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A) Determine a rst order language LV S suited for talking about a vector space and its eld. Hint: use a binary predicate symbol `='. e. LV S -structures interpreting =S by f(s s) : s 2 S g, cf. 10) one has: S j= TV S , S consists of a eld and vector space over this eld. c) De ne the LV S I-structure S of the continuous functions over the eld R of the real numbers. d) Determine LV S -formulas F and G such that the following holds in all LV S Istructures S with S j= TV S : 1. s1 : : : sn 2 S are linear independent , S j= F s1 : : : sn]: 2.

G and x 2= FV(M fGg), then M ` G: Proof. a) The formula (G ! F) ! ((:G ! F) ! F) is boolean valid. 3. b) Both formulas ((G ! H) ! F) ! (:G ! F) as well as ((G ! H) ! F) ! (H ! F ) are boolean valid. 3. c) M ` (9xF ! F) ! G entails M ` :9xF ! G and M ` F ! G by b) Thus we have by an application of the 9-rule also M ` 9xF ! G and obtain M ` G by a). Here we are able to give a positive answer to the rst question: the calculus is complete. For a calculus of a similar type such a result has been observed by K.

3. Let S be an L-structure If is an S -assignment, then B : PA(L) ! 4. Let S be an L-structure and F an L formula. For any S -assignment we have F B = ValS (F ): To prove the lemma we show G 2 PP(F) ) GB = ValS (G ) by an easy induction on the de nition of G 2 PP(F): If G 2 PA(F) the claim follows from the de nition of B . e. G = :G0 G = G1 G2 follow immediately from the induction hypothesis. Since F 2 PP(F) this entails the claim of the proposition. 5. Let F be an L-formula. 6. F G implies F S G: Proof.

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