Duration of the online course: 12 hours and 19 minutes
New
Sharpen rigorous reasoning with this free online logic course—learn proofs, semantics, and compactness, plus exercises to prepare for advanced math.
In this free course, learn about
What a formal system is: syntax, axioms, inference rules, and semantics for mathematics
Building strings/tuples and basic counting in Cartesian powers (e.g., |A^2|)
Sentential logic syntax: connectives, well-formed formulas, and truth assignments
Core propositional laws: implications, tautologies, equivalence, and semantic entailment
Expressive completeness: defining new connectives (e.g., majority) from standard ones
Structural induction and recursion on closures; conditions for well-defined recursion
First-order logic language: terms, atomic formulas, quantifiers, and variable assignments
Free vs bound variables; interpreting terms and formulas in structures
Elementary classes/equivalence, definable sets, and embeddings/homomorphisms preservation
Non-definability techniques using automorphisms and compactness (e.g., graph connectedness)
Compactness applications: linear extensions, graph coloring, König’s lemma, equivalence relations
Proof theory: syntactic implication, logical axioms, substitution, generalization, modus ponens
Soundness, maximal consistent sets, and Gödel completeness via Henkin constants and term models
About the free online 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.
Course content
Video class: 1. Introduction to Mathematical Logic13m
Exercise: Which components together define a formal system suitable for developing mathematics?
Video class: 2. Logic. Strings03m
Exercise: How many tuples are in A^2 when A has 3 elements?
Video class: 3. Logic. The Language Of Sentential Logic09m
Exercise: Identify the well-formed formula in sentential logic
Video class: 4. Logic. Truth Assignments16m
Exercise: When is an implication P -> Q false?
Video class: 5. Logic. Tautological Implication17m
Exercise: Identify the valid tautological equivalence
Video class: 6. Logic. The completeness of the language for sentential logic21m
Exercise: Expressing the majority connective using standard connectives
Video class: 7. Logic. Generating sets out of functions24m
Exercise: Key step to prove top down subset bottom up closure
Video class: 8. Logic. Structural induction and Recursion16m
Exercise: What key condition ensures that a recursive definition on a closure is well defined?
Video class: 9. Logic. The Language of First Order Logic20m
Exercise: Identify the atomic formula in first order logic
Video class: 10. Logic. Structures11m
Exercise: In a c4-structure M, what do the quantifiers 220 and 203 range over?
Video class: 11. Logic. Free Variables14m
Exercise: Identify the set of free variables in the formula: (forall x P(f(x), y)) and (exists y Q(y, z)).
Video class: 12. Logic. Interpretation of Terms10m
Exercise: Understanding variable assignments in first-order logic
Video class: 13. Logic. Interpretation of Formulas12m
Exercise: Semantics of the Universal Quantifier
Video class: 14. Logic. Elementary Classes22m
Exercise: Which class is weakly elementary but not elementary?
Video class: 15. Logic. Elementary Equivalence11m
Exercise: Which pair of structures are elementarily equivalent in the language with 0 and plus only
Video class: 16. Logic. Logical Implication09m
Exercise: Meaning of semantic entailment Gamma entails phi
Video class: 17. Logic. Definable Sets12m
Exercise: Which y satisfy ∃x x × x = y in the structure of real numbers with multiplication
Video class: 18. Logic. Homomorphisms15m
Exercise: Which best describes an embedding between two structures in the same vocabulary?
Video class: 19. Logic. Preservation results22m
Exercise: What do embeddings preserve?
Video class: 20. Logic. Non-definability results using automorphisms13m
Exercise: Using automorphisms to show non-definability in Z with 0 and +
Video class: 21. Logic. Substructures11m
Exercise: Which condition correctly ensures that M is a substructure of N for a common vocabulary
Video class: 22. Logic. Compactness13m
Video class: 23. Logic. An application of compactness20m
Exercise: Compactness and the non axiomatizability of connected graphs
Video class: 24. Logic. Proving non-definability via compactness08m
Exercise: Why is connectedness not first order definable in graphs?
Video class: 25. Logic. Compactness via implication10m
Exercise: Compactness corollary and infinite groups
Video class: 26. Logic. Compactness in Sentential Logic16m
Exercise: Compactness in Sentential Logic: Core Statement
Video class: 27. Logic. An application of Compactness for Sentential Logic20m
Exercise: How does compactness prove that every partial order admits a linearization?
Video class: 28. Logic. Topological Compactness of [0,1]13m
Exercise: Which principle justifies that if every finite subset of Γ is satisfiable, then Γ is satisfiable, allowing construction of x in 0,1 outside the union of rational intervals?
Video class: 29. Logic. Compactness: from Cantor Set to Sentential Logic20m
Exercise: Satisfiability and Cantor set covering
Video class: 30. Logic. Application of compactness to graph coloring29m
Exercise: Which statement captures the compactness-based result about k-colorability of graphs?
Video class: 31. Logic. Proving König's lemma from compactness23m
Exercise: In the compactness-based proof of Koenig lemma for trees in 0-1 sequences, what do the propositional variables A_sigma represent?
Video class: 32. Logic. An application to Compactness to Equivalence Relations16m
Exercise: Compactness and non-elementarity of finite equivalence classes
Video class: 33. Logic. Application of Compactness: fields of characteristic 0 not strongly elementary12m
Exercise: Fields of characteristic 0 and strong elementary classes
Video class: 34. Logic. Syntactical Implication10m
Exercise: Condition for a set to contain all theorems from Gamma
Video class: 35. Logic. Logical Axioms18m
Exercise: When is a term t substitutable for a variable x in a formula alpha for the substitution axiom
Video class: 36. Logic. Tautologies in First Order Logic12m
Exercise: What is a first order tautology as used in axiom schema 1?
Video class: 36. Logic. The Generalization Metatheorem16m
Exercise: Condition for Universal Generalization in a Deduction
Video class: 37. Logic. More Metatheorems18m
Exercise: Which inference rule derives B from A and A implies B
Video class: 38. Logic. Generalization of Constants, change of variables, equality24m
Exercise: Generalization of constants: deriving universal quantification
Video class: 39. Logic. The Soundness theorem09m
Video class: 40. Logic. Maximal Consistent sets of sentences13m
Exercise: Constructing a maximal consistent extension
Video class: 41. Logic. Gödel's Completeness theorem07m
Exercise: Identify the statement that expresses the completeness theorem in first-order logic
Video class: 42. Logic. The Set Of Closed Terms30m
Exercise: Identify the relation E used in the term model construction
Video class: 43. Logic. The Term Model11m
Exercise: What ensures function interpretations on the quotient of closed terms are well defined
Video class: 44. Logic. Term Models for maximal consistent sets of formulas11m
Exercise: Role of maximal consistency in term models
Video class: 45. Logic. Theories that contain term witnesses15m
Exercise: Ensuring completeness in the term model
Video class: 46. Logic. Finishing the proof of completeness15m
Exercise: Purpose of adding Henkin constants in the completeness proof
Video class: 47. Logic. Summing up07m
Exercise: Reconciling Completeness and Incompleteness in First-Order Logic
What is the difference between soundness and completeness in first-order logic?
Soundness means every provable statement is logically valid; completeness means every logically valid statement is provable.
Why is connectedness not first-order definable in graph theory?
By compactness, any finite set of sentences can enforce paths of bounded length, but no first-order theory can enforce a path of some finite length between every pair of vertices.
What does the compactness theorem say in first-order logic?
If every finite subset of a set of first-order sentences has a model, then the entire set has a model.
Ready to get started?Download the app and get started today.
