# Math: Hierarchy ## General ## The foundations of Mathematics ## Math logic ## Set, Relation, Order, Function ## Proof theory ## Type Theory ## Category Theory ## Theory of Computation ## Number Theory ## Graph Theory ## Algebra ## Calculus ## Statistics *** ## (prependix) * Summary * total math debrief, complete math summary * per-topic per-page debriefs * Overview * History of math * Development of math * Math method: rigor, Euclid, axioms, formal system, proofs * Look-ups * Glossary (single page aggregation, short explanations, crosslinks?) * Abbreviations (common, in math) * Shorthands (custom, ad-hoc) * Terminology (article/term per page) * List of symbols, per-topic, iooa * Lists * mathematical objects * mathematical primitives, definitions * mathematical laws, principles, axioms * mathematical theory, theorem, lemma; conjunctions * mathematical structures * mathematical spaces * Fundamentals * phylosophy of math * about math * what is math * meta math * The Fundamentals of Mathematics * History of math * Development of math * Math method Classification - Pure vs applied - Discrete vs continuous ## PART 1: * Fundamentals * History * People * Works, books * The foundations of math ## PART 2: Logic ## PART 3: Set ## PART 4: ## PART 5 ## PART 6 ## APPENDIX * Links * articles * videos * topical links * Index * greek alphabet * glyphs * latex * symbols per math branch *** * The Foundations of Mathematics * Crisis in the Foundations of Mathematics * Demand for axiomatization * Hilbert's program * Logicism * Godel's incompleteness theorem * Candidates for foundation of mathematics * set theory (ZFC) * functions (lambda calculus) * category theory * Mathematical Foundations * Rigorous argumentation * Proofs * Mathematical logic * Logicism * Consistency * Completeness * 03-mf (Math Fundamentals) * (About math fields) * (About discrete math) * The Foundations of Mathematics * Formal system * Incompleteness * Mathematical Logic * Sets * Number theory * Algebra * Calculus * Type theory * Category theory *** ## PART I ## Mathematics * Rigorous argumentation * Axioms * Axiomatic system * Mathematical primitives ## Concepts in Mathematics * Mathematical notion, concept, idea * Mathematical objects * Mathematical structures * Mathematical method ## Fields of Mathematics * The Foundations of Mathematics * Discrete math * Applied Mathematics * Logicism * Metamathematics * Algebra * Calculus * Analysis * Coding Theory * Information Theory * Constructive Mathematics * Computer Science ## Foundations * Foundations of Mathematics * Metamathematics * Hilbert's problems * Logicism * Constructive mathematics * Intuitionism * Gödel's theorems *** ## PART II: SETS ## Sets * Set * Set Notation * Set Membership * Set Cardinality * Set Paradoxes * Set Types * Subset * Powerset * Set Operations * Ordered Pair * Cartesian Product * fundamental sets of numbers ## Naive set theory * Set * Empty set * Element * Enumeration * Extensionality * Finite set * Infinite set * Subset * Power set * Countable set * Uncountable set * Recursive set ## Set Theories * Naïve set theory * Axiomatic set theory * General set theory * Zermelo–Fraenkel set theory * ZF/ZFC * ZFC axioms * Axiom of choice * Kripke–Platek set theory * Von Neumann–Bernays–Gödel set theory * Morse–Kelley set theory * Tarski–Grothendieck set theory ## Relations * Set Relations * Cartesian product * Ordered pair * Relation Types * Binary relations * Recurrence relations * Relation Properties * Reflexivity * Coreflexivity * Irreflexivity * Symmetry * Anti-symmetry * Asymmetry * Transitivity * Anti-transitivity * Equivalence relation * Partial equality * Partial equivalence relation ## Order Theory * Total order * Partial order * Poset ## Functions * functions basics * kinds of functions * inverses * composition * isomorphism * partial functions * functions and relations * Mapping, Domain, Co-domain, Range * Pre-image, Image * Injective * Surjective * Bijective * Domain * Codomain * Image * Pre-image * Map * Injective * Surjective * Bijective *** ## PART III: NUMBER THEORY ## Counting Theory * Counting Theory * Counting Principles * Fundamental counting rules * The Rules of Sum and Product * Pascal's Identity * Pigeonhole Principle * The Inclusion-Exclusion principle * Permutations * Combinatorics * Probability * Probability Axioms * Properties of Probability * Conditional Probability * Bayes' Theorem ## Number theory * number theory * number * numeral * digit * numeral system * radix * properties of numbers * fundamental number sets ## Arithmetic * Arithmetic * Arithmetic operations *** ## PART III: MATHEMATICAL LOGIC ## Mathematical Logic * Logic * Propositions * Implication * Converse, contrapositive, biconditional * Complex statements * Logical Equivalence * Negating propositions * Predicates and Variables * quantifiers * Notation * Negating statements with quantifiers * Binding and scope * Logical Argument * Proposition, Atom, Formula, Hypotesis * Colorally, Axiom, Theorem, Lemma * Declarative statement * Premise, Conclusion * Conditional statement * Antecedent, Consequent * Sufficiency, Necessity * Deductive Argument (necessity) * Deductive argument forms * argument based on mathematics * argument from definition * categorical syllogism * hypothetical syllogism * pure hypothetical syllogism * mixed hypothetical syllogism * disjunctive syllogism * Inductive Argument (probability) * Inductive Argument Forms * prediction * argument from analogy * generalization * argument from authority * argument based on signs * causal inference * Logical Form * Logical Truth * Validity * Soundness * Strength * Cogency * Logic Interpretation * Logical Laws * Law of Identity * Law of Excluded Middle * Tautology * Fallacy * Contradictions * Contingency * Propositional Equivalences * Inverse, Converse, and Contra-positive * Duality Principle * Operators and Postulates * Closure * Associative Laws * Commutative Laws * Distributive Laws * Identity * Inverse * De Morgan's Law * Logical Connectives * negation * conjunction * disjunction * implication * bijection * Normal Forms: CNF, DNF * Rules Of Inference * conjunction elimination * conjunction introduction (1,2) * disjunction elimination, introduction * negation elimination, introduction, double negation * implication elimination, introduction * Modus Ponens, Modus Tollens * Addition * Conjunction * Simplification * MP, MT * Disjunctive Syllogism * Hypothetical Syllogism * Constructive Dilemma * Destructive Dilemma * Sequent Calculus * antecedent * consequent * Structural Rules * Weakening * Exchange * Contraction * Logic Systems * Formal Logic Systems * Symbol * Consistency * Validity * Soundness * Completeness * Well-Orderedness * Syllogistic Logic * Natural Deduction * Predicate Logic - Higher-order Logic - Constructionist Logic - Propositional Logic * Well Formed Formula * Quantifiers - First-order Logic * First-Order Theories * Formal systems * First-order languages and theories * Tautologies * Theorems and rules in first-order theories * Consistency * Consistency and completeness * Extensions by definitions * Interpretations * Herbrand–Skolem theory * Craig's interpolation lemma * Incompleteness theorem * Number-theoretic functions and predicates * Representability * Arithmetizations * The incompleteness theorem * Minimal arithmetic * First-Order Number theory * Recursive extensions * The first-order theory PA - Second-order Logic - Higher-order Logic - Multi-valued Logic - Substructural Logic * Relevant Logic * Linear Logic - Separation Logic - Temporal Logic - Model Logic ## Systems of logic * Formal logic * Formal language * Formation rule * Formal proof * Formal semantics * Well-formed formula * Set * Element * Class * Classical logic * Axiom * Theorem * Logical consequence * Type theory * Symbol * Syntax * Theory * Formal Systems * Deductive system * Axiomatic system * Hilbert style systems * Natural deduction * Sequent calculus * Traditional logic * Proposition * Inference * Argument * Validity * Cogency * Syllogism * Square of opposition * Venn diagram * Propositional calculus * Boolean logic * Boolean functions * Propositional calculus * Propositional formula * Logical connectives * Truth tables * Many-valued logic * Predicate logic * First-order * Quantifiers * Predicate * Second-order * Monadic predicate calculus ## Model theory * Model * Interpretation * Non-standard model * Finite model theory * Truth value * Validity ## Proof theory * Proving a universal statement * Direct proof * Proving existential statements * Disproving a universal statement * Disproving an existential statement * proof methods * Proof by cases * Proof by contrapositive * Formal verification * Automated theorem proving * Proof theory * Formal proof * Deductive system * Formal system * Theorem * Logical consequence * Rule of inference * Syntax *** ## PART IV: ALGEBRA ## Algebra * Elementary Algebra * Abstract Algebra ## Abstract Algebra ## Algebraic Structures * Mathematical induction * Induction * Strong Induction * Recurrence relations * Linear Recurrence Relations * Non-Homogeneous Recurrence Relation and Particular Solutions * Generating Functions * Graph theory * Graph * Graph Models * Types of Graphs * Representation of Graphs * Planar vs. Non-planar graph * Isomorphism * Homomorphism * Euler Graphs * Hamiltonian Graphs * Graph Coloring * Traversal * Trees * Properties * Centers and Bi-Centers of a Tree * Labeled Trees * Unlabeled Trees * Rooted Tree * Binary Search Tree * Spanning Trees * Kruskal's Algorithm * Prim's Algorithm * Group theory * Semigroup * Monoid * Group * Abelian Group * Cyclic Group and Subgroup * Partially Ordered Set (POSET) * Linearly Ordered Set * Hasse Diagram * Lattice * Field * Ring * Semigroup * Group * Field * Ring * Lattice * Boolean algebra * Boolean Functions * Boolean Expressions * Boolean Identities * Boolean Operators * Canonical Forms * Logic Gates * Simplification Of Boolean Functions * Simplification Using Algebraic Functions * Karnaugh Maps * Simplification Using K-map *** ## PART V: Theory of Computation (HIC AVT CS? HIC? CERTVS? NON, HIBERENT TU! NOLI TANGERE TABVLAS MEOS! ### Theory of Computation * Formal language * Automata theory * Computability Theory * Halting Problem ## Lambda Calculi * Lambda Calculus * Lambda Terms * Syntax * Semantics * Alpha conversion * Beta reduction * Lambda Cube * Simply Typed Lambda Calculus * System F * The Simply Typed Lambda Calculus ## Graph theory * Trees * Graphs *** ## PART V: Statistics ## Statistics * theoretical statistics * computational statistics * statistical inference * regression * time series * multivariate analysis * data analysis * Markov chain * Monte Carlo simulation *** ## PART V: Type theory and Category theory (here? or in CS? maybe split? split how? split HOW? SPLIT H-O-W?!)\ (╯°□°）╯︵ ┻━┻ ## Type Theory * Path to Type Theory * Sequent Calculus * Linear Logic * Linear Types * Type Systems * Types of Type Systems * Constructive Type theory * Substructural Type Systems ## Category Theory * Category * Morphism * Functor * Introduction * Categories * Functors * Natural Transformations * Universal Properties * Representability * Yoneda Lemma * Limits and Colimits * Adjunctions * Monads * Kan Extensions * Theorems of Category Theory --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/900-index/980-hierarchy/hierarchy.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.