PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 100(114) (2016), 119–129 DOI: 10.2298/PIM160525001D

COMPLETENESS THEOREM FOR CONTINUOUS FUNCTIONS AND PRODUCT CLASS-TOPOLOGIES

Radosav Djordjević, Vladimir Ristić, and Nebojša Ikodinović

Abstract. We introduce an infinitary logic L_{𝔸}(O^{n} , C^{n})_{n∈ω} which is an extension of L_{𝔸} obtained by adding new quantifiers O^{n} and C^{n}, for every n ∈ ω. The corresponding models are topological class-spaces. An axiomatization is given and the completeness theorem is proved.

1. Introduction

In [7] a topological class logic L_{𝔸}(O, C) appropriate for the study of topologies on proper classes was developed. The logic L_{𝔸}(O, C) is an infinitary logic with new quantifiers O and C. The corresponding semantics consists of a classical first-order structure K whose domain 𝕂 is a class, with addition of two classes 𝕋 and ℂ of subsets of 𝕂 such that (𝕂, 𝕋, ℂ) is a topological class-space introduced in [2] (see Definition 2.1). The intended meaning of Oxφ(x) (Cxφ(x)) is that the set defined by φ(x) belongs to the class 𝕋 (ℂ). The logic L_{𝔸}(O, C) is analogous to the logics L_{ωω} (O) and L_{ω_{1} ω} (O) developed in [14] for ordinary topological spaces. In [13], the author investigated the logics L_{ωω} (O^{n})_{n∈ω} and L_{ω_{1} ω} (O^{n})_{n∈ω} with quantifiers O^{n} for each n ∈ ω in order to give a formal treatment of continuous functions on product topologies.

In this paper, following the ideas from [7] and [13], we continue the study of topological class logics and introduce the logic L_{𝔸}(O^{n}, C^{n})_{n∈ω} appropriate for continuous functions and product class-topologies.

All above mentioned logics are of the form L_{𝔸}(Q_{1}, Q_{2}, …), i.e., they are extensions of an admissible fragment L_{𝔸} of L_{ω_{1} ω} with new quantifiers Q_{1}, Q_{2}, … Here, we give some general remarks and a short (mainly informal) overview of the notions relevant to our consideration.

The original purpose of admissible sets was to generalize classical computability theory from natural numbers to ordinals. The Kripke–Platek set theory, KP for short, is considered as a minimal subsystem of ZF necessary for a good notion of computation. KP arises from ZF by omitting the Power Set Axiom and restricting Separation and Collection to Δ_{0}-formulas. An admissible set is a transitive set 𝔸 such that (𝔸, ∈) is a model of KP. The smallest example of an admissible set is the set of hereditarily finite sets ℍ𝔽 which corresponds to classical computability theory. Another example of an admissible set, important in this paper, is the set ℍℂ of hereditarily countable sets. To emphasize the analogy with computability theory, for an arbitrary admissible set 𝔸, the elements of 𝔸 are called 𝔸-finite, a subset of 𝔸 that is definable in (𝔸, ∈) by Σ-formula, with parameters in 𝔸, is called 𝔸-computable enumerable (𝔸-c.e.), and a set X ⊆ 𝔸 is called 𝔸-computable if both X and 𝔸 ∖ X are 𝔸-c.e.

2010 Mathematics Subject Classification: Primary 03C70; Secondary 03C80, 03B80.

Key words and phrases: infinitary logic; topological class-spaces; completeness.

Supported by the Serbian Ministry of Education and Science (projects 174026 and III 044006).

Communicated by Žarko Mijajlović.

If L is a countable first-order language and 𝔸 is an admissible set such that 𝔸 ⊆ ℍℂ and ω ∈ 𝔸, the admissible fragment L_{𝔸} is the set of all L_{ω_{1} ω} formulas that belongs to 𝔸. L_{𝔸} is an 𝔸-computable set, and is closed under basic syntactical operations. The Barwise compactness theorem is one of the most significant results related to admissible fragments: If 𝔸 is a countable admissible set, and Γ is an 𝔸-c.e. subset of L_{𝔸} such that every 𝔸-finite subset of Γ has a model, then Γ has a model.

Barwise compactness is in the heart of a techniques developed by Rašković in [11]. After that paper, the technique has been applied for proving completeness theorems for many infinitary logics with generalized quantifiers [4–7,12]. Roughly speaking, adding new quantifiers to infinitary logics is one of the most frequent and high acceptable ways to incorporate into the realm of logic those structures whose related concepts are left out of the first-order logic (such as: probability spaces, topological spaces, etc.). Here, our attention is focussed only on n-ary quantifiers (n ⩾ 1) which can be applied to a single formula and bind an n-tuple of variables. More precisely, if L is a first-order language (a vocabulary) and Q is an n-ary quantifier, then the set of formulas L_{ω_{1} ω} (Q) is built in the standard way with the additional formation rule: if φ is a formula and x⃗ is an n-tuple of variables, then Qx⃗φ is a formula. A weak-model for L_{ω_{1} ω} (Q) is a structure of the form (A, q), where A is an L structure and q is a set of subsets of the universe A. The truthfulness of an L_{ω_{1} ω} (Q)-formula in such models is defined inductively in the usual way with the new clause:

(A, q) ⊧ Qx⃗φ iff {a⃗ ∈ A^{n} | (A, q) ⊧ φ[a⃗]} ∈ q.

Of course, there are many concrete interpretations of Q (Q_{∞} “there exist infinitely many"”, Q_{ℵ_{1}} “there exists uncountable many” [10], Q_{⩾r} “to have probability at least r” [9, 12], Q_{open} “to be open” [14], etc.).

Combining a consistency property argument and the Henkin construction yields an L_{ω_{1} ω} (Q)-version of the weak completeness theorem (see [6,11]). In order to obtain stronger completeness theorems, we have to carry out more interesting model constructions according to the intended meaning of a new quantifier. The key step of Rašković’s technique is the construction of a middle model whose all subsets satisfy some desirable conditions being true in a weak model only for L_{ω_{1} ω} (Q)- definable subsets. If K = L ∪ C is introduced in the construction of a weak model (C is a set of new constant symbols and C ∈ 𝔸), then a desirable middle model is obtained by Barwise Compactness applied on a theory of many sorted logic whose language contains (at least) two kinds of variables: X, Y, Z, …, variables for sets, and x, y, z, … variables for urelements. Predicates are E_{n}(x_{1}, …, x_{n}, X), n ⩾ 1, with canonical meaning (x_{1}, …, x_{n}) ∈ X, and Q(X). Constant symbols are A_{φ} for each K_{𝔸}(Q)-formula φ. Finally, when a middle model is properly constructed, it have to be transformed to a strong model. This strategy will be applied in the proof of Completeness theorem for L_{𝔸}(O^{n}, C^{n})_{n∈ω} .

The rest of the paper is organized as follows. Section 2 contains a short overview of topological class-spaces. In Section 3, the syntax and the corresponding semantics of the logic L_{𝔸}(O^{n}, C^{n})_{n∈ω} are described. A sound and complete axiomatic system is given in Section 4. This section contains the main result of the paper, i.e., the completeness theorem for L_{𝔸}(O^{n}, C^{n})_{n∈ω} . We conclude in Section 5.

2. Preliminaries

The notion of topological class-space was introduced in [2], and further developed in [3]. The metatheory is NBG class theory.

Definition 2.1. [2] Let 𝕂 be a class and 𝕋 and ℂ classes of subsets of 𝕂. We call the triple (𝕂, 𝕋, ℂ) a topological class-space if the following axioms are satisfied:

(KT_{1}) if u, v ∈ 𝕋, then u ∩ v ∈ 𝕋;

(KT_{2}) for any i, if u_{j} ∈ 𝕋, j ∈ i, then ⋃ _{j∈i} u_{j} ∈ 𝕋; (KT_{3}) for any x ∈ 𝕂 there exists u ∈ 𝕋 such that x ∈ u; (KT_{4}) if u ∈ 𝕋 and a ∈ ℂ, then u ∖ a ∈ 𝕋; (KC_{1}) if a, b ∈ ℂ, then a ∪ b ∈ ℂ; (KC_{2}) for any i, if a_{j} ∈ ℂ, j ∈ i, then ⋂ _{j∈i} a_{j} ∈ ℂ; (KC_{3}) for any subset x of 𝕂 there exists a ∈ ℂ such that x ⊆ a; (KC_{4}) if u ∈ 𝕋 and a ∈ ℂ, then a ∖ u ∈ ℂ.

Elements of 𝕋 are open subsets, while elements of ℂ are closed subsets of 𝕂.

Many examples of topological class spaces can be found in [2,3]. The following theorem is an answer to the question when an ordered pair (𝔹_{T}, 𝔹_{C}) of classes of subsets of a class 𝕂 defines a class-space over 𝕂.

Theorem 2.1. [3] Let 𝕂 be a proper class and 𝔹_{T} and 𝔹_{C} be classes of subsets of 𝕂 such that

(1) ∀x ∈ 𝕂 ∃s ∈ 𝔹_{T} x ∈ s.

(2) for every subset m ⊆ 𝕂 there is f ∈ 𝔹_{C} such that m ⊆ f.

(3) ∀s ∈ 𝔹_{T} ∀f ∈ 𝔹_{C} (s ∖ f ∈ 𝔹_{T} ∧ f ∖ s ∈ 𝔹_{C}). Then there is the least class-topology (𝕂, 𝕋, ℂ) such that 𝔹_{T} ⊆ 𝕋 and 𝔹_{C} ⊆ ℂ.

If 𝕂 and 𝕃 are classes, recall that a map from 𝕂 into 𝕃 is every class 𝔽 ⊆ 𝕂×𝕃 such that ∀x ∈ 𝕂 ∃_{1}y ∈ 𝕃 (x, y) ∈ 𝔽.

Definition 2.2. [2] Let (𝕂, 𝕋_{𝕂}, ℂ_{𝕂}) and (𝕃, 𝕋_{𝕃}, ℂ_{𝕃}) be class-spaces. A map 𝔽: 𝕂 → 𝕃 is continuous iff the restriction 𝔽 ↾ u to every u ∈ 𝕋_{𝕂} is a continuous function.

Theorem 2.2. [3] Let (𝕂, 𝕋_{𝕂}, ℂ_{𝕂}) and (𝕃, 𝕋_{𝕃}, ℂ_{𝕃}) be class-spaces and let 𝔽: 𝕂 → 𝕃 be a map. Then the following are equivalent:

(1) 𝔽 is continuous.

(2) For all u ∈ 𝕋_{𝕂}, and all w ∈ 𝕋_{𝕃}, u ∩ 𝔽^{−1} (w) ∈ 𝕋_{𝕂}.

(3) For all v ∈ ℂ_{𝕂}, and all w ∈ ℂ_{𝕃}, v ∩ 𝔽^{−1} (w) ∈ ℂ_{𝕂}.

The product of two class-spaces (𝕂_{1}, 𝕋_{1}, ℂ_{1}) and (𝕂_{2}, 𝕋_{2}, ℂ_{2}) is the class-space (𝕂, 𝕋, ℂ), where 𝕂 = 𝕂 _{1} × 𝕂_{2} and the basis for 𝕋 and ℂ are the classes 𝕋_{0} = {π_{1}^{−1} (u) ∩ π_{2}^{−1} (v) | u ∈ 𝕋_{1}, v ∈ 𝕋_{2}} and ℂ_{0} = {π_{1}^{−1} (u) ∩ π_{2}^{−1} (v) | u ∈ ℂ_{1}, v ∈ ℂ_{2}}, where π_{1} : 𝕂 → 𝕂_{1} and π_{2} : 𝕂 → 𝕂_{2} are projection maps. The classes 𝕋_{0} and ℂ_{0} do not satisfy the conditions of Theorem 2.1, so let 𝕋′ be the class of all finite unions of the elements of the class 𝕋_{0}, and ℂ′ be the class of all finite unions of the elements of the class ℂ_{0}. It is obvious that the classes 𝕋′ and ℂ′ are closed under finite intersections. In fact, the classes 𝕋′ and ℂ′ form the base of the topological class-space (𝕂, 𝕋, ℂ). Note that 𝕋_{0} ⊆ 𝕋′ and ℂ_{0} ⊆ ℂ′ and the classes 𝕋′ and ℂ′ satisfy the conditions of Theorem 2.1, and there is the least class-space (𝕂, \overline{𝕋}, \overline{ℂ}) such that 𝕋′ ⊆ \overline{𝕋} and ℂ′ ⊆ \overline{ℂ}. It is not difficult to see that the class \overline{𝕋} = \{∪x | x is a finite intersection of members of 𝕋′\} is exactly the class 𝕋 of the open sets. Similarly, the class \overline{ℂ} = \{∩x | x is a finite union of members of ℂ′\} is the class ℂ of closed sets. Furthermore, each open set O ∈ 𝕋 is a union of sets of the form O_{1}×O_{2}, where O_{1} ∈ 𝕋_{1} and O_{2} ∈ 𝕋_{2}. This fact is of the great importance for our axiomatization given here.

3. Syntax and semantics of the logic L_{𝔸}(O^{n}, C^{n})_{n∈ω}

We assume that 𝔸 is a countable admissible set such that 𝔸 ⊆ ℍℂ and ω ∈ 𝔸. We refer the reader to [1, 8] for a detailed treatment of admissible sets and the infinitary logic L_{𝔸}. Briefly, we note that the set of formulas of L_{𝔸} is the set of all expressions in 𝔸 that are built from atomic formulas using negation, finite or infinite conjunction, and the quantifier ∀.

Let L be a Σ-definable set which contains a set of finitary relation symbols. The infinitary logic L_{𝔸}(O^{n}, C^{n})_{n∈ω} is an extension of L_{𝔸} obtained by adding new quantifiers O^{n}x_{1}, …, x_{n} and C^{n}x_{1}, …, x_{n}, for every n ∈ ω, where x_{1}, …, x_{n} is a tuple of pairwise distinct variables.

Definition 3.1. The set of formulas of L_{𝔸}(O^{n}, C^{n})_{n∈ω} is the least set such that:

(1) Each atomic formula of first-order logic is a formula of L_{𝔸}(O^{n}, C^{n})_{n∈ω} ;

(2) If φ is a formula of L_{𝔸}(O^{n}, C^{n})_{n∈ω} then ¬φ is a formula of L_{𝔸}(O^{n}, C^{n})_{n∈ω} ;

(3) If φ is a formula of L_{𝔸}(O^{n}, C^{n})_{n∈ω} then ∀xφ, ∃xφ, O^{n}x_{1}, …, x_{n}φ and C^{n}x_{1}, …, x_{n}φ are formulas of L_{𝔸}(O^{n}, C^{n})_{n∈ω} ; occasionally, we use the abbreviation Q^{n} x⃗ for Qx_{1} …Qx_{n}, Q ∈ {∀, ∃};

(4) If Φ ∈ 𝔸 is a set of formulas of L_{𝔸}(O^{n}, C^{n})_{n∈ω} with only finitely many free variables, then ⋀ Φ is a formula of L_{𝔸}(O^{n}, C^{n})_{n∈ω} .

Three types of models are relevant for our logic.

Definition 3.2. A weak model for L_{𝔸}(O^{n}, C^{n})_{n∈ω} is a structure (K,𝕋_{n}, ℂ_{n})_{n∈ω} , such that K is a first-order structure of a language L whose universe 𝕂 is a class, and 𝕋_{n}, ℂ_{n} are classes of subsets of 𝕂^{n}.

A middle model for L_{𝔸}(O^{n}, C^{n})_{n∈ω} is a weak model (K,𝕋_{n}, ℂ_{n})_{n∈ω} , such that:

(a) For each (x_{1}, …, x_{n}) ∈ 𝕂^{n} there is U ∈ 𝕋_{n} such that (x_{1}, …, x_{n}) ∈ U;

(b) For every subset X ⊆ 𝕂^{n} there is F ∈ ℂ_{n} such that X ⊆ F;
(c) For all U ∈ 𝕋_{n} and F ∈ ℂ_{n}, U ∖ F ∈ 𝕋_{n} and F ∖ U ∈ ℂ_{n}.

A complete topological class model for L_{𝔸}(O^{n}, C^{n})_{n∈ω} is a weak model (K,𝕋_{n}, ℂ_{n})_{n∈ω} such that for each n ∈ ω, (𝕂^{n}, 𝕋_{n}, ℂ_{n}) is a topological class-space, 𝕋_{n} is the n-th topological product of 𝕋_{1} on 𝕂 and ℂ_{n} is the n-th topological product of ℂ_{1} on 𝕂.

The satisfaction relation ⊧ is defined inductively in the usual way with the new clauses:

(K, 𝕋_{n}, ℂ_{n})_{n∈ω} ⊧ O^{k}x_{1}, …, x_{k}φ[x_{1}, …, x_{k}, b_{1}, …, b_{l}] iff

{(a_{1}, …, a_{k}) | (K, 𝕋_{n}, ℂ_{n})_{n∈ω} ⊧ φ[a_{1}, …, a_{k}, b_{1}, …, b_{l}]} ∈ 𝕋_{k}

for (b_{1}, …, b_{l}) ∈ 𝕂^{l};

(K, 𝕋_{n}, ℂ_{n})_{n∈ω} ⊧ C^{k}x_{1}, …, x_{k}φ[x_{1}, …, x_{k}, b_{1}, …, b_{l}] iff

{(a_{1}, …, a_{k}) | (K, 𝕋_{n}, ℂ_{n})_{n∈ω} ⊧ φ[a_{1}, …, a_{k}, b_{1}, …, b_{l}]} ∈ ℂ_{k}

for (b_{1}, …, b_{l}) ∈ 𝕂^{l}.

4. A complete axiomatization

L_{𝔸}(O^{n}, C^{n})_{n∈ω} has the following set of axioms, where Φ = ⋃ _{n∈ω} Φ_{n},

Φ_{n} = {φ ∈ Φ | φ is a formula of L_{𝔸}(O, C) with n+1 free variables}, and Φ, Φ_{n} ∈ 𝔸 , n ∈ ω:

(1) All axioms schemas for L_{𝔸};

(2) ∀x_{1}, …, x_{n}(φ ↔ ψ) → (O^{n}x_{1}, …, x_{n}φ ↔ O^{n}x_{1}, …, x_{n}ψ); ∀x_{1}, …, x_{n}(φ ↔ ψ) → (C^{n}x_{1}, …, x_{n}φ ↔ C^{n}x_{1}, …, x_{n}ψ);

(3) O^{n}x_{1}, …, x_{n}φ(x_{1}, …, x_{n}) → O^{n}y_{1}, …, y_{n}φ(y_{1}, …, y_{n}); C^{n}x_{1}, …, x_{n}φ(x_{1}, …, x_{n}) → C^{n}y_{1}, …, y_{n}φ(y_{1}, …, y_{n});

(4) O^{n}x_{1}, …, x_{n}(x_{1} ≠ x_{1}); C^{n}x_{1}, …, x_{n}(x_{1} ≠ x_{1});

(5) O^{n}x_{1}, …, x_{n}φ ∧ O^{n}x_{1}, …, x_{n}ψ → O^{n}x_{1}, …, x_{n}(φ ∧ ψ); C^{n}x_{1}, …, x_{n}φ ∧ C^{n}x_{1}, …, x_{n}ψ → C^{n}x_{1}, …, x_{n}(φ ∨ ψ);

(6) O^{n}x_{1}, …, x_{n}φ ∧ C^{n}x_{1}, …, x_{n}ψ → O^{n}x_{1}, …, x_{n}(φ ∧ ¬ψ); O^{n}x_{1}, …, x_{n}φ ∧ C^{n}x_{1}, …, x_{n}ψ → C^{n}x_{1}, …, x_{n}(¬φ ∧ ψ);

(7) ∀yO^{n}x_{1}, …, x_{n}φ(x_{1}, …, x_{n}, y) → O^{n}x_{1}, …, x_{n}∃yφ(x_{1}, …, x_{n}, y); ∀yC^{n}x_{1}, …, x_{n}φ(x_{1}, …, x_{n}, y) → C^{n}x_{1}, …, x_{n}∀yφ(x_{1}, …, x_{n}, y);

(8) ⋀_{φ∈Φ} O^{n}x_{1}, …, x_{n}φ → O^{n}x_{1}, …, x_{n} ⋁ Φ;

⋀_{φ∈Φ} C^{n}x_{1}, …, x_{n}φ → C^{n}x_{1}, …, x_{n} ⋀ Φ;

(9) O^{n}x_{1}, …, x_{n}φ ∧ O^{m}x_{n+1} , …, x_{n+m} ψ → O^{n+m} x_{1}, …, x_{m+n} (φ ∧ ψ); C^{n}x_{1}, …, x_{n}φ ∧ C^{m}x_{n+1} , …, x_{n+m} ψ → C^{n+m} x_{1}, …, x_{n+m} (φ ∧ ψ);

(10) O^{n}x_{1}, …, x_{n}φ(x_{1}, …, x_{n}) → O^{k}x_{i_{1}} , …, x_{i_{k}} φ(x_{σ(1)} , …, x_{σ(n)} ); C^{n}x_{1}, …, x_{n}φ(x_{1}, …, x_{n}) → C^{k}x_{i_{1}} , …, x_{i_{k}} φ(x_{σ(1)} , …, x_{σ(n)} ), where σ: n → n, |σ[n]| = k and σ = {i_{1} < ⋯< i_{k}}; [Informal meaning: The permutation, projection, or consolidation of an open set is open (see [13])]

(11) O^{n}x_{1}, …, x_{n}φ(x_{1}, …, x_{n}) → ∀x_{1}, …, x_{k}O^{n−k} x_{k+1} , …, x_{n}φ(x_{1}, …, x_{n});

[Informal meaning: Any projection of an open set is open.]

(12) ∀^{n} x⃗ ⋁_{m} ⋁_{φ∈Φ_{m+n−1}} ∃^{m} y⃗(O^{n} z⃗φ(z⃗, y⃗) ∧ φ(x⃗, y⃗));

[Axiom corresponds to the condition (KT_{3}).]

(13) ⋀_{n} ⋀_{φ∈Φ_{n+k−1}} ∀^{n} x⃗ ⋁_{m} ⋁_{ψ∈Φ_{m+k−1}} ∃^{m} y⃗(C^{k} z⃗ψ(z⃗, y⃗) ∧ ∀^{k} u⃗(φ(u⃗, x⃗) → ψ(u⃗, y⃗)));

[Axiom corresponds to the condition (KC_{3}).]

(14) O^{m} y⃗ψ(y⃗) → ⋀_{l} ⋀_{θ∈Φ_{n+l−1}} ∀^{l}v⃗(O^{n} x⃗θ(x⃗,v⃗) →

→ O^{m+n−k} z⃗, y_{k+1} , …, y_{m}(∃y_{1}, …, y_{k}(ψ(y⃗) ∧ φ(z⃗, y⃗)) ∧ θ(z⃗,v⃗))); [Informal meaning: φ(z_{1}, …, z_{n}, y_{1}, …, y_{k}) defines an (n, k)-ary continuous relation, i.e., under the restriction to any open set, the inverse image of a slice of an open set is open.]

(15) ⋀_{n} ⋀_{φ∈Φ_{n−1}} ∀^{n} u⃗ ⋁_{k∈ω^{n}} ⋁_{(φ_{1} ,...,φ_{n} )∈Φ_{k_{1}} ×...×Φ_{kn}}  ∃x_{11} …x_{1k_{1}} …∃x_{n1} …x_{nk_{n}}

( O^{n} x⃗φ(x⃗) ∧ φ(u⃗) → ( ⋀_{i=1}^{n} (Oxφ_{i}(x, x_{i}_{1}, …, x_{i}_{k_{i}} ) ∧ φ_{i}(u_{i}, x_{i}_{1}, …, x_{i}_{k_{i}} ))∧

∧∀y_{1}, …, y_{n}( ⋀_{i=1}^{n} φ_{i}(y_{i}, x_{i}_{1}, …, x_{i}_{k_{i}} ) → φ(y_{1}, …, y_{n}))) ),

(for simplicity the formula φ is written without parameters); [Informal meaning: Every open set is the union of open boxes, i.e., if (x_{1}, …, x_{n}) ∈ U ∈ 𝕋_{n}, then there exist V_{1}, …, V_{n} ∈ 𝕋_{1} such that (x_{1}, …, x_{n}) ∈ V_{1} × ⋯× V_{n} ⊆ U.]

(16) ∀^{n} y⃗ ⋀_{k} ⋀_{φ∈Φ_{k+n−1}}  ⋁_{l∈ω^{n}} ⋁_{(φ_{1} ,...,φ_{n} )∈Φ_{l_{1}} ×...×Φ_{ln}}  ∃x_{11} …x_{1l_{1}} …∃x_{n1} …x_{nl_{n}}

( C^{n} x⃗φ(x⃗) ∧ ¬φ(y⃗) → ( ⋀_{i=1}^{n} Oxφ_{i}(x, x_{i}_{1}, …, x_{i}_{l_{i}} ) ∧ φ_{i}(y_{i}, x_{i}_{1}, …, x_{i}_{l_{i}} )) ∧

∧∀x_{1}, …, x_{n}( ⋀_{i=1}^{n} φ_{i}(x_{i}, x_{i}_{1}, …, x_{i}_{l_{i}} ) → ¬φ(x_{1}, …, x_{n})) );

(for simplicity the formula φ is written without parameters); [Informal meaning: If (x_{1}, …, x_{n}) ∉ F ∈ ℂ_{n}, then there exist V_{1}, …, V_{n} ∈ 𝕋_{1} such that (x_{1}, …, x_{n}) ∈ V_{1} × ⋯× V_{n} and (V_{1} × ⋯× V_{n}) ∩ F = ∅ (see [2, 3]).]

and the rules of inferences:

(R_{1}) From φ and φ → ψ, infer ψ. (R_{2}) From φ → ψ, for all ψ ∈ Ψ, infer φ → ⋀_{ψ∈Ψ} ψ.

(R_{3}) From φ, infer ∀x φ.

Axioms (2)–(8), (12) and (13) formalize our notion of topologies on classes and they are analogue to the axioms of the L_{𝔸}(O, C). Axioms (9), (10) and (11) describe products of topological class-spaces. Axiom (14) enforces the constraint that an (n, k)-ary relation defined by a formula φ(z_{1}, …, z_{n}, y_{1}, …, y_{k}) is continuous (see Theorem 2.2, and [13]). Axioms (15) and (16) capture the fact that for each n ∈ ω, 𝕋_{n} is the n-th topological product of 𝕋_{1} on 𝕂, and ℂ_{n} is n-th topological product of ℂ_{1} on 𝕂.

The soundness theorem for L_{𝔸}(O^{n}, C^{n})_{n∈ω} holds since all the axioms represent properties of the topological class-spaces. We prove that the axiomatization is complete with respect to the class of complete topological class models. In the proof we combine Keisler’s construction for the weak completeness theorem in [10], with Sgro’s construction from [14] and Rašković’s middle-strong construction from [11]. Two simple statements will be used in the middle-strong construction:

(1) If Y ⊆ 𝕂 and Y is not open, then there is c ∈ Y such that if U is open and U ⊆ Y then c ∉ U.

(2) If Y ⊆ 𝕂 and Y is not closed, then there is c ∉ Y such that if F is closed and Y ⊆ F then c ∈ F.

Let T be a set of sentences of L_{𝔸}(O^{n}, C^{n})_{n∈ω} such that T is Σ_{1}-definable over 𝔸 and consistent with the axioms of L_{𝔸}(O^{n}, C^{n})_{n∈ω} . Let θ^{O^{k}} and θ^{C^{k}} , k ∈ ω, be the sentences of L_{𝔸}(O^{n}, C^{n})_{n∈ω} introduced for each formula θ(x_{1}, …, x_{k}, y_{1}, …, y_{n}) as follows:

(O) θ^{O^{k}} is ∀^{n} y⃗∃^{k} x⃗(¬O^{k} z⃗θ(z⃗, y⃗) → ⋀_{m} ⋀_{χ∈Φ_{m+k−1}} θ_{χ}^{O^{k}} (x⃗, y⃗)),

where θ_{χ}^{O^{k}} (x⃗, y⃗) is

∀^{m} z⃗((O^{k} u⃗χ(u⃗, z⃗) ∧ ∀^{k}v⃗(χ(v⃗, z⃗) → θ(v⃗, y⃗))) → (θ(x⃗, y⃗) ∧ ¬χ(x⃗, z⃗))).

(C) θ^{C^{k}} is ∀^{n} y⃗∃^{k} x⃗(¬C^{k} z⃗θ(z⃗, y⃗) → ⋀_{m} ⋀_{χ∈Φ_{m+k−1}} θ_{χ}^{C^{k}} (x⃗, y⃗)),

where θ_{χ}^{C^{k}} (x⃗, y⃗) is

∀^{m} z⃗((C^{k} u⃗χ(u⃗, z⃗) ∧ ∀^{k}v⃗(θ(v⃗, y⃗) → χ(v⃗, z⃗))) → (¬θ(x⃗, y⃗) ∧ χ(x⃗, z⃗))).

Note that θ_{χ}^{O^{k}} (x_{1}, …, x_{k}, y_{1}, …, y_{n}) is the set of k-typles (x_{1}, …, x_{k}) wich are in θ, but not in any open subset of θ defined by χ using parameters, and θ^{O^{k}} means that for any parameters, if θ is not open then it is not equal to any union of open sets definable in L_{𝔸}(O^{n}, C^{n})_{n∈ω} . Similarly, θ_{χ}^{C^{k}} (x_{1}, …, x_{k}, y_{1}, …, y_{n}) is the set of k-typles (x_{1}, …, x_{k}) which are in any closed superset of θ defined by χ using parameters, and θ^{C^{k}} means that for any parameters, if θ is not closed then it is not equal to any intersection of closed sets definable in L_{𝔸}(O^{n}, C^{n})_{n∈ω} . It can be shown (as in [7]) that Γ = T ∪ {θ^{O^{k}} , θ^{C^{k}} | θ is a formula in , k ∈ ω} is consistent in L_{𝔸}(O^{n}, C^{n})_{n∈ω} . By Keisler’s construction of weak models, we get a set model with the properties of topological class-spaces (see [7] for more details). In the weak model (K, 𝕋_{n}, ℂ_{n})_{n∈ω} of the theory Γ, the condition (KC_{3}) holds, by Axiom (13), only for L_{𝔸}(O^{n}, C^{n})_{n∈ω} -definable sets. The construction of the middle model extends (KC_{3}) to all subsets of 𝕂^{n}, n ∈ ω.

Theorem 4.1 (Middle Completeness Theorem for L_{𝔸}(O^{n}, C^{n})_{n∈ω} ). Let T be a set of sentences of L_{𝔸}(O^{n}, C^{n})_{n∈ω} such that T is Σ_{1}-definable over 𝔸 and consistent with the axioms of L_{𝔸}(O^{n}, C^{n})_{n∈ω} . Then there is a middle model of T.

Proof. Let K = L∪C be the language introduced in the Keisler construction of a weak model of Γ where C is a set of new constant symbols and C ∈ 𝔸. Let M be a language with two kinds of variables: X, Y, Z, … variables for sets and x, y, z, … variables for urelements. Predicates of our language are E_{n}(x_{1}, …, x_{n}, X), O^{n}(X) and C^{n}(X), n ∈ ω, with canonical meaning (x_{1}, …, x_{n}) ∈ X, X is an open set of n-typles and X closed set of n-typles. Constant symbols are C_{φ} for each formula φ of L_{𝔸}(O^{n}, C^{n})_{n∈ω} . Let S be the following theory of M_{𝔸}:

(1) Axiom of well-definedness

∀X ⋀_{n<m} ¬∃x_{1}, …, x_{n}, y_{n+1} , …, y_{m}(E_{m}(x_{1}, …, x_{n}, y_{n+1} , …, y_{m}, X)∧

∧E_{n}(x_{1}, …, x_{n}, X)),

where {x_{1}, …, x_{n}} ∩ {y_{n+1} , …, y_{m}} = ∅;

(2) Axiom of extensionality ∀x_{1}, …, x_{n} (E_{n}(x_{1}, …, x_{n}, X) ↔ E_{n}(x_{1}, …, x_{n}, Y)) ↔ X = Y;

(3) Axiom of specification

(a) ∀^{n} x⃗ (∀^{m} y⃗y_{1}, …, y_{m}E_{n+m} (x⃗, y⃗, C_{R}) ↔ E_{n+m} (x⃗,c⃗, C_{R})) for each atomic formula R(x_{1}, …, x_{n}, c_{1}, …, c_{m}) of K_{𝔸};

(b) ∀^{n} x⃗ (E_{n}(x⃗, C_{¬φ} ) ↔ ¬E_{n}(x⃗, C_{φ}));

(c) ∀^{n} x⃗ (E_{n}(x⃗, C_{∧Φ} ) ↔ ⋀_{φ∈Φ} E_{n}(x⃗, C_{φ}));

(d) ∀^{n} x⃗ (E_{n}(x⃗, C_{∀xφ} ) ↔ ∀xE_{1+n} (x, x⃗, C_{φ}));

(e) ∀^{n} x⃗ (E_{n}(x⃗, C_{∃xφ} ) ↔ ∃xE_{1+n} (x, x⃗, C_{φ}));

(f) ∀^{m} x⃗ (E_{m}(x⃗, C_{O^{n} y⃗ φ} ) ↔ ∃X(O^{n}(X) ∧

∧ ∀^{n} y⃗(E_{n}(y⃗, X) ↔ E_{n+m} (y⃗, x⃗, C_{φ}))));

(g) ∀^{m} x⃗ (E_{m}(x⃗, C_{C^{n} y⃗ φ} ) ↔ ∃X(C^{n}(X) ∧

∧ ∀y⃗(E_{n}(y⃗, X) ↔ E_{n+m} (y⃗, x⃗, C_{φ}))));

(4) Axiom of subbases

(a) ∀^{n} x⃗ ∃X (O^{n}(X) ∧ E_{n}(x⃗, X)); [ see L_{𝔸}(O^{n}, C^{n})_{n∈ω} -axiom (12)]

(b) ∀X ∃Y (∃^{n} x⃗E_{n}(x⃗, X) → C^{n}(Y) ∧ ∀^{n} x⃗(E_{n}(x⃗, X) → E_{n}(x⃗, Y))); [ see L_{𝔸}(O^{n}, C^{n})_{n∈ω} -axiom (13)]

(c) ∀X ∀Y ∀Z (O^{n}(X) ∧ C^{n}(Y) → ((∀^{n} x⃗ (E_{n}(x⃗, Z) ↔

↔ ¬E_{n}(x⃗, Y) ∧ E_{n}(x⃗, X)) → O^{n}(Z))));

[ see L_{𝔸}(O^{n}, C^{n})_{n∈ω} -axiom (6)]

(d) ∀X ∀Y ∀Z (O^{n}(X) ∧ C^{n}(Y) → ((∀^{n} x⃗ (E_{n}(x⃗, Z) ↔

↔ ¬E_{n}(x⃗, X) ∧ E_{n}(x⃗, Y)) → C^{n}(Z))));

[ see L_{𝔸}(O^{n}, C^{n})_{n∈ω} -axiom (6)]

(5) Axioms of complete topological product and continuity (a) ∀X ∀Y ∀Z (O^{n}(X) ∧ O^{m}(Y) ∧ ∀^{n+1} x⃗ (E_{n+m} (x⃗, Z) ↔

↔ E_{n}(x_{1}, …, x_{n}, X) ∧ E_{m}(x_{n+1} , …, x_{n+m} , Y)) → O^{n+m} (Z)); [ see L_{𝔸}(O^{n}, C^{n})_{n∈ω} -axiom (9)]

(b) ∀X ∀Y (O^{n}(X) ∧ ∀x_{i_{1}} , …, x_{i_{k}} (E_{k}(x_{i_{1}} , …, x_{i_{k}} , Y) ↔

↔ E_{n}(x_{σ(1)} , …, x_{σ(n)} , X)) → O^{k}(Y)),

σ: {1, …, n} → {1, …, n} and the range of σ is {i_{1} < i_{2} < ⋯< i_{k}}; Similarly for the predicates C^{n}; [ see L_{𝔸}(O^{n}, C^{n})_{n∈ω} -axiom (10)]

(c) ∀X ∀Y (O^{n}(X) ∧ ∀^{k} x⃗ (∃y_{k+1} , …, y_{n} E_{n}(x⃗, y_{k+1} , …, y_{n}, X) →

→ ∀x_{k+1} , …, x_{n} (E_{n−k} (x_{k+1} , …, x_{n}, Y) ↔

↔ E_{n}(x_{1}, …, x_{n}, X))) → O^{n−k} (Y));

[ see L_{𝔸}(O^{n}, C^{n})_{n∈ω} -axiom (11)]

(d) ∀X∀^{n} x⃗( O^{n}(X)∧E_{n}(x⃗, X) → ∃Y_{1} …∃Y_{n} ( ⋀_{i=1}^{n} ( O(Y_{i})∧E_{1}(x_{i}, Y_{i})) ∧

∧ ∀^{n} y⃗ ( ⋀_{i=1}^{n} E_{1}(y_{i}, Y_{i}) → E_{n}(y⃗, X)) ) );

[ see L_{𝔸}(O^{n}, C^{n})_{n∈ω} -axiom (15)]

(e) ∀X∀^{n} x⃗( C^{n}(X)∧¬E_{n}(x⃗, X) → ∃Y_{1} …∃Y_{n}( ⋀_{i=1}^{n} ( O(Y_{i})∧E_{1}(x_{i}, Y_{i})) ∧

∧ ∀^{n} y⃗ ( ⋀_{i=1}^{n} E_{1}(y_{i}, Y_{i}) → ¬E_{n}(y⃗, X)) ) );

[ see L_{𝔸}(O^{n}, C^{n})_{n∈ω} -axiom (16)]

(f) ∀X ∀Y ∀Z (O^{m}(X) ∧ O^{n}(Z) →

→ (∀z_{1}, …, z_{n}, y_{k+1} , …, y_{m} ((∃y_{1}, …, y_{k}(E_{m}(y_{1}, …, y_{m}, X) ∧

∧ E_{n+k} (z_{1}, …, z_{n}, y_{1}, …, y_{k}, C_{φ})) ∧ E_{n}(z_{1}, …, z_{n}, Z) ↔ ↔ E_{m+n−k} (z_{1}, …, z_{n}, y_{k+1} , …, y_{m}, Y)) → O^{m+n−k} (Y)))),

for each (n, k)-ary relation φ; [ see L_{𝔸}(O^{n}, C^{n})_{n∈ω} -axiom (14)]

(6) Axioms which are transformations of all axioms φ of K_{𝔸}(O^{n}, C^{n})_{n∈ω} ∀x_{1}, …, x_{n} E_{n}(x_{1}, …, x_{n}, C_{φ});

(7) Axioms of realizability of all sentences φ in Γ ∀x E_{1}(x, C_{φ}).

The theory S is Σ-definable over 𝔸 and each S_{0} ⊆ S, S_{0} ∈ 𝔸 has a standard model since the axiom

⋀_{n} ⋀_{φ∈(S′_{0})_{n+k−1}} ∀^{n} x⃗ ⋁_{m} ⋁_{ψ∈(S′_{0})_{m+k−1}} ∃^{m} y⃗(C^{k} z⃗ψ(z⃗, y⃗) ∧ ∀^{k} u⃗(φ(u⃗, x⃗) → ψ(u⃗, y⃗)))

holds in the weak model (K, 𝕋_{n}, ℂ_{n})_{n∈ω} , which can be transformed to a standard model for S_{0} (see [7]), where S_{0} ⊆ S′_{0}, S′_{0} ∈ 𝔸 is closed under substitution of the constant symbol from 𝕂 and disjunction, and

(S′_{0})_{n} = {φ ∈ S′_{0} | φ has n + 1 free variables }.

It follows by means of the Barwise compactness theorem (see [1]) that S has a standard model B, which can be transformed to a middle model (K, 𝕋_{n}, ℂ_{n})_{n∈ω} of Γ, similarly as in the logic L_{𝔸}(O, C). □

Let (K, 𝕋′_{n}, ℂ′_{n})_{n∈ω} be the middle model of T where T is Σ_{1}-definable over 𝔸 consistent with the axioms of L_{𝔸}(O^{n}, C^{n})_{n∈ω} . Classes 𝕋′_{n} and ℂ′_{n} for each n ∈ ω satisfy conditions 2.1, so we can form the classes

\overline{𝕋}_{n} = \{x | x is a finite intersection of members of 𝕋′_{n}\}

and

\overline{ℂ}_{n} = \{x | x is a finite unionof members of ℂ′_{n}\}.

As in the logic L_{𝔸}(O, C) we can prove that

(K, 𝕋′_{n}, ℂ′_{n})_{n∈ω} ≡_{L_{𝔸}(O^{n} ,C^{n} )_{n∈ω}} (K, 𝕋_{n}, ℂ_{n})_{n∈ω} ,

where 𝕋_{n} = \{⋃ x | x ∈ \overline{𝕋}_{n}\} and ℂ_{n} = \{⋂ x | x ∈ \overline{ℂ}_{n}\} for each n ∈ ω using the sentences θ^{O^{k}} and θ^{C^{k}} for each formula θ(z_{1}, …, z_{n}, y_{1}, …, y_{n}) of L_{𝔸}(O^{n}, C^{n})_{n∈ω} . The given structure will be a complete topological class model for T. We have thus proved the following theorem.

Theorem 4.2 (Completeness Theorem for L_{𝔸}(O^{n}, C^{n})_{n∈ω} ). If T is Σ_{1}-defin-able over 𝔸 and consistent with the axioms of L_{𝔸}(O^{n}, C^{n})_{n∈ω} , then there is a complete topological class model of T.

5. Conclusion

We introduced the logic L_{𝔸}(O^{n}, C^{n})_{n∈ω} related to topological class-spaces and suitable for studying continuous functions and product class-topologies. A sound and complete axiomatization is given, and the completeness theorem is proved. The proof illustrates the power of Rašković’s middle-model method in producing new results. In general, it is worth noticing that Barwise compactness and the great expressive power of infinitary logics can provide more subtle models that can be obtained from ordinary compactness of the first-order logic.

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Faculty of Science (Received 02 03 2016) University of Kragujevac (Revised 03 05 2016) Kragujevac Serbia

rdjordjevic@kg.ac.rs

Faculty of Pedagogy University of Kragujevac Jagodina Serbia

vladimir.ristic@pefja.kg.ac.rs

Faculty of Mathematics University of Belgrade Belgrade Serbia

ikodinovic@matf.bg.ac.rs