By William S. Hatcher and Mario Bunge (Auth.)
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Extra resources for The Logical Foundations of Mathematics, Edition: 1st
Since a theorem of a predicate calculus is obtained from the axioms by our rules of inference, it follows that the theorems must be universally valid (again, we skirt an induction on the length of the proof of the theorem). This situation is clearly analogous to our system P in which the rules of inference (namely modus ponens) preserved the property of being tautological, and all of our axioms were tautologies. It then followed that all theorems of P were tautologies. The converse, that all tautologies of P were theorems of P, was stated but not proved, Likewise, we state, but do not prove: 30 FIRST-ORDER LOGIC THEOREM 1.
If every finite subset of X has a model, then every finite subset of X is consistent. The set X is thus consistent, since it is not inconsistent on any finite subset. But every consistent set X has a model, and our theorem is established. The compactness theorem has many useful applications to algebra and analysis. The interested reader should consult A. Robinson  and . Finally, we state (without proof) a modern form of the famous Lowenhein-Skolem theorem: THEOREM 10. If a system F has a model, then it has a finite or denumerable model; that is, a model (D, g) in which the set D is finite or denumerable.
In other words, the values of a sequence at indices corresponding to variables which are not free in A do not affect the satisfaction of A by s. Thus, even though sequences are infinite in length, we are never really concerned with more than a finite number of values at any given time since any formula has only a finite number of free variables. Using these observations, we now want to sketch the inductive proof that any closed wff X in any system F is either true or false under any given interpretation (D, g) of F.