The Language Associated with a Relation
As explained in the introduction, we are not introducing the language, or formulas, as primitive concepts. This is because the back-and-forth conditions introduced in Chapter 1 are enough to express the basic notions of model theory efficiently. A Fraïsséan extremist would find it pointless to speak of these formulas; at most, he would consider them a heuristic disguise for the study of local isomorphisms. We could compare him to a teacher who, in an elementary class on integral calculus, introduced Lebesgue’s space L 1 as the completion of such-and-such a normed space, and remarked only at the end of the course that it happened to turn out that the elements of L 1 correspond to integrable functions. I shall not adopt such an extreme position, and am now ready to speak of formulas, which are quite handy in some cases for seeing whether relations are elementarily equivalent or not! But as I am not treating them as a principal notion, I shall allow myself to define them quickly, giving short shrift to those petty details of no mathematical content that can discourage the readers of the opening pages of a logic text.
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