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As you look at the statement of the Soundness Theorem, you can see that it is explicitly trying to relate the syntactical notion of deducibility $$\left( \vdash \right)$$ with the semantical notion of logical implication $$\left( \models \right)$$. The first major result of Chapter 3, the Completeness Theorem, will also relate these two notions and will in fact show that they are equivalent. Then the Compactness Theorem (which is really a quite trivial consequence of the Completeness Theorem) will be used to construct some mathematical models with some very interesting properties.