Topological Beth Model and its Application to Functionals of High Types

Email: farida.kachapova@aut.ac.nz Abstract: Based on the definition of Beth-Kripke model by Dragalin, we describe Beth model from the topological point of view. We show the relation of the topological definition with more traditional relational definition of Beth model that is based on forcing. We apply the topological definition to construct a Beth model for a theory of intuitionistic functionals of high types and to prove its consistency.


Introduction
The studies of metamathematical properties of nonclassical theories are based on a variety of models such as topological models, Beth model and Kripke model. Moschovakis developed a topological model (Moschovakis, 1973) for intuitionistic second-order arithmetic and developed semantics (Moschovakis, 1987) for a theory of lawless sequences. Van Dalen (1978) constructed a Beth model for intuitionistic analysis. In his book (Dragalin, 1987) Dragalin studied a general Beth-Kripke model (BK-model) that combines forcing from Beth model and realizability from Kripke model. The applications of BK-model in (Dragalin, 1987) include different versions of intuitionistic arithmetic and analysis. Most applications of the aforementioned models are for intuitionistic sequences of natural numbers (e.g. choice sequences and lawless sequences). In (Kachapova, 2014;2015) we created a Beth model for intuitionistic functionals of high types: 1functionals (sequences of natural numbers), 2functionals (sequences of 1-functionals), ..., (n + 1)functionals (sequences of n-functionals). That model was based on the relational definition of Beth model by van Dalen (1978).
In this study we describe the general concept of Beth model from the topological point of view. The topological version has a simpler definition than the relational version and is consistent with the definition of a general algebraic model for an axiomatic theory as in (Dragalin, 1987). In this study we apply the topological version of Beth model to the intuitionistic theory SLP of high-order functionals from (Kachapova, 2015), including lawless functionals and the "creating subject". It can be seen that the topological version of Beth model simplifies some constructions and consistency proofs. In metamathematical proofs we use classical logic.

Preliminary Concepts
The introductory theory in this section follows the book by Dragalin (1987).

Definition 2.1
A logical-mathematical language of first order (or logical-mathematical language, or just language in short) is defined as a sequence Ω = 〈Srt, Cnst, Fn, Pr〉, where 1) Srt is a non-empty set of sorts of objects and for each sort π ∈ Srt there is a countable collection of variables of this sort; 2) Cnst is the set of all constants of the language; 3) Fn is the set of all functional symbols of the language; 4) Pr is the set of all predicate symbols of the language.
In the language Ω we can construct terms, atomic formulas and formulas as usual.

Definition 2.2
Axiomatic theory (or just theory in short) is defined as Th = 〈Ω, l, A〉, where each of the three objects is described as follows.
1) Ω is a logical-mathematical language. 2) l is the logic of the theory. We will use only the intuitionistic logic HPC (Heyting's predicate calculus).
3) A is some set of closed formulas (i.e., formulas without parameters) of the language Ω; it is called the set of non-logical axioms of Th. When axioms are stated as non-closed formulas, it means that they must be closed by universal quantifiers over all parameters.
The notation Th ⊢ϕ (formula ϕ is derivable in the theory Th) means that ϕ is derivable in the logic l from a finite subset of the axiom set A.

Definition 2.3
A pair 〈B, ≤〉 is called a pseudo Boolean algebra (p.B.a.) if B is a set, ≤ is a binary relation on B and they satisfy the following 9 conditions.
For any a, b ∈ B there exists an element a ∧ b ∈ B such that: For any a, b ∈ B there exists an element a ∨ b ∈ B such that: For any a, b∈B there exists an element (a ⊃ b) ∈ B such that: 9) There exists an element ⊥ ∈ B such that for any a ∈ B: ⊥ ≤ a.

Definition 2.4
Suppose 〈B, ≤〉 is a p.B.a., W ⊆ B and a ∈ B.

1) a is denoted ∧W and is called the intersection
2) a is denoted ∨W and is called the union or disjunction of W if a) (∀c ∈ W)(c ≤ a) and b) for any d ∈ B, (∀c ∈ W)(c ≤ d) ⇒ a ≤ d.

Algebraic Model of a Language
The following is definition of an algebraic model with constant domains. For brevity we call it just an algebraic model.

Definition 2.6
An algebraic model of the language Ω is a sequence A = , , , , B D Cnst Fn Pr of objects defined as follows.
1) B is a complete p.B.a.
2) To each sort π of the language Ω the function D assigns a non-empty set D π , which is called the domain of elements of sort π.
3) To each constant of sort π the function Cnst assigns an element c ∈ D π . 4) Function Fn assigns values to functional symbols of Ω in the following way. Let f be a functional symbol of sort π with arguments of sorts π 1 , ..., π k . Then for any Fn satisfies the following conditions (1) and (2). (1) If q ≠ q', then: ,..., , , ,..., , . k k Fn f q q q Fn f q q q′ ∧ = ⊥ (2) 5) If P is a predicate symbol of Ω with arguments of sorts π 1 , ..., π k , then for any An evaluated term is a term of the language Ω, in which all parameters are replaced by elements from suitable domains. An evaluated formula is a formula of the language Ω, in which all parameters are replaced by elements from suitable domains.

Definition 2.7
Suppose t is an evaluated term of sort π and q∈ D π . The set ||t∼q|| is defined by induction on the complexity of t. T if c q t q otherwise For an evaluated term t of sort π: For an evaluated formula ϕ, ||ϕ|| is defined by induction on the complexity of ϕ. 1) Suppose ϕ is an atomic formula P(t 1 , ..., t k ), where each t i is an evaluated term of sort π i , i = 1, ..., k. Then In particular, if each t i is an element p i ∈ i D π , then ||ϕ|| = Pr (P, p 1 ,..., p k ).
2) If ϕ is a closed formula, then

Definition 2.11
A pair Y = 〈X, S〉 is called a topological space if it satisfies the following conditions: Elements of S are called open sets and S is called the topology on X.

Definition 2.12
Suppose Y = 〈X, S〉 is a topological space. A collection H of subsets of X is called a base of the topology S if every A ∈ S can be written as a union of elements of H.
Then we say that H generates the topology S.

Lemma 2.13
Suppose H is a collection of subsets of X. H is a base of some topology on X if it satisfies the following two conditions: Lemma 2.14 Suppose Y = 〈X, S〉 is a topological space. Then 〈S, ⊆〉 is a complete p.B.a. with the operations given by:

Definition 2.15
Suppose Y = 〈B, ≤〉 is a p.B.a. and C : B→B.
1) C is called a completion operator on Y if the following 4 conditions are satisfied for any a, b ∈ B: 2) An element a∈B is called complete if C a = a.
3) Denote Cl(B) the set of all complete elements of B.

Lemma 2.16
Suppose Y = 〈B, ≤〉 is a p.B.a. and C is a completion operator on Y. Denote Y + = 〈Cl(B), ≤〉. Then Y + is also a p.B.a. with the operations given by: , then Y + is also a complete p.B.a. and

Beth Model
The following definitions of Beth frame, Beth algebra and Beth model are modified from the definitions of Beth-Kripke frame, Beth-Kripke algebra and Beth-Kripke model given in the book (Dragalin, 1987). Some accompanying lemmas and theorems are proven here; other proofs can be found in (Dragalin, 1987).

Beth Frame
A tree is a set M with partial order ≤ such that for any x∈M the set {y∈M | y > x} is a well-ordered set. We fix a tree 〈M, ≤〉 till the end of this section.
The triple 〈M, ≤, Q〉 is called a Beth frame. A is open in the order topology

Beth Algebra
Follows from the definition. □

Definition 3.4
For any U ⊆ M denote: The operator C defined above has the following properties.
It is a complete p.B.a. The operator C from Definition 3.4 is a completion operator.
Thus, by Lemma 2.16, Y + = 〈Cl(Op(M)), ⊆〉 is a complete p.B.a. It is denoted B(M, ≤, Q) and is called the Beth algebra generated by the Beth frame 〈M, ≤, Q〉.
Theorem 3.7. Operations in Beth Algebra.
Operations in the Beth algebra are given by:

Proof
Follows from Lemma 2.16. □ Lemma 3.8 1) For any subsets A and B of the set M: 2) For any collection W of subsets of M: To prove the inverse, it is sufficient to show that CA∪CB ⊆ C (A∪B). Since 2) is proven similarly. □

Beth Model
Beth model is a particular case of an algebraic model. We will use the notation f: a b, which means that f is a partial function from set a to set b. The notation Z↓ means that the object Z is defined.

Definition 3.9
A Beth model for a language Ω = 〈Srt, Cnst, Fn, Pr〉 is an algebraic model 2) Before defining Fn we define a function Fn .
To each α ∈ M and each functional symbol f (x 1 ,..., x k ) with sort π and arguments of sorts π 1 , ..., π k , respectively, the function Fn assigns a partial such that for any is an open set and (4) Clearly, Fn satisfies the conditions (1) and (2) of the Definition 2.6 of an algebraic model.
3) Before defining Pr we define a function Pr .

Definition 3.10
For any α and evaluated term t of sort π we define t [α] by induction on the complexity of t.
is not always defined.
The set ||t ~ q|| was defined in Definition 2.7 for a general algebraic model. Next lemma specifies it for the Beth model.

Lemma 3.11
Suppose t is an evaluated term of sort π and q∈D π . Then Proof 1) Proof is by induction on the complexity of t.
If t is an element of D π , then {α | t [α] = q} is either T or ⊥; in both cases it is an open set.
When t is a constant c, the proof is similar.
Suppose t is f (t 1 , ..., t k ), where each t i is an evaluated term of sort π i , i = 1, ..., k.
Then 2) Proof is by induction on the complexity of t.
If t is an element of D π , then t [α] = t. So each side of the equality is , .
T if t q otherwise Suppose t is f (t 1 , ..., t k ), where each t i is an evaluated term of sort π i , i = 1, ..., k. Then by Lemma 3.8.2) and (6). □

1) Monotonicity of forcing:
3) Suppose ϕ is an atomic formula P(t 1 ,..., t k ), where each t i is an evaluated term of sort π i , i = 1, ..., k. Then , where x is a variable of sort π

Proof
In the Beth model ||ϕ|| ∈Cl(Op(M)), so ||ϕ|| is both an open set and a complete element.

1) Follows from the fact that ||ϕ|| is an open set 2) Follows from the fact that ||ϕ|| is a complete element 3) Follows from Lemma 3.13
The rest follow from Definition 2.9 and Theorem 3.7. □

Axiomatic Theory of Functionals of High Types
As an application of the topological definition of Beth model, we construct a Beth model for an intuitionistic theory SLP of functionals of high types. We introduced this theory in (Kachapova, 2015) and constructed its Beth model using the van Dalen's relational definition (Van Dalen, 1978). The topological definition outlined here simplifies some parts of the construction and proofs.
We define the theory SLP in three steps: We introduce axiomatic theories L, LP and SLP. In these theories variables have superscripts for types. A superscript for a variable is usually omitted when the variable is used for the second time or more in a formula (so its type is clear).

Axiomatic Theory L
The language of theory L has the following variables: x, y, z, ... over natural numbers (variables of type 0) and variables of type n (n ≥ 1): , , ,...  Constants: 0 of type 0 and for each n ≥ 1 a constant K n (an analog of 0 for type n).
Predicate symbols: = n for each n ≥ 0. Terms and n-functionals are defined recursively as follows.
1. Every numerical variable is a term 2. Constant 0 is a term 3. Every variable of type n is an n-functional 4. Constant K n is an n-functional 5. If t is a term, then St is a term 6. If t 1 and t 2 are terms, then t 1 +t 2 and t 1 ·t 2 are terms 7. If Z is an n-functional, then N n (Z) is an n-functional (a successor of Z) 8. If Z is a 1-functional, t is a term, then Ap 1 (Z, t) is a term 9. If Z is an (n +1)-functional, t is a term, then Ap n+1 (Z, t) is an n-functional Ap n (Z, t) is interpreted as the result of application of functional Z to term t. We also denote Ap n (Z, t) as Z(t).
Here 1-functional is interpreted as a function from natural numbers to natural numbers and (n+1)functional is interpreted as a function from natural numbers to n-functionals.
Atomic formulas: where t and τ are terms; where Z and V are n-functionals (n ≥ 1).
Formulas are constructed from atomic formulas using logical connectives and quantifiers. For a formula ϕ its sort sort(ϕ) is the maximal type of parameters in ϕ; it is 0 if ϕ has no parameters.
The theory L has intuitionistic predicate logic HPC with equality axioms and the following non-logical axioms.
The axioms 1 -4 define arithmetic at the bottom level.
The axioms 5 and 6 describe K n and N n as analogs of zero and successor function, respectively, on level n. 7. Principle of primitive recursive completeness of lawlike functions: where t is any term containing only variables of type 0 and variables over lawlike 1-functionals. Denote L s the fragment of the theory L with types not greater than s. The language of L 1 has one type of functionals and is essentially the language of the intuitionistic analysis FIM.

Axiomatic Theory LP
This theory is obtained from L by adding predicate symbols and axioms for the "creating subject".
Gödel numbering of symbols and expressions can be defined for the language of L. For an expression q we denote ⌞q⌟ the Gödel number of q in this numbering.
The language of theory LP is the language of L with an extra predicate symbol ( ) . , X Pv z X ϕ for every formula ϕ of L, which has all its parameters in the list X ; here X is a list of variables X 1 , ..., X k . Traditionally this symbol is denoted ⊢ z ϕ ( ) X ; it means that the formula ϕ ( ) X has been proven by the "creating subject" at time z. Axioms of the theory LP are all axioms of L, where the axiom schemata are taken for the formulas of the new language, and the following three axioms; in all of them ϕ is an arbitrary formula of L.

Axiomatic Theory SLP
We can introduce finite sequences in LP and for any k-functional F use the notation F (n) = < F(0), ..., F(n − 1) >. We consider the following three axioms for lawless functionals.
(LL1) The axiom of existence of lawless functionals: The principle of open data: where ϕ is a formula of LP, sort(ϕ) ≤ n and ϕ does not have non-lawlike parameters of type n other than H n . Denote (LL) a conjunction of the closures of (LL1), (LL2) and (LL3).
For an m-functional F of language LP and n ≤ m we denote F ( ) ( ) 0 ... 0 n as F(0) n for brevity. We consider the following two choice axioms.
(C1) The axiom of choice for numbers: where ϕ is a formula of LP and m ≥ max(sort(ϕ), 1). (C2) The axiom of choice with uniqueness: where ϕ is a formula of LP and m ≥ max(sort(ϕ), n +1). Denote (C) a conjunction of the closures of (C1) and (C2).
Bar induction axiom: Axiomatic theory SLP is defined by: In the name of the theory SLP, S stands for strong, L for lawless and P for proof (relating to the "creating subject").
The fragment SLPs is defined by:

Application of the Topological Beth Model
As an application, we construct a Beth model for any fragment SLP s (s ≥1) of the theory SLP. This is sufficient for the proof of consistency of SLP, since any formal proof in the theory SLP is finite and therefore it is a proof in some fragment SLP s .
The symbol * denotes the concatenation function: x x x y y y x x x y y y * = For a sequence x, 〈x〉 n denotes its n th element and x (n) denotes the initial segment of x of length n for any n < lth(x). So if x = 〈x 0 , x 1 , ..., x m−1 〉 and n ≤ m, then x (n) = 〈x 0 , x 1 , ..., x n −1 〉.
For a function f on natural numbers, f (n) denotes the initial segment of f of length n, that is the sequence 〈f (0), f (1), ..., f (n−1)〉.
For a function f of two variables denote f (x) = λy. f (x,y).
Next we introduce a few notations for a fixed set b.
1. b (n) is the set of all sequences of elements of b that have length n. 2. b * is the set of all finite sequences of elements of b. 3. On the set b * a partial order ≤ is defined by the following: ≤ if x is an initial segment of y.
With this order b * is a tree growing down; its root is the empty sequence < >.
Suppose 〈d, ≤〉 is a partially ordered set and f: (d×ω) c.
1. f is called monotonic on d if for any x, y∈d, 2. f is called complete on d if for any path S in d, ⋃{f (x) | x ∈ S} is a total function on ω.

Beth Model
To construct a Beth model B s for the language L s , we will specify a tree 〈M, 〉, a domain function D and functions Cnst , Fn and Pr . 1) First we introduce a triple of objects 〈a k , d k , k 〉 by induction on k; here k is a partial order on d k .
• a 0 = ω, the set of all natural numbers a }; thus, each element of d 0 is a sequence of length 1 • 0 is generated by the order ≤ on * 0 a 〈x〉 0 〈y〉 if x ≤ y, that is the sequence y is an initial segment of the sequence x.
For 〈x〉 ∈d 0 we denote lh(〈x〉) = lth(x) and call it the length of 〈x〉.
For k ≥ 1: and f is complete and monotonic}; , we denote lh(x) = m and call it the length of x.
We take M = d s-1 with partial order s-1 , which we denote just .
With this order M is a subtree of the direct product of trees a 0 * , ..., a s−1 * . M is a non-countable tree growing down. Its root is ,..., . s ε = < < > < > > 1) Next we define a domain for each sort of variables. a) Domain for natural numbers is a 0 = ω. b) Domain for k-functionals is a k (k = 1, 2, ..., s). Elements of c k are called k-permutations. For any kpermutation ξ we define ν k (ξ) as the function f ∈ a k such that for any n ∈ ω, x ∈ d k−1 :
2) Next we define Cnst , that is interpretation C for each constant C. 3) Next we define Fn . This means defining a partial function h [α] for each functional symbol h and α ∈M.
Thus, at each node α the values of a k-functional f depend not on the entire α but only on its first k components.
Due to the definition of a k , the conditions (4) and (5)

Definition 5.1
Consider a formula ϕ of L s with all its parameters in the list X 1 , ..., X k , which is denoted X in short. For brevity we denote the predicate ( ) . , X Pv z X ϕ as ( ) 0 , P z X . We define: where, q is the list q 1 , ..., q k .
Let us show that Pr (P 0 , n, q ) is an open set in the order topology. Suppose α∈ Pr (P 0 , n, q ). Then there is γ such that α ≺ γ & lh (γ) = n & γ ⊩ ϕ ( q ). Then for any β ≺ α we have β ≺ γ, so β ∈ Pr (P 0 , n, q ). The language of SLP s is the same as the language LP s , so the extended model B s is also a Beth model for the language of SLP s .

Proof
Proof is by induction on the length of derivation of ϕ.
Here we provide proofs only for the case of the axioms (CS1) − (CS3) for the "creating subject". We prove these in all detail to illustrate our application of the topological Beth model. Other, more technical proofs can be found in (Kachapova, 2014;2015). , , . P n q P n q ε ∨ ¬ ⊩ By Theorem 3.15.5) it is equivalent to: Consider a path S∈ Q(ε). Fix α∈ S with lh(α) = n. There are two cases: α ⊩ ϕ(q) or α ⊭ ϕ(q) (we use classical logic in metamathematics).