Free online courseMathematical Logic
Duration of the online course: 12 hours and 19 minutes
New course
Explore Mathematical Logic in this free course covering key concepts in calculus, such as sentential logic, compactness, Gödel's theorem, and more.
In this free course, learn about
- Foundations of Sentential Logic
- First-Order Logic: Syntax and Semantics
- Definability, Structures, and Model-Theoretic Notions
- Compactness and Its Applications
- Proof Systems and Metatheorems
- Gödel Completeness and Term Models
Course Description
Delve into the comprehensive world of Mathematical Logic with this free online course tailored for those interested in enhancing their understanding of calculus-based logic. Designed for basic study levels, this course meticulously covers an array of subjects structured to provide a thorough foundation in logic. Starting with an introduction to mathematical logic, the course progresses through critical concepts such as sentential logic and truth assignments.
Students will explore the completeness of sentential logic, structural induction, recursion, and the language of first-order logic. The course takes a deep dive into complex topics like elementary classes, logical implication, and definable sets. Alongside, there is a significant focus on structural analysis, including homomorphisms and preservation results.
The course discusses the nuance of non-definability results using automorphisms and substructures. Learners will benefit from an extensive study of compactness and its applications, with sections dedicated to the compactness in sentential logic, topological compactness, and its relevance to graph coloring.
A critical section on syntactical and logical axioms equips students to understand the intricacies of tautologies in first-order logic, metatheorems, and the soundness theorem. The exploration of Gödel's completeness theorem and term models will further enhance the analytical prowess of the participants.
Concluding with an insightful summary, this course is an excellent opportunity for learners to grasp the complexities of mathematical logic and its various applications in calculus. Whether you're looking to solidify your mathematical foundation or expand your knowledge, this course is a valuable resource for academic and personal growth.
Course content
- Video class: 1. Introduction to Mathematical Logic 13m
- Exercise: Which components together define a formal system suitable for developing mathematics?
- Video class: 2. Logic. Strings 03m
- Exercise: How many tuples are in A^2 when A has 3 elements?
- Video class: 3. Logic. The Language Of Sentential Logic 09m
- Exercise: Identify the well-formed formula in sentential logic
- Video class: 4. Logic. Truth Assignments 16m
- Exercise: When is an implication P -> Q false?
- Video class: 5. Logic. Tautological Implication 17m
- Exercise: Identify the valid tautological equivalence
- Video class: 6. Logic. The completeness of the language for sentential logic 21m
- Exercise: Expressing the majority connective using standard connectives
- Video class: 7. Logic. Generating sets out of functions 24m
- Exercise: Key step to prove top down subset bottom up closure
- Video class: 8. Logic. Structural induction and Recursion 16m
- Exercise: What key condition ensures that a recursive definition on a closure is well defined?
- Video class: 9. Logic. The Language of First Order Logic 20m
- Exercise: Identify the atomic formula in first order logic
- Video class: 10. Logic. Structures 11m
- Exercise: In a c4-structure M, what do the quantifiers 220 and 203 range over?
- Video class: 11. Logic. Free Variables 14m
- 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 Terms 10m
- Exercise: Understanding variable assignments in first-order logic
- Video class: 13. Logic. Interpretation of Formulas 12m
- Exercise: Semantics of the Universal Quantifier
- Video class: 14. Logic. Elementary Classes 22m
- Exercise: Which class is weakly elementary but not elementary?
- Video class: 15. Logic. Elementary Equivalence 11m
- Exercise: Which pair of structures are elementarily equivalent in the language with 0 and plus only
- Video class: 16. Logic. Logical Implication 09m
- Exercise: Meaning of semantic entailment Gamma entails phi
- Video class: 17. Logic. Definable Sets 12m
- Exercise: Which y satisfy ∃x x × x = y in the structure of real numbers with multiplication
- Video class: 18. Logic. Homomorphisms 15m
- Exercise: Which best describes an embedding between two structures in the same vocabulary?
- Video class: 19. Logic. Preservation results 22m
- Exercise: What do embeddings preserve?
- Video class: 20. Logic. Non-definability results using automorphisms 13m
- Exercise: Using automorphisms to show non-definability in Z with 0 and +
- Video class: 21. Logic. Substructures 11m
- Exercise: Which condition correctly ensures that M is a substructure of N for a common vocabulary
- Video class: 22. Logic. Compactness 13m
- Video class: 23. Logic. An application of compactness 20m
- Exercise: Compactness and the non axiomatizability of connected graphs
- Video class: 24. Logic. Proving non-definability via compactness 08m
- Exercise: Why is connectedness not first order definable in graphs?
- Video class: 25. Logic. Compactness via implication 10m
- Exercise: Compactness corollary and infinite groups
- Video class: 26. Logic. Compactness in Sentential Logic 16m
- Exercise: Compactness in Sentential Logic: Core Statement
- Video class: 27. Logic. An application of Compactness for Sentential Logic 20m
- 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 Logic 20m
- Exercise: Satisfiability and Cantor set covering
- Video class: 30. Logic. Application of compactness to graph coloring 29m
- Exercise: Which statement captures the compactness-based result about k-colorability of graphs?
- Video class: 31. Logic. Proving König's lemma from compactness 23m
- 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 Relations 16m
- Exercise: Compactness and non-elementarity of finite equivalence classes
- Video class: 33. Logic. Application of Compactness: fields of characteristic 0 not strongly elementary 12m
- Exercise: Fields of characteristic 0 and strong elementary classes
- Video class: 34. Logic. Syntactical Implication 10m
- Exercise: Condition for a set to contain all theorems from Gamma
- Video class: 35. Logic. Logical Axioms 18m
- 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 Logic 12m
- Exercise: What is a first order tautology as used in axiom schema 1?
- Video class: 36. Logic. The Generalization Metatheorem 16m
- Exercise: Condition for Universal Generalization in a Deduction
- Video class: 37. Logic. More Metatheorems 18m
- Exercise: Which inference rule derives B from A and A implies B
- Video class: 38. Logic. Generalization of Constants, change of variables, equality 24m
- Exercise: Generalization of constants: deriving universal quantification
- Video class: 39. Logic. The Soundness theorem 09m
- Video class: 40. Logic. Maximal Consistent sets of sentences 13m
- Exercise: Constructing a maximal consistent extension
- Video class: 41. Logic. Gödel's Completeness theorem 07m
- Exercise: Identify the statement that expresses the completeness theorem in first-order logic
- Video class: 42. Logic. The Set Of Closed Terms 30m
- Exercise: Identify the relation E used in the term model construction
- Video class: 43. Logic. The Term Model 11m
- 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 formulas 11m
- Exercise: Role of maximal consistency in term models
- Video class: 45. Logic. Theories that contain term witnesses 15m
- Exercise: Ensuring completeness in the term model
- Video class: 46. Logic. Finishing the proof of completeness 15m
- Exercise: Purpose of adding Henkin constants in the completeness proof
- Video class: 47. Logic. Summing up 07m
- Exercise: Reconciling Completeness and Incompleteness in First-Order Logic
This free course includes:
12 hours and 19 minutes of online video course
Digital certificate of course completion (Free)
Exercises to train your knowledge
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