Mathematical Logic

Computability theory studies which mathematical problems can be solved using an algorithm. You will understand the limits of computation through models like Turing machines, explore the halting problem, and distinguish between decidable and undecidable problems.

Formal systems are abstract frameworks consisting of an alphabet, grammar rules, axioms, and inference rules used to derive theorems. You will understand how these systems model logical reasoning and how to analyze their consistency, completeness, and soundness.

Gödel’s incompleteness theorems establish the inherent limitations of formal mathematical systems. You will understand how these theorems prove that any consistent axiomatic system capable of doing basic arithmetic contains true statements that cannot be proven within that system.

This topic examines the axiomatic foundations and logical systems that underpin mathematical reasoning. You will understand the historical development of set theory, proof theory, and model theory, and how they address questions of mathematical truth and consistency.

Mathematical logic explores the formal systems of reasoning, proof, and computation. Learners will understand propositional and predicate logic, model theory, and the foundational limits of mathematics, including Gödel's incompleteness theorems.

Modal logic extends classical logic to represent statements about necessity, possibility, belief, and time. You will understand how to use modal operators, construct Kripke semantics, and apply these formalisms to computer science, philosophy, and artificial intelligence.

Model theory investigates the relationship between formal mathematical languages and their interpretations, or models. You will understand how to analyze the truth of logical formulas within specific mathematical structures, such as groups, fields, and graphs.

Predicate logic, or first-order logic, extends propositional logic by incorporating quantifiers and variables to express complex relationships. You will understand how to formalize mathematical statements, construct proofs, and analyze the validity of arguments containing quantifiers.

Proof techniques are formal methods used to establish the mathematical truth of statements. You will understand how to construct rigorous arguments using direct proof, contradiction, contraposition, mathematical induction, and case analysis to solve diverse mathematical problems.

Propositional logic is the branch of logic that studies truth-functional combinations of statements. You will understand how to construct truth tables, evaluate logical equivalence, and use inference rules to analyze the validity of arguments without analyzing internal sentence structure.

Set theory axioms define the foundational rules for constructing and manipulating mathematical sets. You will understand the Zermelo-Fraenkel axioms, the Axiom of Choice, and how these principles resolve paradoxes to provide a rigorous basis for all mathematics.