Box diagrams

Completeness Theorem

If is unsatisfiable, there is a refutation of .

Compactness Theorem

If is unsatisfiable, there is an unsatisfiable finite subset of .

Sko-Lo

If is satisfiable, there is an interpretation with an enumerable domain such that .

Lemma I

For any set , there is a canonical derivation from .

matches

Lemma II

What is it; how the proof of it works.

`` So let us suppose that is unsatisfiable. By Lemma I, there is a canonical derivation from . Since is unsatisfiable and is a subset of all the sentences appearing in , there is no model of the set of all such sentences. By Lemma II, then, there is no model of the set of quantifier-free sentences in that matches . If we could prove a proposition to the effect that

(+)

if every finite subset of is satisfiable then some interpretation maches

then we should have proved the completeness theorem, for we should know that some finite set of the quantifier-free sentences in is unsatisfiable" (p. 135).

Let's make sure we get this

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