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Extra info for An introduction to mathematical logic

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F and x 2= FV(G) so one has S j= G ! 8xF b) Is the condition x 2= FV(G) necessary? Prove your claim. 4. 5. Hint: Show rst that for a term s also sx (t) is a term with FV(sx (t)) FV(t) (FV(s) n fxg): Do we have in general FV(Fx(t)) = FV(t) (FV(F) n fxg)? 5. Let F be an L-formula, S an L-structure and an S -assignment. 6. 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.

8. 2. For a formula F of a Tait-language F is de ned by induction on the de nition of the formulas. 1. (Pt1 : : : tn) = (Pt1 : : :tn ) (Pt1 : : :tn ) = (Pt1 : : :tn ) 2. (F ^ G) = ( F _ G) (F _ G) = ( F ^ G) 3. (8xF ) = (9x F) (9xF) = (8x F) As an easy consequence of the de nition we get the syntactical property ( F) = F: The intended meaning of F was that of :F: To see that we really met this intention we prove the following proposition. 3. Let S be an L-structure, an S -assignment and F an L-formula.

Therefore assume that there is a set M of LGT -sentences such that for any group G G j= M i G has the torsion property. e. = is interpreted standardly. 10. Take T be AxGT plus the sentences :(cn = 1) for every n 1 44 I. Pure Logic where c is a new constant symbol. But then every nite set of T M has a model since there are only nitely many sentences containing the new constant symbol. All we have to do is to nd a group with the torsion property in which c can be interpreted by an element of order N, where N is bigger than all n such that :(cn = 1) occurs in the given nite set of sentences.