Mathematical Logic Overview

Milestones in Mathematical Logic

• Gödel's Incompleteness Theorem marks a milestone in recursion theory and proof theory.

Areas of Contemporary Mathematical Logic

• According to the Handbook of Mathematical Logic, the four areas of contemporary mathematical logic are:
  • Model theory
  • Proof theory
  • Set theory
  • Recursion theory

Emergence of Mathematical Logic

• Mathematical logic emerged as a separate subfield of mathematics in the mid-19th century.

Historical Development of Logic

• Ancient Greek culture developed theories of logic in history.
• Greek methods particularly focused on dialectics.

Category Theory

• Category theory is often proposed as a foundation for mathematics.
• Category theory uses categorical and functorial methods.

Pre-Emergence of Mathematical Logic

• Logic preceded the emergence of mathematical logic as a subfield of philosophy.

Mathematical Logic Inclusions

• Sometimes, proof theory and set theory are included as part of mathematical logic.

Subareas of Mathematical Logic

• The major subareas of mathematical logic are:
  • Model theory
  • Proof theory
  • Set theory
  • Recursion theory

Axiomatic Frameworks

• The study of axiomatic frameworks for geometry, arithmetic, and analysis began in the late 19th century.

Early 20th-Century Mathematical Logic

• The study of early 20th-century mathematical logic was shaped by Hilbert's program to prove the consistency of foundational theories.

Formalization of Mathematics

• The work of von Neumann, Gödel, and Bernays showed that almost all ordinary mathematics can be formalized in terms of sets.

Resolution of Consistency Problem

• The results of Kurt Gödel, Gerhard Gentzen, and others provided a partial resolution to the consistency problem.

Common Subject Matter

• The common subject matter for mathematical logic includes the study of logical structures and formal systems.