In these first two chapters we have developed a vocabulary for talking about mathematical structures, mathematical languages, and deductions. Chapter 2 has focused on deductions, which are supposed to be the formal equivalents of the mathematical proofs that you have seen for many years. We have seen some results, such as the Deduction Theorem, which indicate that deductions behave like proofs behave. The Soundness Theorem shows that deductions preserve truth, which gives us some comfort as we try to justify in our minds why proofs preserve truth.
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.