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Extra info for Extensions As Representative Objects In Frege's Logic
Example text
Groundwork The next result follows immediately from the previous exercise. Both results are easily proved by induction on the complexity of formulas. See [Chang & Keisler 1973]. 24 Lemma. 25 Definition. Let E C M be linearly ordered by <, which need not be a definable relation in M. The ordered sequence (E, <) is a sequence of order indiscernibles if for any φ(x) and any pair e, e' of finite sequences from E, which are both in increasing order, (= φ(e) <-* φ(e'). The following result is proved using the compactness theorem and Ramsey's theorem.
7 it is rather easy to see that if φ(x) is strongly minimal in T then the dependence relation defined by algebraic closure allows us to assign a dimension to the strongly minimal set. If T is not No-categorical, it is easy to conclude that infinitely many distinct finite dimensions are possible. Thus when φ contains no parameters one concludes that T has NO countable models. 34, some finite inessential extension of every NI but not N0-categorical theory has infinitely many countable models. To prove that T itself has infinitely many models is much more difficult.
Formula of the form 0(0; 6 — y) or is inconsistent. 10 Exercise. p. formula without parameters then φ(M) is a submodule of Mn. p. p. definable subgroups where the meet of two such subgroups is their intersection and the join is defined by (φ + ψ)(v) = (3w)φ(w) Λ ψ(v — w). The following exercise is crucial to our development. 11 Exercise. p. formula φ and any α, 6, either 0(z;α) and 0(x;6) are equivalent or they are contradictory. Here is the main theorem of this section. 12 Theorem. Fix an R- module M.