Compactness, completeness, Lowenheim-Skolem theorems, nonstandard models, and limitations of first order logic.
Compactness and completeness are two central results that explain why first order logic is both powerful and limited, since they connect formal proof, semantic truth, and the existence of models in a precise way.
The chapter begins with the compactness theorem, which says that a set of first order sentences has a model whenever every finite subset has a model, and this result shows that global satisfiability can often be detected through finite pieces.
The completeness theorem connects syntax with semantics by showing that every logically valid first order sentence can be proved in a suitable formal proof system, so semantic consequence and formal derivability coincide.
The Lowenheim-Skolem theorems are then introduced to describe how models of different cardinalities arise, and these results show that first order theories often have models much smaller or much larger than one might first expect.
Applications to algebra illustrate how compactness and related methods can be used to construct models with prescribed finite behavior, transfer local conditions into global conclusions, and reveal the expressive limits of first order axiomatization.
The chapter also introduces nonstandard models, especially in arithmetic, where structures satisfy the same first order sentences as the usual natural numbers while containing additional elements that do not correspond to ordinary finite numbers.
Finally, the limitations of first order logic are examined, including its inability to characterize some important structures uniquely up to isomorphism, and these limitations explain why stronger logics or additional set theoretic tools are sometimes needed.