implies C M , HA + ECT ⊢ i ⊥ . Hence S ( M , C M , HA + ECT ) = ∅ . This means that K ( M , C M , HA + ECT ) has only one node r such that M r = M . Note that M = PA , so r (cid:13) HA and this completes the proof. ⊣ On binary Kripke models for intuitionistic first-order logic
In this section, we will prove that every countable rooted Kripke model K (there exists a node k in K such that for every k in K , k ≤ k ′ ) can be transformed to a Kripke model K ′ with the infinitefull binary tree as Kripke frame such that K and K ′ force the same sentences. This was known forthe case of finite Kripke models of intuitionistic propositional logic (see Theorem 2.21 and Corollary2.22 of [9]), but to best of our knowledge it was not mentioned for the case of Kripke models ofintuitionistic first-order logic in the literature. The transformation for Kripke models of intuitionisticfirst-order logic can be done in the same way that was done for the case of finite Kripke modelsof intuitionistic propositional logic, but for the sake of completeness we will state the theorem andprove it in this section.Let Γ = { , } and Γ ∗ be the set of all finite binary strings (including empty string λ ). For every x, y ∈ Γ ∗ , x (cid:22) y iff x is a prefix of y . Lemma 4.1
Let K = ( K, ≤ , M ) be a countable rooted Kripke model in a language σ . Then there isan onto function f : Γ ∗ → K , such that:1. K ′ = (Γ ∗ , (cid:22) , M ′ ) is a Kripke model where M ′ is defined as M ′ x = M f ( x ) for every x ∈ Γ ∗ ,2. for every k ∈ K , for every σ ( M k ) sentence φ , and for every x ∈ Γ ∗ such that f ( x ) = k , x (cid:13) φ iff k (cid:13) φ . Proof.
Without loss of generality, we can assume ( K, ≤ ) is a tree (see Theorem 6.8 in the secondchapter of [20]) with the root r . Also, we can assume that for every k ∈ K , there is a k ′ ∈ K differentfrom k such that k ≤ k ′ . This is true because for every k ∈ K that does not have relation with anyother nodes, we can put an infinite countable path above k such that the classical structure of everynode in this path is M k . This transformation does not change the sentences that were forced in theoriginal model. For every k ∈ K , define neighbor of k as N k = { k ′ ∈ K | k ≤ k ′ ∧ k = k ′ ∧ ∀ k ′′ ∈ K ( k ≤ k ′′ ∧ k ′′ ≤ k ′ → k = k ′′ ∨ k ′ = k ′′ ) } . For every k ∈ K , fix an onto function g k : N → N k such that for every k ′ ∈ N k , { n ∈ N | g k ( n ) = k ′ } is infinite. Now we define f inductively with a sequence of partial function f ⊂ f ⊂ ... and thenwe put f = S n ∈ N f n . Put f ( λ ) = r . For a function h , let Dom ( h ) be domain of h . Let A n = { x ∈ Γ ∗ | x ∈ Dom ( f n ) , x Dom ( f n ) , x Dom ( f n ) } . Now f n +1 is defined inductively from f n as follows: f n +1 ( x ) = f n ( x ) x ∈ Dom ( f n ) f n ( y ) x = y m , for some y ∈ A n , m ∈ N g f n ( y ) ( m ) x = y m , for some y ∈ A n , m ∈ N . It is easy to see that
Dom ( f ) = Γ ∗ . Claim 4.2
For every k ∈ K , for every x ∈ Γ ∗ if f ( x ) = k , then { k ′ ∈ K | k ≤ k ′ } = { f ( y ) ∈ K | y ∈ Γ ∗ , x (cid:22) y } . This claim is easy to prove considering the definition of f and the fact that g k functions enumerateneighbors infinitely many times.Using this claim, we can finish the proof. The proof goes by induction on the complexity of φ . We will only mention a nontrivial case in the induction steps. All other cases can be treatedsimilarly. Let φ := ψ → η and k (cid:13) ψ → η . Let x ∈ Γ ∗ be such that f ( x ) = k . Suppose for some y (cid:23) x , we know y (cid:13) ψ . So by the induction hypothesis, f ( y ) (cid:13) ψ and by Claim 4.2, we know f ( y ) ≥ k , hence f ( y ) (cid:13) η , therefore by the induction hypothesis we get y (cid:13) η , so x (cid:13) φ . ⊣ orollary 4.3 There exists a Kripke model of HA with (Γ ∗ , (cid:22) ) as the Kripke frame that is notlocally I ∆ . Proof.
Let K be a rooted Kripke model with the root r in a language σ . Let U be a countableset of sentences of σ . It is easy to see that K can be represented by a suitable two-sorted classicalstructure M K such that:1. For every φ ∈ U , ” r (cid:13) φ ” is first-order definable in M K by the sentence φ F .2. For every φ ∈ U , ” M r | = φ ” is first-order definable in M K by the sentence φ M .By applying the downward L¨owenheimSkolem theorem on M K we get a countable substructure of M K like M ′ K such that:1. M ′ K is a representation of a countable rooted Kripke model in the language σ .2. For every φ ∈ U , M K | = ψ iff M ′ K | = ψ , for ψ ∈ { φ F , φ M } .Let K ( M , T M ) be the rooted Kripke model from Corollary 3.12. Let U = HA ∪ { ϕ } where ϕ is aninstance of ∆ induction that fails in the classical structure of the root of K ( M , T M ). Following thesame argument on K ( M , T M ) and U , we get a countable rooted Kripke model K ′ of HA that is notlocally I ∆ . Hence applying Lemma 4.1 on K ′ finishes the proof. ⊣ Problem 1.1 can be asked about other theories than HA . One can ask the same question aboutarithmetic over sub-intuitionistic logic too. One of these logics is Visser’s Basic logic, and its ex-tension Extended Basic logic. The model theory of arithmetic over these logics were investigated in[16, 5, 6]. From the point of view of Problem 1.1, it is proved in [4] that every irreflexive node ina Kripke model of BA (Basic Arithmetic) is locally I ∃ +1 . So In general, every irreflexive node in aKripke model of the natural extension of BA such as EBA (Extended Basic Arithmetic) is locally I Σ (see Corollary 3.33 in [6]). Also it is proved in [6] that every Kripke model of EBA is locally Th Π ( I Σ ) + Th Π ( PA ). Note that every Kripke model of HA is also a Kripke model of BA and EBA . So Corollary 3.12 applies to these theories too, and this solves Problem 1.1 for these theories.Furthermore, this shows that the known positive results are the best we can get for BA and EBA .Focusing on the proof of Theorem 2.8, we essentially use
ECT for proving the existence and dis-junction properties of T M . We do not know whether ECT is essential for such a model construction,so we have the following question: Problem 5.1
Does theory HA + Diag ( M ) has the existence property for every M | = Th Π ( PA ) ? An important problem which we could not answer is the following:
Problem 5.2
Is there any Kripke model K (cid:13) HA such that for every node k in K , M k = PA ? Another unsolved question in the direction of completeness with respect to locally PA Kripkemodels is the following:
Problem 5.3
Does HA have completeness with respect to its class of locally PA Kripke models?
By the result of [8], for every sentence φ such that HA i φ , there exists a locally PA Kripke model K such that K (cid:13) φ , but this result does not say anything about whether K is a Kripke model of HA or not.We call a rooted tree Kripke frame ( K, ≤ ), a PA -frame iff for every Kripke model K (cid:13) HA withframe ( K, ≤ ), K is locally PA . Let F PA be the set of all PA -frames. We know that semi narrowrooted tree Kripke frames are in F PA . On the other hand, by Corollary 4.2 infinite full binary treeis not in F PA . So we have the following question: Problem 5.4
Is there a nice characterization of F PA ? cknowledgment We are indebted to Mohammad Ardeshir and truly grateful to him for his careful guidance, invaluableacademic teachings, and many invaluable discussions that we have had during the studies in theDepartment of Mathematical Sciences of the Sharif University of Technology which had a clearimpact on our academic life. We also thank him for fruitful discussions about this work. We aregrateful to Mohsen Shahriari for fruitful discussions about this work and also reading the draft ofthis paper and pointing out a gap in the proof of Corollary 4.3. We are grateful to Pavel Pudl´akfor fruitful discussions about this work and also reading the draft of the paper and pointing outsome English errors in it and also comments which led to a better presentation of the work. Wethank Emil Jeˇr´abek for a discussion about this work and also reading the draft of the paper andhis comments on it. We also thank Sam Buss, Fedor Pakhomov, and Albert Visser for discussionsabout this work and answering our questions. The first model construction was done when theauthor was at the Sharif University of Technology. The second construction was proved while theauthor was in the Institute of Mathematics of the Czech Academy of Sciences. This research waspartially supported by the project EPAC, funded by the Grant Agency of the Czech Republic underthe grant agreement no. 19-27871X.
References [1] M. Abiri, M. Moniri, M. Zaare,
From forcing to satisfaction in Kripke models of intuitionisticpredicate logic , Logic Journal of the IGPL, 26-(5) (2018), 464474.[2] M. Abiri, M. Moniri, M. Zaare,
Forcing and satisfaction in Kripke models of intuitionisticarithmetic , Logic Journal of the IGPL, 27-(5) (2019), 659670.[3] Z. Adamowicz,
A recursion-theoretic characterization of instances of B Σ n provable in Π n +1 ( N ),Fundamenta Mathematicae, 129 (1988), 213-236.[4] M. Ardeshir, B. Hesaam. Every Rooted Narrow Tree Kripke Model of HA is Locally PA , Math-ematical Logic Quarterly, 48-(3) (2002), 391-395.[5] M. Ardeshir, B. Hesaam, An introduction to basic arithmetic , Logic Journal of the IGPL, 16-(1)(2008), 113.[6] M. Ardeshir, E. Khaniki, M. Shahriari,
Provably total recursive functions and MRDP theoremin Basic Arithmetic and its extensions , arXiv: 2003.01603 (2020).[7] M. Ardeshir, W. Ruitenburg, S. Salehi,
Intuitionistic axiomatizations for bounded extensionKripke models , Annals of Pure and Applied Logic, 124-(13) (2003), 267-285.[8] S. R. Buss,
Intuitionistic validity in T-normal Kripke structures , Annals of Pure and AppliedLogic, 59-(3) (1993), 159-173.[9] A. Chagrov, M. Zakharyaschev,
Modal Logic , Oxford Logic Guides, Vol. 35, Oxford: ClarendonPress, (1997).[10] H. Friedman,
Classically and intuitionistically provably recursive functions , In: Mller G.H.,Scott D.S. (eds) Higher Set Theory, Lecture Notes in Mathematics, vol 669, Springer, Berlin,Heidelberg, (1978).[11] E. Jeˇr´abek,
On models of Th Π ( PA ), Mathoverflow, https://mathoverflow.net/a/300514/83598 ,(2020-06-15).[12] Z. Markovi, On the structure of Kripke models of Heyting Arithmetic , Mathematical LogicQuarterly, 39-(1) (1993), 531-538. 1613] M. Mojtahedi,
Localizing finite-depth Kripke models , Logic Journal of the IGPL, 27-(3) (2019),239-251.[14] M. Moniri, H -theories, fragments of HA and PA -normality , Archive for Mathematical Logic,41-(1) (2002), 101-105.[15] T. Po lacik, Partially-Elementary Extension Kripke Models: A Characterization and Applica-tion , Logic Journal of IGPL, 14-(1) (2006), 7386.[16] W. Ruitenburg,
Basic predicate calculus , Notre Dame Journal of Formal Logic, 39-(1) (1998),1846.[17] T. A. Slaman, Σ n -bounding and ∆ n -induction , Proceedings of the American MathematicalSociety, 132 (2004), 2449-2456.[18] C. Smory´nski, Applications of Kripke models , In: Troelstra A.S. (eds) Metamathematical In-vestigation of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, vol 344.Springer, Berlin, Heidelberg, (1973).[19] C. Smory´nski,
Self-Reference and Modal Logic , Universitext, Springer, New York, (1985).[20] A.S. Troelstra, D. van Dalen,
Constructivism in Mathematics, Vol I , North Holland, Amster-dam, (1988).[21] D. van Dalen, H. Mulder, E. C. W. Krabbe, A. Visser,