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