By Wolfram Pohlers (author), Thomas Glaß (editor)
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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. If we have F G this means (F $ G)B = t for all boolean assignments B . 4. 7. Let M be a set of L-formulas. We say: 29 a) M is sententially consistent if there is a boolean assignment B such that F B = t for all F 2 M: b) M is nitely sententially consistent if every nite subset of M is sententially consistent. c) M is maximally nitely sententially consistent if M is nitely sententially consistent and maximal, which means that for any nitely sententially consistent set M0 with M0 M it is M0 = M: For example M = fA ^ :Ag is sententially inconsistent (not sententially consistent), since we have for any boolean assignment B (A ^ :A)B = B (A) ^ ( : B (A)) = f: The following lemma gives a useful characterisation of maximally nitely sententially consistent sets of formulas.
X z (transitive) 3. x y ^ y x ! t. e. e. 9x2X 8y2X(x y ! 15 using Zorn's lemma. c) Let M be an sententially consistent set. 3. e. formulas F with F B = t for all boolean assignments B : a) (A ! (B ! C)) ! ((A ! B) ! (A ! C)) b) ((A ! B) ^ (A ! C)) ! (A ! 5. 5 The Compactness Theorem for First Order Logic The aim of this section is to extend the compactness theorem for propositional logic to full rst order logic. The compactness theorem is due to Kurt Godel (1930) and Anatolii I. Mal'cev 1909, y1967] (1936).
The formulas A1 : : : An are the antecedents while S1 : : : Sm are the succedents of the sequent. Gentzen used the notions of sequents to prove his famous `Hauptsatz' (1935) which roughly says that for every derivable formula F there is already a derivation without detours, which means that it only uses sub-formulas of F: Of course this cannot be true in full generality. Since there are only nitely many sub-formulas of F this would give us a decision procedure. We are going to explore what a derivation without detours really means.
An introduction to mathematical logic by Wolfram Pohlers (author), Thomas Glaß (editor)