## Tags - Part of: [[Metamathematics]] [[Mathematics]] [[Philosophy of mathematics]] [[Logic]] [[Philosophy of mathematics]] - Related: [[Russel’s paradox]] - Includes: [[Philosophy of mathematics]] [[Set Theory]] [[ZFC]] [[Category Theory]] [[Classical logic]] [[Constructivism]] [[Computationalism]] [[Finitism]] [[Topos theory]] - Additional: ## Definitions - Study of the [[Philosophy|philosophical]] and [[Logic|logical]] and/or [[Algorithm|algorithmic]] basis of mathematics, or, in a broader sense - Mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics ## Main resources - [Foundations of mathematics - Wikipedia](https://en.wikipedia.org/wiki/Foundations_of_mathematics) - <iframe src="https://en.wikipedia.org/wiki/Foundations_of_mathematics" allow="fullscreen" allowfullscreen="" style="height:100%;width:100%; aspect-ratio: 16 / 5; "></iframe> ## Written by AI (may include factually incorrect information) - The foundations of mathematics is a field of study that focuses on the underlying theoretical frameworks and logical structures of mathematics. It seeks to understand and formalize the rules and principles that form the basis of mathematical reasoning and knowledge. Here's a comprehensive list of various branches and topics within the foundations of mathematics: ### 1. Mathematical [[Logic]] - Propositional Logic - Predicate Logic - Modal Logic - Temporal Logic - Fuzzy Logic - Intuitionistic Logic - Non-classical Logics ### 2. [[Set Theory]] - Naive Set Theory - Axiomatic Set Theory (Zermelo-Fraenkel, Von Neumann-Bernays-Gödel) - Ordinal Theory - Cardinal Theory - Infinitary Combinatorics - Large Cardinals ### 3. [[Model theory]] - First-Order Model Theory - Model-Theoretic Algebra - Model-Theoretic Semantics - Nonstandard Models - Ultrafilters and Ultraproducts ### 4. Proof Theory - Formal Proofs and Deduction - Constructive Mathematics - Cut-Elimination and Normalization - Reverse Mathematics - Consistency Proofs - Computational Proof Theory ### 5. [[Category Theory]] - Abstract Categories - Functors and Natural Transformations - Topos Theory - Monoidal Categories - Categorical Algebra ### 6. Computability and Complexity Theory - Turing Machines - Recursive and Recursively Enumerable Sets - Church-Turing Thesis - Complexity Classes - NP-Completeness - Decidability and Undecidability ### 7. Formal Semantics - Formal Languages - Syntax-Semantics Interface - Semantics of Programming Languages - Formal Specification Languages ### 8. Philosophy of Mathematics - Platonism - Formalism - Intuitionism - Logicism - Constructivism - Nominalism - Structuralism - Mathematical Realism and Anti-Realism ### 9. History of Foundations of Mathematics - Development of Axiomatic Systems - Historical Analysis of Foundational Crises - Contributions of Key Mathematicians (e.g., Cantor, Gödel, Hilbert) ### 10. Non-Standard Analysis - Hyperreal Numbers - Internal Set Theory - Applications to Calculus and Analysis ### 11. Type Theory - Lambda Calculus - Intuitionistic Type Theory - Dependent Type Theory - Homotopy Type Theory ### 12. Formal Number Theory - Peano Arithmetic - Diophantine Equations - Gödel's Incompleteness Theorems ### 13. Foundations of Geometry - Axiomatic Systems for Geometry - Non-Euclidean Geometry - Projective and Affine Geometry ### 14. Axiomatic Foundations - Axiomatization of Various Mathematical Theories - Consistency and Independence Proofs - Inter-Axiomatic Relations ### 15. Algorithmic Foundations - Algorithm Theory - Foundations of Computer Science - Theoretical Aspects of Algorithms The foundations of mathematics is an intellectually rich and challenging field that not only provides the framework for mathematical thought but also intersects with computer science, philosophy, and logic. It is fundamental for understanding the nature and limitations of mathematical knowledge.