## Tags - Part of: [[Meta]] [[Mathematics]] [[Foundations of mathematics]] - Related: - Includes: [[Foundations of mathematics]] - Additional: ## Definitions - Study of [[Mathematics]] itself using mathematical methods ## Main resources - [Metamathematics - Wikipedia](https://en.wikipedia.org/wiki/Metamathematics) <iframe src="https://en.wikipedia.org/wiki/Metamathematics" allow="fullscreen" allowfullscreen="" style="height:100%;width:100%; aspect-ratio: 16 / 5; "></iframe> ## AI # The Gigantic Map of Metamathematics --- Metamathematics is the study of mathematics itself using mathematical methods. It explores the foundations, structures, and implications of mathematical theories, often through the lens of formal logic and proof theory. Below is an extensive map outlining the vast landscape of metamathematics. --- ## 1. **Mathematical Logic** ### A. **Propositional Logic** - **Syntax and Semantics** - **Logical Connectives**: AND, OR, NOT, IMPLIES, IFF - **Truth Tables** - **Tautologies and Contradictions** - **Normal Forms**: Conjunctive (CNF) and Disjunctive (DNF) - **Resolution Principle** ### B. **Predicate Logic (First-Order Logic)** - **Quantifiers**: Universal (∀) and Existential (∃) - **Predicates and Functions** - **Domains and Interpretations** - **Gödel's Completeness Theorem** - **Compactness Theorem** - **Löwenheim-Skolem Theorems** ### C. **Higher-Order Logic** - **Second-Order Logic** - **Type Theory** - **Expressive Power vs. Decidability** ### D. **Modal Logic** - **Necessity (□) and Possibility (◇)** - **Kripke Semantics** - **Temporal Logic** - **Deontic Logic** - **Epistemic Logic** ### E. **Non-Classical Logics** - **Intuitionistic Logic** - **Fuzzy Logic** - **Paraconsistent Logic** - **Linear Logic** - **Relevance Logic** - **Quantum Logic** --- ## 2. **Proof Theory** ### A. **Formal Proof Systems** - **Hilbert Systems** - **Natural Deduction** - **Sequent Calculus** - **Tableau Methods** ### B. **Proof Transformations** - **Normalization** - **Cut-Elimination** - **Consistency Proofs** ### C. **Gödel's Incompleteness Theorems** - **First Incompleteness Theorem** - **Second Incompleteness Theorem** ### D. **Ordinal Analysis** - **Proof-Theoretic Ordinals** - **Strength of Formal Systems** ### E. **Automated Theorem Proving** - **Resolution Methods** - **Proof Assistants**: Coq, Isabelle, Lean --- ## 3. **Model Theory** ### A. **Structures and Interpretations** - **Signatures and Languages** - **Elementary Equivalence** - **Isomorphisms** ### B. **Ultraproducts and Ultrafilters** - **Łoś's Theorem** - **Nonstandard Analysis** ### C. **Stability Theory** - **Stable, Superstable, and Unstable Theories** - **Forking and Independence** ### D. **Definability and Quantifier Elimination** - **Algebraically Closed Fields** - **Real Closed Fields** ### E. **Categoricity** - **Morley's Categoricity Theorem** - **ℵ₀-Categoricity** --- ## 4. **Set Theory** ### A. **Axiomatic Set Theories** - **Zermelo-Fraenkel Set Theory (ZF)** - **Axiom of Choice (AC)** - **Von Neumann-Bernays-Gödel Set Theory (NBG)** - **Alternative Theories**: New Foundations (NF), Kripke-Platek Set Theory ### B. **Ordinal and Cardinal Numbers** - **Transfinite Induction** - **Cardinal Arithmetic** - **Continuum Hypothesis (CH)** ### C. **Large Cardinals** - **Inaccessible, Measurable, and Supercompact Cardinals** - **Inner Model Theory** ### D. **Forcing and Independence Proofs** - **Cohen's Method of Forcing** - **Independence of CH and AC** ### E. **Descriptive Set Theory** - **Borel Hierarchy** - **Projective Sets** - **Determinacy** --- ## 5. **Computability Theory (Recursion Theory)** ### A. **Models of Computation** - **Turing Machines** - **λ-Calculus** - **Recursive Functions** ### B. **Decidability and Undecidability** - **Halting Problem** - **Rice's Theorem** - **Entscheidungsproblem** ### C. **Degrees of Unsolvability** - **Turing Degrees** - **Turing Reducibility** - **Post's Problem** ### D. **Algorithmic Randomness** - **Kolmogorov Complexity** - **Martin-Löf Randomness** - **Chaitin's Ω Number** ### E. **Computable Analysis** - **Effective Procedures in Analysis** - **Computability over the Real Numbers** --- ## 6. **Formal Languages and Automata Theory** ### A. **Chomsky Hierarchy** - **Regular Languages and Finite Automata** - **Context-Free Languages and Pushdown Automata** - **Context-Sensitive Languages and Linear Bounded Automata** - **Recursively Enumerable Languages and Turing Machines** ### B. **Grammars and Parsing** - **Backus-Naur Form (BNF)** - **Parsing Algorithms** ### C. **Automata Theory** - **Deterministic and Non-Deterministic Automata** - **Minimization and Equivalence** ### D. **Computational Complexity** - **P vs. NP Problem** - **Complexity Classes**: P, NP, co-NP, PSPACE, EXPTIME - **Reducibility and Completeness** --- ## 7. **Foundations of Mathematics** ### A. **Philosophical Foundations** - **Logicism**: Frege, Russell - **Formalism**: Hilbert - **Intuitionism**: Brouwer - **Platonism and Nominalism** - **Structuralism** ### B. **Alternative Foundations** - **Constructive Mathematics** - **Reverse Mathematics** - **Univalent Foundations and Homotopy Type Theory** ### C. **Set-Theoretic Foundations** - **ZFC as a Foundation** - **Cumulative Hierarchy** ### D. **Category-Theoretic Foundations** - **Categories, Functors, Natural Transformations** - **Topos Theory** - **Structuralism in Mathematics** --- ## 8. **Philosophy of Mathematics** ### A. **Nature of Mathematical Objects** - **Abstract Entities** - **Mathematical Realism vs. Anti-Realism** ### B. **Epistemology** - **A Priori Knowledge** - **Mathematical Intuition** ### C. **Mathematical Truth** - **Objectivity and Intersubjectivity** - **Social Constructivism** ### D. **Mathematical Practice** - **Role of Proofs** - **Visualization and Heuristics** - **Computer-Assisted Proofs** --- ## 9. **Category Theory** ### A. **Basic Concepts** - **Objects and Morphisms** - **Functors and Natural Transformations** - **Duality Principle** ### B. **Universal Properties** - **Limits and Colimits** - **Adjoint Functors** - **Yoneda Lemma** ### C. **Monoidal Categories** - **Tensor Products** - **Braided and Symmetric Monoidal Categories** ### D. **Topos Theory** - **Grothendieck Topoi** - **Logical Interpretation** ### E. **Higher Category Theory** - **2-Categories and n-Categories** - **Infinity Categories** --- ## 10. **Type Theory** ### A. **Simply Typed λ-Calculus** - **Types and Functions** - **Curry-Howard Correspondence** ### B. **Dependent Type Theory** - **Martin-Löf Type Theory** - **Proof Assistants**: Coq, Agda ### C. **Homotopy Type Theory** - **Univalence Axiom** - **Higher Inductive Types** ### D. **Applications** - **Formal Verification** - **Programming Language Design** --- ## 11. **Non-Classical Logics** ### A. **Intuitionistic Logic** - **Constructive Reasoning** - **Heyting Algebras** ### B. **Linear Logic** - **Resource Sensitivity** - **Proof Nets** ### C. **Paraconsistent Logic** - **Handling Contradictions** - **Dialetheism** ### D. **Modal and Temporal Logics** - **Dynamic Logic** - **Epistemic Temporal Logic** --- ## 12. **Algebraic Logic** ### A. **Boolean Algebras** - **Stone Representation Theorem** - **Relation to Propositional Logic** ### B. **Heyting Algebras** - **Algebraic Semantics for Intuitionistic Logic** ### C. **Modal and Relation Algebras** - **Algebraic Structures for Modal Logic** ### D. **Cylindric Algebras** - **Algebraic Approach to Predicate Logic** --- ## 13. **Proof Complexity** ### A. **Length of Proofs** - **Proof Size in Different Systems** - **Lower Bounds** ### B. **Complexity Classes of Proofs** - **Frege Systems** - **Extended Frege Systems** ### C. **Propositional Proof Systems** - **Resolution** - **Cutting Planes** --- ## 14. **Reverse Mathematics** ### A. **Subsystems of Second-Order Arithmetic** - **RCA₀, WKL₀, ACA₀, ATR₀, Π₁¹-CA₀** ### B. **Classification of Theorems** - **Equivalences between Systems and Theorems** --- ## 15. **Set-Theoretic Topology** ### A. **Cardinal Functions** - **Weight, Density, Character** ### B. **Continuum Hypothesis in Topology** - **Impact on Topological Spaces** --- ## 16. **Mathematical Structures and Theories** ### A. **Universal Algebra** - **Algebraic Structures** - **Varieties and Equational Theories** ### B. **Nonstandard Analysis** - **Hyperreal Numbers** - **Internal Set Theory** --- ## 17. **Applications of Metamathematics** ### A. **Computer Science** - **Formal Verification** - **Programming Language Semantics** ### B. **Mathematical Physics** - **Logic in Quantum Mechanics** ### C. **Cryptography** - **Complexity Theory** --- ## 18. **Historical Development** ### A. **Ancient to 19th Century** - **Aristotelian Logic** - **Euclidean Geometry** ### B. **20th Century Advances** - **Hilbert's Program** - **Gödel's Contributions** ### C. **Contemporary Trends** - **Computational Logic** - **Interdisciplinary Approaches** --- ## 19. **Educational Aspects** ### A. **Teaching Logic** - **Curriculum Development** - **Interactive Theorem Proving in Education** --- ## 20. **Interdisciplinary Connections** ### A. **Linguistics** - **Formal Semantics** - **Montague Grammar** ### B. **Cognitive Science** - **Logic and Reasoning** --- ## 21. **Contemporary Research Areas** ### A. **Quantum Computing and Logic** - **Quantum Algorithms** - **Quantum Logic** ### B. **Logic and Games** - **Game Semantics** - **Logic Games** --- ## 22. **Open Problems and Conjectures** ### A. **P vs. NP Problem** ### B. **Continuum Hypothesis** ### C. **Large Cardinal Hypotheses** --- ## 23. **Metalogic** ### A. **Metatheorems** - **Soundness and Completeness** - **Compactness** ### B. **Definability and Interpolation** - **Craig Interpolation Theorem** --- ## 24. **Logic in Computer Science** ### A. **Temporal and Dynamic Logics** - **Model Checking** - **Verification of Concurrent Systems** ### B. **Logic Programming** - **Prolog** - **Answer Set Programming** --- ## 25. **Formal Epistemology** ### A. **Bayesian Epistemology** - **Probability in Reasoning** ### B. **Belief Revision** - **AGM Theory** --- ## 26. **Advanced Topics in Set Theory** ### A. **Inner Model Theory** ### B. **Determinacy Axioms** --- ## 27. **Advanced Topics in Model Theory** ### A. **Model Theory of Fields** - **Differential and Difference Fields** ### B. **Stability and Simplicity Theories** --- ## 28. **Computability in Analysis** ### A. **Effective Descriptive Set Theory** ### B. **Computable Structure Theory** --- ## 29. **Connections with Algebra and Number Theory** ### A. **Decision Problems in Group Theory** ### B. **Logic and Diophantine Equations** --- ## 30. **Historical Figures and Contributions** ### A. **Kurt Gödel** - **Incompleteness Theorems** ### B. **Alan Turing** - **Computability Theory** ### C. **Alonzo Church** - **λ-Calculus** ### D. **David Hilbert** - **Formalism and Hilbert's Program** --- ## 31. **Logic and Probability** ### A. **Probabilistic Logics** - **Reasoning under Uncertainty** ### B. **Algorithmic Randomness** --- ## 32. **Metamathematics of Fuzzy Logic** ### A. **Degrees of Truth** - **t-Norms and s-Norms** --- ## 33. **Metalogic of Artificial Intelligence** ### A. **Knowledge Representation** - **Ontologies** - **Semantic Web** --- ## 34. **Ethics and Logic** ### A. **Deontic Logic** - **Formalizing Ethical Statements** --- ## 35. **Logic in Multi-Agent Systems** ### A. **Epistemic Logic** - **Common Knowledge** - **Distributed Knowledge** --- ## 36. **Advanced Proof Theory** ### A. **Structural Proof Theory** - **Deep Inference** ### B. **Proof Interpretations** - **Realizability** --- ## 37. **Formal Ontology** ### A. **Ontology Engineering** - **Formal Representation of Concepts** --- ## 38. **Mathematical Logic and Geometry** ### A. **Geometric Logic** - **Sheaf Models** ### B. **O-Minimal Structures** --- ## 39. **Logic and Computation** ### A. **Concurrency Theory** - **Process Calculi** ### B. **Logic in Programming Languages** - **Type Systems** --- ## 40. **Metamathematics and Education** ### A. **Curriculum Development** - **Incorporating Logic into Mathematics Education** --- This map represents a comprehensive overview of metamathematics, covering foundational theories, advanced topics, historical developments, and interdisciplinary connections. Each area is a rich field of study with ongoing research and significant implications for both mathematics and related disciplines.