Completeness theorem for probability models with finitely many valued measure

Abstract The aim of the paper is to prove the completeness theorem for probability models with finitely many valued measure.


Introduction
Probability logics introduced by H. J. Keisler are logics appropriate for the study of structures of the form (A, µ) arising in Probability Theory, where A is a rst order structure and µ is a probability measure on A. The reader can nd detailed presentations on the host of probability logics in [9] and the monograph [11]. The basic probability logic L AP is similar to the in nitary logic L A except that instead of the ordinary quanti ers ∀x and ∃x, the logic L AP possesses the probability quanti ers (Px > r).
In this paper using the ideas from [2,5,10] we introduce logic L n AP which is complete for Σ de nable theories with respect to the class of probability models with nitely many valued measure. Let us note that our work could be seen as the rst step towards the widening of application frame for probability logics since in applied mathematics one often deals with (very large but) nite phenomena.

L n AP logic
The main result which enable us to prove the corresponding Completeness Theorem is the following theorem (see [2]). The logic L n AP has all the axiom schemas and rules of inference of L AP (listed in [9,11]) as well as the following axiom of nitely many valued measure: where Φn ∈ A and Φn = {φ : φ has n free variables}.
Completeness theorem will be proven by combining a consistency property argument, such as that of [9] or [6], and a weak-middle-strong construction, such as that of [10]. We need two sorts of auxiliary structures.
De nition 2.2. (i) A weak structure for L n AP is a structure (A, µn) n≥ such that each µn is a nitely additive probability measure on A n with each singleton measurable and the set Using a consistency property similarly as in [6] or [9] we prove that Σ de nable theory of L n AP is consistent if and only if it has a weak model in which each theorem of L n AP is true. Let C ∈ A be a set of new constant symbols introduced in this Henkin construction and let K = L ∪ C.

Theorem 2.3. (Middle Completeness Theorem) A Σ de nable theory T of K n AP is consistent if and only if it has a middle model in which each theorem of K n
AP is true.

Proof.
In order to prove that consistent Σ de nable theory T of K n AP has a middle model, we introduce language M with three sorts of variables, such as that of [10]: X, Y , Z, . . . variable for sets, x, y, z, . . . variable for urelements and r, s, t, . . . variables for reals from [ , ]. The predicates of M are ≤ for reals, En( x, X) for n ≥ and x = x , . . . , xn (with the canonical meaning x ∈ X) and µ(X, r) (with the meaning µ(X) = r). The constant symbols are set constant symbols Xφ, for each φ ∈ K n AP and r for each r ∈ [ , ] ∩ A. The functional symbols are + and · for reals.
Let S be the rst-order theory of M A which has the following list of formulas: 5. Axiom of nitely many valued measure where µ(X) > r is the formula (∃s)(s > r ∧ µ(X, s)); 6. Axioms for an Archimedean eld (for real numbers); 7. Axioms which are transformations of axioms of K n AP (∀ x)En( x, Xφ), where φ is an axiom of K n AP ; 8. Axiom of realizability of T (∀x)E (x, Xφ), for each sentence φ in T. A standard structure for M A is the structure The theory S is Σ de nable over A. To prove that S is consistent it is enough, by Barwise Compactness Theorem (see [1]), to show that S ⊆ S, S ∈ A has a standard model. First, note that a weak structure (A, µn) for K n AP can be transformed into a standard structure by taking: holds in the weak model (A, µn), where S ⊆ S , S ∈ A is the closure for the substitution of constant symbols from C and disjunction and (S )n = {φ ∈ S : φ has n free variables}, it follows that is the standard model for S and S , too. Lastly, note that a standard model B of S can be transformed into a middle model B of T by taking: n ( x, X R ) for an n-ary relational symbol R ∈ L, µ B n (X) = r i µ B (X, r) for X ∈ P(B).
It follows from the Loeb-Hoover-Keisler construction (see [6,9,11]) that the axiom of nitely many valued measures implies that (*) holds for all internal sets in the nonstandard superstructure. The property (*) also holds for all Loeb measurable sets because these can be approximated by internal ones. Thus, it follows from Theorem 2.1. that each middle model in which all theorems of L n AP hold is elementary equivalent to a probability model for L n AP . As a consequence of the preceding we obtain the following theorem.

Theorem 2.4. (Completeness Theorem for L n AP ) A Σ de nable theory T of L n AP is consistent if and only if T has a probability model with nitely many valued measure.
Finally, let us note that structure (A, µ), where µ is a nitely many valued probability measure, cannot be axiomatized so that extended completeness theorem holds. The following example of a countable consistent theory T in L n AP does not have a probability model with nitely many valued measure. Example. Let L = {R (x), R (x), . . .} be a ∆ de nable set which is not a subset of an element of A, and let φ , φ , . . . be an enumeration of all formulas from L n AP . Then there exists the rst predicate, denoted by Rφ n (x), not occurring in φ , . . . , φn; otherwise L ⊆ TC(φ ) ∪ . . . ∪ TC(φn) ∈ A, which would imply that L ∈ A as a ∆ de nable set.
It is obvious that the countable theory