
	Alexander Sakharov

Foundations of Mathematics

- Textbook / Reference -

with contributions by Bhupinder Anand, Harvey Friedman, Haim Gaifman, Vladik Kreinovich, Victor Makarov, Grigori Mints, Karlis Podnieks, Panu Raatikainen, Stephen Simpson,
featured in the Computers/Mathematics section of Science Magazine NetWatch

This is an online resource center for materials that relate to foundations of mathematics (FOM). It is intended to be a textbook for studying the subject and a comprehensive reference. As a result of this encyclopedic focus, materials devoted to advanced research topics are not included. The author has made his best effort to select quality materials on www.

This reference center is organized as a book as opposed to an encyclopedia, dictionary, directory, or link collection. This page represents book's contents page. One can use this page to study the foundations of mathematics by reading topics following the links in their order or jumping over certain chapters. Where appropriate, topics covered in the referred web resource are listed under the link. In particular, it is done if the resource covers more than the respective section heading and title suggest. Presumably, this is the only anchor page one needs to navigate all math foundations topics. I believe you can even save some $$ because the materials listed here should be sufficient, and you do not have to buy a book or two.

The links below are marked in order to indicate the type of material:
   ^intro - definitions and other introductory material such as entries in online encyclopedias
   ^more - materials giving a broader coverage of the respective subject
   ^most - advanced material giving an in-depth coverage of the subject
   ^topic - supplementary material covering particular topics or particular views of the subject
   ^quick - quick reference type of material such as lecture slides
Links with similar contents are grouped, and the groups are marked by a vertical bar on the right.

As you know, www links become obsolete pretty quickly. Unfortunately, this especially applies to the kind of materials compiled on this page. For instance, lecture handouts are often removed once the course is finished. Please keep in mind that despite my maintenance efforts, this www reality just kicks in.

Please select link navigation style:
'Frame' leaves the left column in place (default).
'New' opens a new window.
'Jump' replaces the entire window.

What is Foundations of Mathematics?

Foundations of Mathematics - Article from Wikipedia    ^intro
What Is a "Foundation" for Mathematics? - by Roger B. Jones    ^intro
Branch foundations, fundamental concepts, logical foundations
What Is Mathematics: Gödel's Theorem and Around - by Karlis Podnieks    ^more
Platonism, the mathematical method, axioms, formal theories, Hilbert's program

History

History of the Foundations of Mathematics - by Roger B. Jones ^intro
Greece, axiomatic method, 20-th century schools, logical foundations, etc
Historical notes - from Stephen Wolfram, A New Kind of Science ^topic
See also: Personalities

Hilbert's Program

Hilbert's Program - Article from Stanford Encyclopedia of Philosophy ^more
Finitary point of view, formalism, historical development of Hilbert's program

Naive Set Theory

Naive Set Theory - Article from MathWorld    ^intro
Naive Set Theory - Article from Wikipedia    ^intro
A Primer on Sets, Classes - Lecture Slides by Stanley N. Burris    ^quick
Set membership, equality, subsets, operations on sets, Russell's paradox
Russell's paradox
See also: Paradoxes

Cantor's Diagonal Method

Cantor Diagonal Method - Article from MathWorld ^intro
Cantor's Diagonal Argument - Article from Wikipedia ^intro

Paradoxes

Paradox - Article from MathWorld ^intro

Russell's Paradox

Russell's Antinomy - Article from MathWorld    ^intro
Russell's Paradox - Article from Stanford Encyclopedia of Philosophy    ^more
Russell's Paradox - Article from The Internet Encyclopedia of Philosophy    ^more

Burali-Forti Paradox

Burali-Forti Paradox - Article from MathWorld ^intro
Burali-Forti Paradox - Article from Wikipedia ^intro

Cantor's Paradox

Cantor's Paradox - Article from MathWorld ^intro
Cantor's Paradox - Article from Wikipedia ^intro

Formal Systems

Formalized Mathematics - by John Harrison ^intro
History and philosophy, formalization, practical issues

Formal Language

Formal Language - Article from MathWorld ^intro
Formal Language - Article from Wikipedia ^intro

Axioms and Inference Rules

Axiom - Article from MathWorld    ^intro
Axiom - Article from Wikipedia    ^intro
Axiom Schema - Article from MathWorld    ^intro
Syllogism - Article from MathWorld    ^intro

Axiomatic Method

Axiomatic System - Article from Wikipedia ^intro
The Axiomatic Method - by Roger B. Jones ^intro

Proof Theory

Proof Theory - Article from Wikipedia    ^intro
An Introduction to Proof Theory by Samuel R. Buss    ^most
Proof theories of propositional logic, first-order logic, intuitionistic logic, linear logic
The Foundations of Mathematics - by David Hilbert    ^topic
Axioms of logic and arithmetic, discussion of various approaches to formalization of mathematics and its branches
Manipulating Proofs - by Jonathan P. Seldin    ^quick
Propositional logic: natural deduction, normalization, sequents, cut elimination
Proof Theory [PS] by Helmut Schwichtenberg    ^most
Natural deduction, typed lambda calculus, normalization, computational content of proofs
See also: Natural Deduction and Sequent Calculus

Axiomatizations

Axiomatization - Article from Wikipedia ^intro

Set Theory

Set Theory - Article from Stanford Encyclopedia of Philosophy ^more
The essence and origins of set theory, continuum hypothesis, axiom of choice

Zermelo-Fraenkel Axioms

Zermelo-Fraenkel Axioms - Article from MathWorld ^intro
Zermelo-Fraenkel Axioms - Article from Wikipedia ^intro

Neumann-Bernays-Gödel Set Theory

Von Neumann-Bernays-Gödel Set Theory - Article from MathWorld ^intro

Continuum Hypothesis

Continuum Hypothesis - Article from MathWorld    ^intro
Continuum Hypothesis - Article from Wikipedia    ^intro
The Continuum Hypothesis - by Nancy McGough    ^topic
Assumptions, style, terminology, mathematics of the continuum
The Continuum Hypothesis - by Hugh Woodin    ^most

Geometry

Foundations of geometry - Article from Wikipedia ^intro

Euclid's Postulates

Euclid's Postulates - Article from MathWorld ^intro

Hilbert's Axioms

Hilbert's Axioms - Article from MathWorld    ^intro
A Version of Hilbert's axioms for the Euclidean plane - by Ian Chiswell    ^topic
Axioms of incidence, betweenness, congruence, parallelism, Dedekind's axiom
Parallel Postulate - Article from MathWorld    ^topic

Non-Euclidean Geometries

Non-Euclidean Geometry - Article from MathWorld ^intro

Arithmetic

Peano Axioms - Article from MathWorld    ^intro
First-Order Proof Theory of Arithmetic - by Samuel R. Buss    ^most
Fragments of arithmetic, incompleteness, witnessing theorems
The Dedekind-Peano Number System - Lecture Slides by Stanley N. Burris    ^quick
Arithmetic - by Stanley N. Burris    ^more
First-order arithmetic and Peano arithmetic

Other Formal Systems

Algebraic Systems: Group, Ring, Field, Lattice - Articles from MathWorld ^intro
A Course in Universal Algebra - by Stanley N. Burris and H.P. Sankappanavar ^most
Lattices, examples of algebras, homomorphisms, varieties, term algebras, free algebras, boolean algebras, connections with model theory
See also: Logic Systems

Mathematical Logic

See also: Axioms and Inference
Classical Logic - Article from Stanford Encyclopedia of Philosophy    ^more
Language, deduction, semantics, meta-theory
Introduction to Mathematical Logic - by Vilnis Detlovs and Karlis Podnieks    ^most
Propositional and predicate logic, completeness theorems, normal forms, resolution method, unsolvability
A Short Introduction to Logic [PPT] - by Stefano Berardi    ^quick
Propositional and predicate logic, Godel's completeness theorem, strong normalization
Logic - Lecture Notes [PS] - by Josef Schicho    ^topic
Formal theories, propositional and predicate logic, the completeness theorem, automatic theorem proving, other predicate theories

Propositional Calculus

Propositional Calculus - Article from MathWorld    ^intro
Propositional Logic - Lecture Slides by Stanley N. Burris    ^quick
Comments on propositional proof systems - by Stanley N. Burris    ^topic
Theorem checkers and other algorithms, Frege/Hilbert propositional calculi, Gentzen-style calculi
Deduction Theorem - Article from MathWorld    ^topic

First-Order Predicate Calculus

First-Order Logic - Article from MathWorld    ^intro
First-Order Logic - Article from Wikipedia    ^intro
First-Order Logic - Lecture Slides by Stanley N. Burris    ^quick
Logical Laws - by Alex Sakharov    ^topic
Interpretation - Article from MathWorld    ^topic
Deduction Theorem - Article from MathWorld    ^intro
See also: Natural Deduction and Sequent Calculus

Normal Forms

Conjunctive Normal Form - Article from MathWorld    ^intro
Disjunctive Normal Form - Article from MathWorld    ^intro
Prenex Normal Form - Article from MathWorld    ^intro

Herbrand Theorem

Herbrand's Theorem - Article from MathWorld ^intro
On Herbrand's Theorem - by Samuel R. Buss ^most
Weak and strong forms of the theorem, Herbrand's original version

Intuitionistic Logic

Intuitionistic Logic - Article from MathWorld    ^intro
Intuitionistic Logic - Article from Wikipedia    ^intro
Intuitionistic Logic - Article from Stanford Encyclopedia of Philosophy    ^more
Intuitionistic first-order predicate logic, Kripke semantics
A Brief Introduction to The Intuitionistic Propositional Calculus - by Stuart A. Kurtz    ^intro
Intuitionistic proofs, Curry-Howard isomorphism, semantics
Kripke Semantics [PS] - by Jan Smith    ^intro
See also: Natural Deduction and Sequent Calculus

Higher-Order Logic

Second-Order Logic and Foundations of Mathematics - by Jouko Vaananen ^more
Informal reasoning, formal languages of the first-order and second-order logics, semantics

Modal Logic

Modal Logic - by John McCarthy ^intro
Modal Logic - Article from Stanford Encyclopedia of Philosophy ^more
Modal logics, deontic Logics, temporal logics, conditional logics and relationships between them, quantifiers in modal logic

Equational Logic

Equational Logic - Article from MathWorld    ^intro
Equational Logic - Lecture Slides by Stanley N. Burris    ^quick
Terms, semantics, algebras, substitution, unification, term-rewriting systems
Knuth-Bendix Completion Algorithm - Article from MathWorld    ^topic

Logic Systems

Constructive Logics. Part I: A Tutorial on Proof Systems and Typed lambda-Calculi - by Jean Gallier ^most
Natural deduction, Gentzen's sequent calculi, calculi transformations, cut elimination
See also: Proof Theory

Natural Deduction

Natural Deduction - by Frank Pfenning    ^more
Intuitionistic and classical natural deduction, localizing hypotheses
Natural Deduction: Some Recent Developments - by Jan von Plato    ^most
Natural deduction in intuitionistic and classical logics, normalization
Logic - by Helmut Schwichtenberg    ^more
Formal languages, natural deduction, normalization, permutative conversions

Sequent Calculus

Sequent Calculus - Article from Wikipedia    ^intro
Sequent Calculus Primer - by Alex Sakharov    ^intro
Definition, proof theory, sample derivations, formulations, automated deduction
Sequent Calculus - by Frank Pfenning    ^more
Relationship between sequent calculus and natural deduction, cut elimination
Proof Normalization: Gentzen's Hauptsatz - by Amos Omondi    ^more
Cut elimination (Hauptsatz), sharpened Hauptsatz, subformula property

Completeness and Consistency

Inconsistent Mathematics - Article from Stanford Encyclopedia of Philosophy    ^more
Gödel's Completeness Theorem - Article from MathWorld    ^intro
Gödel's Completeness Theorem - Article from Wikipedia    ^intro

Gödel's Incompleteness Theorems

Gödel's First Incompleteness Theorem - Article from MathWorld    ^intro
Gödel's Second Incompleteness Theorem - Article from MathWorld    ^intro
Gödel's Incompleteness Theorems - Article from Wikipedia    ^intro
Gödel's Incompleteness Theorem - by Dale Myers    ^intro
Cantor's power-set theorem, paradoxes, Tarski's undefinability of truth, Godel's two incompleteness theorems
What Is Mathematics: Gödel's Theorem and Around - by Karlis Podnieks    ^more
Platonism, the mathematical method, axioms, formal theories, Hilbert's program

Intuitionism

Intuitionism - Article from Wikipedia    ^intro
Lectures on Intuitionism - Historical introduction and Fundamental Notions - by Luitzen Brouwer (Cambridge Lectures on Intuitionism given in 1951)    ^topic
Gödel's Functional (Dialectica) Interpretation - by Jeremy Avigad and Solomon Feferman    ^most
Dialectica interpretation of arithmetic, interpretation of analysis, term model, non-constructive interpretations, interpretation of theories of ordinals
See also: Intuitionistic Logic

Constructive Mathematics

Constructive Mathematics - Article from Stanford Encyclopedia of Philosophy    ^more
Constructive interpretation of logic, intuitionistic mathematics, recursive constructive mathematics, Bishop's mathematics, Martin-Lof's type theory
Notes on the Foundations of Constructive Mathematics - by Joan Moschovakis    ^most
Background, formal systems and semantics for intuitionistic logic, intuitionistic logic in mathematics
Relationships between Constructive, Predicative and Classical Systems of Analysis - by Solomon Feferman    ^more
Constructive and predicative redevelopments of mathematics, formal systems for Bishop constructive mathematics and for predicativity

Model Theory

Model Theory - Article from MathWorld    ^intro
Model Theory - Article from Wikipedia    ^intro
Model Theory - Article from Stanford Encyclopedia of Philosophy    ^more
Model-theoretic definitions and consequences, expressive strength of languages
First-Order Model Theory - Article from Stanford Encyclopedia of Philosophy    ^more
First-order languages, elementary maps, compactness theorem, diagram lemma, Lyndon interpolation theorem, omitting types theorem
Fundamentals of Model Theory - by William Weiss and Cherie D'Mello    ^most
Compactness, Löowenheim-Skolem theorems, diagrams, embedding, model completeness, model completions
Models - by Helmut Schwichtenberg    ^more
Structures for classical logic, Beth structures for minimal logic, completeness of minumal, intuitionistic, and classical logic, uncountable languages, basics of model theory

Löwenheim-Skolem Theorem

Löwenheim-Skolem Theorem - Article from MathWorld ^intro

Skolem Paradox

Skolem Paradox - Article from MathWorld ^intro

Computability

Computability - by Helmut Schwichtenberg ^more
Register machines, elementary functions, normal form, recursive functions

Computational Models

Turing Machine - Article from MathWorld    ^intro
Recursive Functions - Article from MathWorld    ^intro
Markov Algorithms - Article from Wikipedia    ^intro
The Church-Turing Thesis - Article from MathWorld    ^intro
The Church-Turing Thesis - Article from Stanford Encyclopedia of Philosophy    ^more
Misunderstandings of the thesis, Turing's remarks

Unsolvability

Unsolvability - Article from MathWorld    ^intro
Godel Number - Article from MathWorld    ^intro
Recursively Enumerable Set - Article from MathWorld    ^intro
Creative Set - Article from MathWorld    ^topic

Lambda Calculus

Lambda Calculus - Article from Wikipedia ^intro
The Lambda-calculus, Combinatory Logic, and Type Systems - by Roger B. Jones ^intro

Category Theory

Category Theoretic Perspectives on the Foundations of Mathematics - by Roger B. Jones ^intro
Category Theory - from Stanford Encyclopedia of Philosophy ^more
Definitions, history, philosophical significance

Philosophy of Mathematics

The Modern Development of The Foundations of Mathematics in The Light of Philosophy - by Kurt Gödel (Collected Works, Oxford University Press, 1981) ^more
The Philosophy of Mathematics and Hilbert's Proof Theory - by Paul Bernays ^most
Development of math conceptions, math in logic, formal abstraction, infinity, intuitionism, logicism, Hilbert's proof theory

General Information

FOM (Foundations of Mathematics) - Automated email list for discussing foundations of mathematics
Metamath Home Page
Metamath is a language for expressing theorems and their proofs

Reference

Foundations of Mathematics - list of articles from MathWorld
Stanford Encyclopedia of Philosophy - table of contents

Personalities

Georg Cantor - from The MacTutor History of Mathematics archive

Georg Cantor - from Wikipedia

David Hilbert - from The MacTutor History of Mathematics archive

David Hilbert - from Wikipedia

Kurt Gödel - from The MacTutor History of Mathematics archive

Alan Turing - from The MacTutor History of Mathematics archive

Alan Turing - from Stanford Encyclopedia of Philosophy

Paul Bernays - from The MacTutor History of Mathematics archive

The Bernays Project - This site contains a lot of Bernays works in PDF format. See contents.

Arend Heyting - from The MacTutor History of Mathematics archive

Stephen Kleene - from The MacTutor History of Mathematics archive

Stephen Kleene - from Wikipedia

Luitzen Brouwer - from The MacTutor History of Mathematics archive

Luitzen Brouwer - from Stanford Encyclopedia of Philosophy

Alonzo Church - from The MacTutor History of Mathematics archive

Alonzo Church - from Wikipedia

Bertrand Russell - from The MacTutor History of Mathematics archive

Bertrand Russell - from Stanford Encyclopedia of Philosophy

Gerhard Gentzen - from The MacTutor History of Mathematics archive

Ernst Zermelo - from The MacTutor History of Mathematics archive

Adolf Fraenkel - from The MacTutor History of Mathematics archive

Thoralf Skolem - from The MacTutor History of Mathematics archive

Leopold Löwenheim - from The MacTutor History of Mathematics archive

Emil Post - from The MacTutor History of Mathematics archive

Haskell Curry - from The MacTutor History of Mathematics archive

Andrey Kolmogorov - from The MacTutor History of Mathematics archive

Andrey Markov - from Laboratory of Mathematical Logic of Steklov Institute of Mathematics at St.Petersburg

Alfred Tarski - from The MacTutor History of Mathematics archive

Giuseppe Peano - from The MacTutor History of Mathematics archive

Jacques Herbrand - from The MacTutor History of Mathematics archive

Need to relax? Try brain teasers. I would recommend those marked 'cool'.