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Let r,A be finite sets of propositional formulas. The following are equivalent : (i) Every truth assignment v making all cp E true makes at least one $ E A true. (ii) There is a derivation of t A using the above axioms and rules. The proof of (ii) 3 (i) is easy by induction on the length of the derivation. For the proof of (i) 3 (ii), start with a pair ( r , A ) satisfying (i). We attempt to build a derivation of r FA by working backwards. At each step, we work on a formula in U A of maximal length, breaking it apart by means of one of the rules.

Diagrams and compactness . . . . . Lowenheim-Skolem theorems . . . . . Recursively saturated models . . . . . Large and small models . . . . . . Stable theories . . . . . . . . Model-theoretic forcing . . . . . . Infinite formulas and extra quantifiers . . . References . . . . . . . . . . . . . . . . . . . . . . . ; . . . . . . . . . . . . . . . . . . . . . . . . 48 49 57 63 69 73 82 89 95 101 HANDBOOK OF MATHEMATICAL LOGIC Edited by J .

It seems somehow more to the point, however, to treat the laws of thought behind these quantifiers separately. Let L be a fixed first-order language. All formulas below are first-order formulas of L, and all terms t are terms of L. Recall our convention in Section 3 about writing c p ( t / u ) , the result of replacing u by t in cp, only in case the variables in t do not occur bound in cp. We write cp(t) for c p ( t / u ) below. Axiom Schemata of H (1) All tautologies, (2) A11 equality axioms, (3) All formulas of either of the forms Rules of Inference of H (1) (Modus Ponens) From (cp + 4 ) and cp infer 4, CH.