Model theory can look like a catalogue of definitions: structures, languages, theories, types, saturation. Its real power is a small set of transport principles that let you move from local information to global objects, and then back again. The two engines that show up everywhere are compactness and ultraproducts. Compactness turns “every finite fragment is […]

Set theory sits in an unusual position. On one hand, it supplies a common language for most of mathematics. On the other, it contains statements that cannot be decided from standard axioms alone. Forcing is the central method for proving such independence results. It is often described in metaphors: “adding a new real,” “extending the […]

Proof theory is sometimes introduced as the study of formal proofs for their own sake. Its real role in logic and foundations is more structural: it measures how strong a theory is by analyzing what kinds of deductions the theory supports, what normal forms its proofs can be reduced \to, and what well-founded principles are […]

Proof systems and computability can look like separate worlds. Proof theory studies derivations, cut elimination, normalization, and the shape of formal reasoning. Computability studies what can be effectively decided, computed, or reduced to something else. The bridge between them is one of the deepest unifying ideas in logic and foundations: proof is a form of […]

Type theory is often introduced as a catalogue of formalisms: simply typed \lambda calculus, dependent types, universes, inductive families. The fastest route to competence is different. Treat type theory as a discipline of proof engineering in which every statement is a judgment, every proof is a derivation tree, and the core lemmas are about how […]

Logic and foundations contain two kinds of results that feel opposite. Some theorems deliver completeness: every valid statement has a proof, every consistent set has a model, every derivation can be normalized. Other theorems deliver limits: there are true statements that cannot be proved, decision procedures that cannot exist, and questions that cannot be settled […]