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By John Stuart Mill

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Additional info for A System of Logic Ratiocinative and Inductive, Part II (The Collected Works of John Stuart Mill - Volume 08)

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Xn ) ↔ ϕ(x1 , . . , xn )). 1 That is, compact but not necessarily Hausdorff. 32 3. Quantifier elimination The resulting theory, the Morleyisation T m of T , has quantifier elimination. Many other properties of a theory are not affected by Morleyisation. So T is complete if and only if T m is; similarly for κ-categoricity and other properties we will define in later chapters. A prime structure of T is a structure which embeds into all models of T . The following is clear. 2. A consistent theory T with quantifier elimination which possesses a prime structure is complete.

Xn ] such that f(a, da, . . 7). We may find some b with f(b, db, . . , d n b) = 0 and g(b, ba, . . , d n−1 b) = 0 for all g ∈ K [x0 , . . , xn−1 ] \ 0 in an elementary extension of F2 . The field isomorphism from K1 = K (a, . . , d n a) to K2 = K (b, . . , d n b) fixing K and taking d i a to d i b takes the derivation of F1 restricted to K(a, . . , d n−1 a) to the derivation of F2 restricted to K (b, . . , d n−1 b). 3 implies that K1 and K2 are closed under the respective derivations, and that K1 and K2 are isomorphic over K as differential fields.

Let K be a field. Then the theory of all infinite K -vector spaces has quantifier elimination and is complete. Proof. , a subspace) of the two infinite K -vector spaces V1 and V2 . Let ∃y (y) be a simple existential L(A)-sentence which holds in V1 . Choose a b1 from V1 which satisfies (y). If b1 belongs to A, we are finished since then V2 |= (b1 ). If not, we choose a b2 ∈ V2 \ A. Possibly we have to replace V2 by an elementary extension. The vector spaces A + Kb1 and A + Kb2 are isomorphic by an isomorphism which maps b1 to b2 and fixes A elementwise.

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