Logic and Foundations

Formal systems, set theory, model theory, type theory, and proof theory.

foundation tier

Logic and Foundations. Formal systems, set theory, model theory, type theory, and proof theory. This page collects canonical references that organise the subject and provide entry points to its main techniques.

Foundations and canonical references

The standard treatments of logic and foundations approach the subject from complementary angles. Enderton, A Mathematical Introduction to Logic (2001) is the anchor reference for the subject and lays out the core definitions, theorems, and worked examples that practitioners return to. Ebbinghaus, Mathematical Logic (1994) gives a parallel, more proof-oriented exposition of the same material and is widely used as a graduate text.

Open methodological questions for logic and foundations include sharpening the bridges between foundational theory and computational practice, extending classical results to broader or more structured settings, and integrating the techniques surveyed above with adjacent mathematical disciplines. The references listed in this page are the entry points that current work builds on.

Prerequisites

Sources

  • textbook · primary · 2001
    A Mathematical Introduction to Logic
    enderton-2001
  • textbook · primary · 1994
    Mathematical Logic
    ebbinghaus-1994, flum-1994, thomas-1994

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  1. 01

    Propositional Logic

    Boolean connectives, truth tables, and propositional proof systems.
  2. 02

    First-Order Logic

    Predicates, quantifiers, completeness, and compactness theorems.
  3. 03

    Proof Systems

    Natural deduction, sequent calculus, and Hilbert systems.
  4. 04

    Gödel's Incompleteness Theorems

    Coding, fixed points, and the limits of formal systems.
  5. 05

    Computability Theory

    Turing machines, recursive functions, and the arithmetic hierarchy.
  6. 06

    Model Theory

    Structures, theories, types, and stability.
  7. 07

    Proof Theory

    Cut elimination, ordinal analysis, and Gentzen-style proofs.
  8. 08

    Type Theory

    Simply typed, dependent, and homotopy type theories.
  9. 09

    Formalized Mathematics

    Mathematical libraries in Lean, Coq, Isabelle, and Mizar.
  10. 10

    Categorical Logic

    Logic via categories: classifying topoi and Lawvere theories.
  11. 11

    Set Theory

    Axiomatic set theory, cardinality, ordinals, and forcing.
  12. 12

    Axiomatic Set Theory

    ZFC, NBG, and Morse–Kelley axiomatizations.

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