D'Math University | Education & Specialist Mathematics
MSc Mathematical Logic
A programme of exceptional depth and intellectual rigour, exploring the formal structure of mathematical reasoning itself. From Gödel's incompleteness theorems and Turing's computability to model theory, type theory, and non-classical logics — this MSc is for those who want to understand the very limits of what can be proved, computed, and known.
Programme Overview
The Logic Behind Mathematics
Mathematical logic is the study of formal systems — of what it means to prove something, what structures satisfy a given set of axioms, which functions are computable, and which truths transcend any formal system. This MSc covers all four classical branches — proof theory, model theory, set theory, and computability — while also engaging with cutting-edge developments in type theory, categorical logic, and non-classical logics with direct applications to computer science and AI safety research.
- Four classical pillars: Proof theory, model theory, set theory, and computability — all covered with full rigour
- Modern extensions: Type theory, categorical logic, modal logic, and their applications to programming language theory
- Gödel in depth: A full module dedicated to the incompleteness theorems, their proofs, and their philosophical consequences
- AI connections: Non-classical logics and type theory directly relevant to AI alignment and formal verification
The Most Fundamental Mathematics
While other branches of mathematics study particular structures — numbers, spaces, functions — mathematical logic studies the process of mathematical reasoning itself. It asks: what are the rules that govern valid inference? Are those rules complete? Can every mathematical truth be proved? Is there a boundary between the computable and the uncomputable? These are questions that have shaped the twentieth century's deepest intellectual crises and continue to animate the most important problems in mathematics and computer science.
- Reverse mathematics: Exploring exactly which axioms are needed to prove classical theorems
- Independence results: Statements that are neither provable nor disprovable from standard axioms (ZFC)
- Formal verification: Using type theory and proof assistants like Coq and Lean in assessed coursework
- Career versatility: Graduates enter academia, AI research, formal verification, and cryptography
Core Modules
Classical Logic & Proof Theory
Sequent calculi, natural deduction, cut-elimination, and the Curry-Howard correspondence linking proofs to programs. The structural theory of valid reasoning.
Model Theory
The study of mathematical structures satisfying formal theories. Compactness, Löwenheim-Skolem, quantifier elimination, and stability — the tools of model-theoretic algebra.
Set Theory (ZFC & Beyond)
Axiomatic set theory, ordinals, cardinals, the axiom of choice and its equivalents. Large cardinal axioms, forcing, and independence results including Cohen's proof on the Continuum Hypothesis.
Computability & Recursion Theory
Turing machines, Church-Turing thesis, the halting problem, degrees of unsolvability, and the arithmetical hierarchy. The mathematics of what computers can and cannot do.
Modal & Temporal Logic
Kripke semantics, possible world models, normal modal logics. Applications to formal epistemology, verification of concurrent systems, and dynamic logic.
Type Theory & Categorical Logic
Simply typed and dependent type theories, the propositions-as-types principle, and topos theory as a foundation for mathematics. Practical work with Coq and Lean proof assistants.
Non-Classical Logics
Intuitionistic logic and its BHK interpretation, paraconsistent logics rejecting ex contradictione, relevance logic, and fuzzy logic — challenging classical assumptions about negation and truth.
Gödel's Theorems (Deep Dive)
A full module: arithmetisation of syntax, construction of the Gödel sentence, the first and second incompleteness theorems in full proof, and their impact on Hilbert's programme and beyond.
Reverse Mathematics
Which axioms are actually needed? The calibration of classical theorems to their exact logical strength within the Big Five subsystems of second-order arithmetic.
Logic Dissertation
Original research in any area of mathematical logic. Past students have worked on proof complexity, non-standard models, and the logic of AI reasoning systems.
Click any course to view its objective and learning outcomes.
LOG 501 Mathematical Logic +
Objective
To establish the foundations of mathematical logic.
Learning Outcomes
- Apply propositional and predicate logic.
- Use natural deduction and sequent calculus.
- Apply soundness and completeness theorems.
- Use Henkin construction.
- Apply compactness theorem.
AND OR NOT XOR -> <->
LOG 502 Set Theory +
Objective
To study Zermelo-Fraenkel set theory rigorously.
Learning Outcomes
- Apply ZF axioms.
- Use ordinals and cardinals.
- Apply transfinite induction.
- Discuss axiom of choice.
- Use forcing in independence proofs.
AND OR NOT XOR -> <->
LOG 503 Model Theory +
Objective
To study models of first-order theories.
Learning Outcomes
- Apply Löwenheim-Skolem theorems.
- Use elementary substructures.
- Apply ultraproducts.
- Use saturation and stability.
- Apply quantifier elimination.
LOG 504 Recursion Theory +
Objective
To study computability and degrees of unsolvability.
Learning Outcomes
- Apply Turing machines and Church-Turing thesis.
- Use recursive and recursively enumerable sets.
- Apply degree theory.
- Use priority arguments.
- Discuss arithmetical hierarchy.
LOG 505 Proof Theory +
Objective
To study the structure of formal mathematical proofs.
Learning Outcomes
- Apply natural deduction and sequent calculi.
- Use cut elimination.
- Apply ordinal analysis.
- Discuss intuitionistic logic.
- Use realizability.
AND OR NOT XOR -> <->
LOG 506 Category Theory +
Objective
To study categorical foundations of mathematics.
Learning Outcomes
- Apply categories, functors and natural transformations.
- Use limits and colimits.
- Apply adjunctions.
- Use monoidal categories.
- Apply category theory to logic.
LOG 507 Type Theory +
Objective
To study formal type systems and dependent types.
Learning Outcomes
- Apply simply-typed lambda calculus.
- Use Curry-Howard correspondence.
- Apply dependent type theory.
- Use Coq or Agda.
- Discuss homotopy type theory.
LOG 508 Modal & Intensional Logics +
Objective
To study modal logics and their applications.
Learning Outcomes
- Apply Kripke semantics.
- Use modal logics K, T, S4, S5.
- Apply temporal and dynamic logic.
- Use possible-worlds semantics.
- Discuss intensionality.
LOG 509 Non-Classical Logics +
Objective
To study logics beyond classical first-order.
Learning Outcomes
- Apply intuitionistic logic.
- Use linear logic.
- Apply many-valued logics.
- Use fuzzy logic.
- Discuss substructural logics.
LOG 510 Computability & Complexity +
Objective
To classify problems by computational difficulty.
Learning Outcomes
- Apply Turing machines.
- Use P, NP, co-NP, PSPACE classes.
- Apply polynomial reductions.
- Use approximation algorithms.
- Discuss randomised complexity.
LOG 511 Foundations of Mathematics +
Objective
To examine the philosophical and logical foundations of mathematics.
Learning Outcomes
- Discuss Gödel's incompleteness theorems.
- Apply ZF and ZFC.
- Use category-theoretic foundations.
- Discuss large cardinals.
- Apply reverse mathematics.
LOG 512 Master's Research Project +
Objective
To complete an original logic research project.
Learning Outcomes
- Identify a research-quality problem.
- Apply rigorous logical methods.
- Survey relevant literature.
- Write a 15,000-word dissertation.
- Defend orally to a logic panel.
Career Outcomes
Mathematical Logician
Academic research positions in mathematics or logic departments — publishing in journals such as the Journal of Symbolic Logic, Annals of Pure and Applied Logic, and the Journal of Mathematical Logic. A small but globally active research community.
Theoretical Computer Scientist
Research in computational complexity, programming language theory, and the semantics of programming languages. Logic is the mathematical backbone of theoretical computer science and is increasingly valued in industry research labs.
Formal Verification Engineer
Using type theory and model checking to verify that critical software systems — aerospace, financial, medical — are provably correct. A highly specialised and exceptionally well-compensated career path.
AI Alignment Researcher
Applying formal logic, type theory, and computability theory to the problem of specifying and verifying the behaviour of AI systems. One of the most urgent and high-profile emerging research areas globally.
Academic (Philosophy / CS / Math)
Mathematical logicians are genuinely tri-disciplinary — equally at home in mathematics, computer science, and philosophy departments. This versatility opens academic career paths unavailable to more narrowly trained specialists.
Type Theory Specialist
Working at the frontier of programming language design and proof assistant development — organisations such as the Lean4 team, Agda developers, and Coq community actively seek graduates with type theory expertise.
Top Universities for Mathematical Logic
Why D'Math University for Mathematical Logic?
Uncompromising Rigour
This is not a survey course. Every theorem is proved in full. Students leave with genuine technical mastery of the four classical branches of mathematical logic and the ability to engage with current research literature independently.
Connections to the Frontier
Our modules are continuously updated to reflect the explosive growth of type theory and categorical logic in programming language research, and the emerging role of formal logic in AI safety — areas where demand massively outstrips supply of trained researchers.
Tri-Disciplinary Community
Students engage with faculty and visiting researchers from mathematics, computer science, and philosophy. Seminars frequently feature speakers from all three disciplines, reflecting the true nature of mathematical logic as a boundary-crossing field.
Proof Assistant Training
Hands-on experience with Coq and Lean4 throughout the programme. This practical dimension distinguishes our graduates in the formal verification and programming language theory job market.
A programme at the absolute frontier of pure mathematics and theoretical computer science. Places are strictly limited.