Logic & Foundations

The rules of mathematical reasoning itself — what constitutes a proof, what can be proven, and the limits of formal systems.

Logic & Foundations is the starting point of mathematics — the rules of reasoning, the limits of formal systems, and the question of what can be known with certainty. The sub-topics below are ordered so each builds on the ones before it.

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  1. 01

    Propositional Logic

    The simplest formal system — truth values, logical connectives, and the algebra of propositions.
  2. 02

    First-Order Logic

    Extending propositions with predicates, quantifiers, and variables — the standard language of mathematics.
  3. 03

    Proof Systems

    Formalizing deduction — natural deduction, sequent calculus, and the relationship between provability and truth.
  4. 04

    Model Theory

    The study of mathematical structures through the lens of formal languages — compactness, categoricity, and types.
  5. 05

    Computability Theory

    What can be computed in principle — Turing machines, the halting problem, and the limits of algorithmic reasoning.
  6. 06

    Axiomatic Set Theory

    The formal foundation of mathematics — ZFC axioms, ordinals, cardinals, and the continuum hypothesis.
  7. 07

    Gödel's Incompleteness Theorems

    The most profound results in mathematical logic — no consistent formal system can prove all truths about arithmetic.
  8. 08

    Proof Theory (Advanced)

    Analyzing proofs as mathematical objects — ordinal analysis, cut-elimination, and reverse mathematics.
  9. 09

    Type Theory

    The modern frontier — propositions as types, proofs as programs, and the unification of logic with computation.
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