This article is a rewriting of my Ph.D. Thesis, supervised by Professor G. Sabbagh, and incorporates a suggestion from Professor B. Poizat. My main result can be crudely summarized (but see below for detailed statements) by the equality: first-order theory of F[Xi]i∈I = weak second-order theory of F.
§I.1. Conventions. The letter F will always denote a commutative field, and I a nonempty set. A field or a ring (A; +, ·) will often be written A for short. We shall use symbols which are definable in all our models, and in the structure of natural numbers (N; +, ·):
— the constant 0, defined by the formula Z(x): ∀y (x + y = y);
— the constant 1, defined by the formula U(x): ∀y (x · y = y);
— the operation ∹ x − y = z ↔ x = y + z;
— the relation of division: x ∣ y ↔ ∃ z(x · z = y).
A domain is a commutative ring with unity and without any zero divisor.
By “… → …” we mean “… is definable in …, uniformly in any model M of L”.
All our constructions will be uniform, unless otherwise mentioned.
§I.2. Weak second-order models and languages. First of all, we have to define the models Pf(M), Sf(M), Sf′(M) and HF(M) associated to a model M = {A; ℐ) of a first-order language L [CK, pp. 18–20]. Let L1 be the extension of L obtained by adjunction of a second list of variables (denoted by capital letters), and of a membership symbol ∈. Pf(M) is the model (A, Pf(A); ℐ, ∈) of L1, (where Pf(A) is the set of finite subsets of A. Let L2 be the extension of L obtained by adjunction of a second list of variables, a membership symbol ∈, and a concatenation symbol ◠.