Mathematical Logic by Antonio Montalban

About this course

Build the kind of precision that makes mathematics feel coherent instead of mysterious. This free online course in Mathematical Logic develops the formal tools used to state definitions clearly, analyze arguments step by step, and prove results with confidence. If you have ever felt that a proof works but you could not explain exactly why, logic gives you a language for making every assumption explicit and every inference accountable.

You will start by learning how formal systems are put together: symbols, strings, and the rules that turn raw notation into meaningful statements. From there, you will practice reasoning in sentential logic using truth assignments and tautological implication, gaining intuition for what makes an argument valid and how complex expressions are built and evaluated. That foundation becomes a bridge to first-order logic, where quantifiers, structures, and variable assignment let you talk precisely about mathematical objects rather than just truth tables.

As the course progresses, the focus shifts from basic syntax and semantics to the deeper ideas that power modern mathematics. You will examine elementary classes and elementary equivalence, connect semantic entailment to mathematical consequence, and learn how definability works—what can and cannot be captured inside a given language. The course also highlights structure-preserving maps such as homomorphisms and embeddings, showing how logical descriptions interact with algebraic and relational viewpoints.

A major theme is compactness, a cornerstone principle with surprising applications. You will see how compactness can be used to prove results that feel far removed from logic at first glance, and how it underpins classic non-definability arguments. Later, the course moves toward proof theory: syntactical implication, axioms, metatheorems, soundness, maximal consistent sets, and ultimately Gödel’s completeness theorem, tying together what is provable and what is true in a model.

Throughout, exercises are integrated to help you move from recognition to mastery—training you to manipulate formulas correctly, reason about models, and justify each step of an argument. By the end, you will have a clearer picture of how formal reasoning supports calculus, algebra, and higher mathematics, and you will be better prepared for proof-based courses and theoretical work in math or computer science.