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By Wolfram Pohlers (author), Thomas Glaß (editor)

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2. 3. 4. 56 I. 8, is the following: If we have derived T 9xF Fx(t) we can conclude T 9xF: Now we give some technical results about the Tait-calculus which will be proven in the exercises. 5. a) T b) c) d) e) In a T F and F~ is obtained from F by renaming bounded variables implies ~ F: implies T x (t): (Structural rule) T implies T ;: (_-inversion) T F0 _ F1 implies T F0 F1: (^-inversion) T F0 ^ F1 implies T F0 and T F1: rst step we show the soundness of the Tait-calculus. We de ne T : = f F : F 2 g S j= i S j= _ and prove the following lemma.

X z < y z) A structure S j= TOF is called Archimedian ordered of for any s 2 S there is an n 2 IN such that S j= x < 1| + :{z: : + 1} s] n-times a) Prove that there is no theory T such that for all LI-structures S (cf. 6) S j= T , S is an Archimedian ordered eld. 6. Let L be a rst order language. De ne a theory T such that for any Lstructure S a) S j= T , S has 3 elements. b) S j= T , S has in nitely many elements. c) Is F an L-sentence such that for all in nite L-structures S one has S j= F, then there is an m > 0, such that S 0 j= F for all L-structures S 0 with at least m elements.

4. Propositional Properties of First Order Logic 33 the smallest in nite cardinal. @1 is the next one and so on. Sets whose cardinality is @0 are called countable. More details concerning cardinals and their arithmetic can be found in the appendix. Let us return to the propositional properties of rst order languages. We want to show that we can extend a nitely sententially consistent set to a maximally nitely sententially consistent set. 14. Let M be a nitely sententially consistent set. Then there is a maximally nitely sententially consistent set M comprising M.

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An introduction to mathematical logic by Wolfram Pohlers (author), Thomas Glaß (editor)

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