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

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