we can introduce other operational symbols by definition, such as the other standard connectives ∧, ∨ and ↔.
(4) In a general way, let Fn := {hf, α1, α2, . . . , αar(f)i : f ∈ fλ ∧αi ∈ Fn−1}, then
(5) F = S
n∈ω Fn.
We can prove that A = hF, fλi is a free algebra by adapting the proof given in Barnes & Mack (1975, p.5). We call a free algebra constructed this way a language.
Thus we can see how a language can be constructed in ZFC. The cases of first-order usual languages, higher-order languages and even infinitary languages can be treated the same way, although they will demand more details. Anyway, they can be treated as being certain free algebras. Thus, when we mention certain languages to speak of structures in the next section, they can be considered as constructed within ZFC.
6.5 Languages for speaking of structures
Of course we aim at to speak of structures and of all the objects of a scale. To do that in an adequate way, we consider two ba- sic infinitary languages, termed (cf. da Costa and Rodrigues) Lωωω(R) (or simply Lω(R)) and Lωωκ(R).
In general, an infinitary language Lηµκ, with κ < µ being infinite cardinals (or ordinals) and 1 ≤ η ≤ ω, enables us to consider conjunctions and disjunctions of n ≤ µ formulas
and blocks of quantifiers with m < κ many quantifiers. The superscript η indicates the order of the language (first-order, second order, etc.). In both casesRis the set of the constants of the language. Thus, in Lωµω(R) (ω < µ) we may have infinitely many conjunctions and disjunctions of formulae, but blocks of quantifiers with finitely many quantifiers only. Lωωω(R) is a higher-order language, suitable for type theory (higher-order logic). Standard first-order languages are of the kind L1ωω, so is L∈. Put in a more precise way,
Definition 6.5.1 (Order of a language) A language Lnµκ, with 1 ≤ n < ω, is called a language of order n. A language Lωµκ is said to be or order ω.
A language of order n contains only types of order t ≤ n and quantification of variables of types having order ≤ n − 1 (da Costa & Rodrigues 2007, p.8).
In order to exemplify how we can define a higher-order lan- guage using a frst-order language (such as L∈), let us sketch the language Lωω
1ω(R), but we could consider whatever infini- tary languageLωµκ, provided that the involved cardinals exist in ZFC (for instance, we couldn’t use an inaccessible cardinal).3
The primitive symbols of Lωω
1ω(R) are the following ones:4 (i) Sentential connectives: ¬, ∧, ∨, →, V
, and W .
3As we have seen in the last chapter, there are cardinals whose existence cannot be proved in ZFC, provided this theory is consistent. Inaccessible cardinals belong to this class.
4Of course we could use the above schema of free algebras to characterize this language, but this would demand a lot of artificiality and will not conduce to nothing really relevant. The important thing is to acknowledge that the languages we will consider can be treated as free algebras.
6.5. LANGUAGES FOR SPEAKING OF STRUCTURES 89
(ii) Quantifiers: ∀ and ∃
(iii) For each type t, a family of variables of type t whose cardinal is ω.
(iv) Primitive relations: for any type t, a collection of con- stants of that type (possibly some of them may be empty).
The collection of these constants form the set R.
(v) Parentheses: left and right parentheses (‘(’ and ‘)’), and comma (‘,’).
(vi) Equality: =t of type t = ht1,t2i, with t1 and t2 of the same type.
Variables and constants of type t areterms of that type. If T is a term of type ht1, . . . ,tni and T1, . . . ,Tn are terms of types t1, . . . ,tn respectively, then T(T1, . . . ,Tn) is anatomic formula.
If T1 and T2 are terms of the same type t, then T1 =ht1,t2i T2 is an atomic formula. We shall write T1 = T2 for this last for- mula, leaving the type of the identity relation implicit. If α, β, αi are formulas (i = 1, . . . ), then ¬α, α ∧ β, α ∨ β, α → β, Vαi, and Wαi are formulas. Then, we are able to write for- mulas with denumerably many conjunctions and disjunctions.
Furthermore, if X is a variable of type t, then ∀Xα and ∃Xα are also formulas (and only finite blocks of quantifiers are al- lowed). These are the only formulas of the language. The concepts of free and bound variables and other syntactic con- cepts can be introduced as usual.
Now let E = hD,rιi be a structure, where rι ∈ R, that is, the primitive relations of the structure are chosen among the con- stants of our language Lωω
1ω(R). Then, Lωω
1ω(R) can be taken as a language forE = hD,rιi, provided that κD = ω (recall that κD is the cardinal associated to E). Still working within (say) ZFC, we can define an interpretation ofLωω
1ω(R) inE = hD,rιi in an obvious way, so as what we mean by a sentence S of Lωω
1ω(R) (a formula without free variables) being true in such a structure in the Tarskian sense, that is,
E |= S. (6.4)
In the same vein, we can define the notion of validity. A sentence S is valid, and we write
|= S, if E |= S for every structure E.
Important to emphasize that we are describing the language Lωω
1ω(R) using the resources of some set theory such as ZFC.
This way, we can speak of denumerable many variables, for instance, in a precise way. In this sense, any symbol ofLωωκ(R), as we have remarked already, can be seen as a name for a set.
Thus, ‘(’ (the left parenthesis), for example, names a set.
6.5.1 The language of a structure
Now let E = hD,rιi be a structure, while rng(rι) denote the range of rι. Remember that rι stands for a sequence of rela-