Theoretical / proof-based
Mathematical logic & set theory · Undergraduate · Math
Learning objectives from the Mathematical logic & set theory syllabus, grouped by unit. Click an objective for study materials.
Topics typically covered
Click a topic for the full text and related unit practice.
Undergraduate Mathematical Logic & Set Theory — scope drawn from open logic/set theory notes and typical US intro logic syllabi.
Propositional and predicate logic
- Syntax and semantics of propositional logic — Syntax and semantics of propositional logic
- Truth tables and tautologies — Truth tables and tautologies
- Logical equivalence — Logical equivalence
- Normal forms: CNF and DNF — Normal forms: CNF and DNF
- Predicate logic — Predicate logic
- Interpretation — Interpretation
- Validity and satisfiability — Validity and satisfiability
- Logical consequence — Logical consequence
Proof systems and completeness
- Natural deduction or Hilbert-style proof systems (introduction) — Natural deduction or Hilbert-style proof systems (introduction)
- Soundness — Soundness
- Completeness theorems (statements) — Completeness theorems (statements)
- Compactness theorem — Compactness theorem
- Its consequences (introduction) — Its consequences (introduction)
- Löwenheim–Skolem theorems (overview) — Löwenheim–Skolem theorems (overview)
- Undecidability — Undecidability
- Gödel's incompleteness theorems (introduction) — Gödel's incompleteness theorems (introduction)
Axiomatic set theory
- Zermelo–Fraenkel axioms (ZFC) — Zermelo–Fraenkel axioms (ZFC)
- The cumulative hierarchy — The cumulative hierarchy
- Ordinals and cardinals — Ordinals and cardinals; cardinality and Cantor's theorem
- Axiom of choice — Axiom of choice
- Well-ordering (equivalents overview) — Well-ordering (equivalents overview)
- Construction of number systems from set theory — Construction of number systems from set theory
- Russell's paradox — Russell's paradox
- Limitations of naive set theory — Limitations of naive set theory
Learning objectives
Click an objective for study materials.
Propositional and predicate logic
- Syntax and semantics of propositional logic — Syntax and semantics of propositional logic
- Truth tables and tautologies — Truth tables and tautologies
- Logical equivalence — Logical equivalence
- Normal forms: CNF and DNF — Normal forms: CNF and DNF
- Predicate logic — Predicate logic
- Interpretation — Interpretation
- Validity and satisfiability — Validity and satisfiability
- Logical consequence — Logical consequence
Proof systems and completeness
- Natural deduction or Hilbert-style proof systems (introduction) — Natural deduction or Hilbert-style proof systems (introduction)
- Soundness — Soundness
- Completeness theorems (statements) — Completeness theorems (statements)
- Compactness theorem — Compactness theorem
- Its consequences (introduction) — Its consequences (introduction)
- Löwenheim–Skolem theorems (overview) — Löwenheim–Skolem theorems (overview)
- Undecidability — Undecidability
- Gödel's incompleteness theorems (introduction) — Gödel's incompleteness theorems (introduction)
Axiomatic set theory
- Zermelo–Fraenkel axioms (ZFC) — Zermelo–Fraenkel axioms (ZFC)
- The cumulative hierarchy — The cumulative hierarchy
- Ordinals and cardinals — Ordinals and cardinals; cardinality and Cantor's theorem
- Axiom of choice — Axiom of choice
- Well-ordering (equivalents overview) — Well-ordering (equivalents overview)
- Construction of number systems from set theory — Construction of number systems from set theory
- Russell's paradox — Russell's paradox
- Limitations of naive set theory — Limitations of naive set theory
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Propositional and predicate logic
Syntax and semantics of propositional logic
Coming soon - Proof systems and completeness
Natural deduction or Hilbert-style proof systems (introduction)
Coming soon - Axiomatic set theory
Zermelo–Fraenkel axioms (ZFC)
Coming soon
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