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 Britannica Concise
intro
Foundations of Mathematics - Article from Wikipedia
intro
What is Foundations of Mathematics? - by Stephen Simpson
intro
The hierarchy of concepts, mathematics, foundations
What Is a "Foundation" for Mathematics? - by Roger B. Jones
intro
Branch foundations, fundamental concepts, logical foundations
Foundations of Mathematics: Past, Present, and Future [PS] - by Harvey M. Friedman
more
FOM and Computer Science, crucial developments in FOM
Mathematical Logic and Foundations - Article from The Mathematical Atlas
intro
History, applications, subfields of FOM, reference materials, software
Platonism, Intuition and The Nature of Mathematics - by Karlis Podnieks
topic
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
A Century of Controversy over the Foundations of Mathematics - Gregory Chaitin's Lecture
more
Crisis in set theory, paradoxes, Hilbert's formal theories, metamathematic,
incompleteness, uncomputability, randomness
Historical notes - from Stephen Wolfram, A New Kind of Science
topic
A History of Set Theory - from School of Mathematical and Computational Sciences, University of St. Andrews
topic
Cantor, Bolzano, Crelle, Dedekind, Russell, and others
History of Constructivism in The 20th Century - by A.S.Troelstra
more
Finitism, predicativism and semi-intuitionism, Brouwerian intuitionism,
intuitionistic logic, arithmetic, analysis, constructive recursive mathematics,
Bishop\u2019s constructivism
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
Partial Realizations of Hilbert's Program - by Stephen G. Simpson
most
Program explication, reverse mathematics, Q&A
Hilbert's Program Revisited - by Panu Raatikainen
most
New interpretations of Hilbert's program
and recent research relevant to the 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
Naive Set Theory Is Inconsistent - by Curtis Brown
topic
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
Cantor's Paradox
Formal Systems
Formal System - from Britannica Concise
intro
The Formalization of Mathematics - by Harvey M. Friedman
more
Formalization goals, formal set theory, syntax and semantics of formal theories
Formal Languages and Systems - by Heinrich Herre and Peter Schroeder-Heister
more
Formal language, grammars, deductive systems, logical calculi
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
Inference Rule - Article from FOLDOC
intro
Syllogism - Article from MathWorld
intro
Axiomatic Method
Axiomatic System - Article from Wikipedia
intro
Axiomatic Method - Article from Britannica Concise
intro
The Axiomatic Method - by Roger B. Jones
intro
The Axiomatic Method - by Anthony Aaby
quick
Classical logic, Hilbert's formulation, intuitionistic logic
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
Highlights in Proof Theory [compressed PS] - by Solomon Feferman
more
Hilbert's finitary proof theory, Gentzen's L-calculi, countably infinitary methods
Structural Proof Theory [PS] - by Roy Dyckhoff
more
Natural deduction, sequent calculi
The Foundations of Mathematics - by David Hilbert
topic
Axioms of logic and arithmetic, discussion of various approaches
to formalization of mathematics and its branches
Proof Theory on the eve of Year 2000 - by Solomon Feferman
topic
Answers to 10 questions about proof theory given by theorists
Proof Interpretations and The Computational Content of Proofs [PS] - by Ulrich Kohlenbach
topic
Intuitionistic logics, realizability, D-interpretation, negative translation, A-translation
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
The Structure of An Axiomatized Theory - by Mary Leng
quick
Sample axiomatic theories, axioms
Set Theory
Set Theory - Article from Stanford Encyclopedia of Philosophy
more
The essence and origins of set theory, continuum hypothesis, axiom of choice
Axiomatic Set Theory - by Karlis Podnieks
topic
Cantor's set theory and its formalization, Zermelo-Fraenkel axioms
Zermelo-Fraenkel Axioms
Neumann-Bernays-Gödel Set Theory
Von Neumann-Bernays-Gödel Set Theory - Article from MathWorld
intro
Von Neumann-Bernays-Gödel Set Theory - Article from PlanetMath
intro
Continuum Hypothesis
Continuum Hypothesis - Article from MathWorld
intro
Continuum Hypothesis - Article from Wikipedia
intro
Around the Continuum Problem - by Karlis Podnieks
topic
Counting infinite sets, axioms of constructibility and determinateness,
Ackermann's set theory
The Continuum Problem - by Nancy McGough
topic
Assumptions, style, terminology, mathematics of the continuum
The Continuum Hypothesis - by Hugh Woodin
most
Geometry
Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and
Non-Euclidean Geometries - by Marvin J. Greenberg
more
Incidence, betweenness, congruence, continuity, parallelism, hyperbolic geometry
Euclid's Postulates
Euclid's Postulates - Article from MathWorld
intro
Definitions, Postulates & Common Notions - Euclid's Elements - by Robert Campbell
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
Hilbert's Axioms - by Robert Campbell
intro
Parallel Postulate - Article from MathWorld
topic
Non-Euclidean Geometries
Non-Euclidean Geometry - Article from MathWorld
intro
Non-Euclidean Geometry - Article at the web site of University of St. Andrews
more
Overview of contributions of Bolyai, Lobachevsky, Gauss, Riemann and others
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
First Order Arithmetic - by Karlis Podnieks
topic
Peano axioms, first-order arithmetic, arithmetic in other formal theories
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 Problem Course in Mathematical Logic - by Stefan Bilaniuk
most
Propositional logic, first-order logic, computability, incompleteness
A Short Introduction to Logic - 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
Computational Logic - Lecture Handouts by Sergei N. Artemov
quick
Classical and intuitionistic propositional logic, models,
Gentzen proof systems for intuitionistic and modal logic, Kripke completeness theorem, embedding,
natural derivations, typed lambda calculus, Curry-Howard isomorphism, combinatory logic
Propositional Calculus
Propositional Calculus - Article from MathWorld
intro
Propositional Logic Terms and Symbols - by Peter Suber
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 Predicate Calculus - Article from MathWorld
intro
First-Order Predicate Calculus - 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
Replacement Theorem - by R. Grandy
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
Normal Forms and Skolem Functions - by Anthony Aaby
intro
Negational normal form, conjunctive normal form, disjunctive normal form,
prenex normal form, Skolem normal form
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
Constructive (Intuitionistic) Logic - by Anthony A. Aaby
quick
Syntax, Kripke model, analytic tableau
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
Second and Higher-Order Logic - by Robert Harper
intro
Second-order quantification, second-order predicate logic
Higher Order Logic - by Hans de Nivelle
quick
Modal Logic
Modal Logic - Article from FOLDOC
intro
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
Modal Logic - Article by Anthony Aaby
quick
Syntax, semantics, models
Equational Logic
Equational Logic - Article from MathWorld
intro
Introduction to Equational Logic - by David Gries
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 [PS] - by Jean Gallier
most
Natural deduction, Gentzen's sequent calculi, calculi transformations, cut elimination
See also:
Proof Theory
Natural Deduction
Natural Deduction - Article from FOLDOC
intro
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
Natural Deduction for First-Order Logic - by Hans de Nivelle
quick
From Natural Deduction to Sequent Calculus - Slides by Christoph Benzmüller
quick
Sequent Calculus
Sequent Calculus - Article from Wikipedia
intro
Sequent Systems (Gentzen) - by Anthony Aaby
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
Sequent Calculi for First-Order Logic - by Hans de Nivelle
quick
Sequent Calculi for First-Order Logic - Slides by Rajeev Gore
quick
Proof Normalization: Gentzen's Hauptsatz - by Amos Omondi
more
Cut elimination (Hauptsatz), sharpened Hauptsatz, subformula property
Completeness and Consistency
Consistent - Article from PlanetMath
intro
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
General Setting for Incompleteness - by Anthony Aaby
topic
Gödel's and Tarski's theorems, abstract provability system
Gödel's Incompleteness Theorem
Gödel's Incompleteness Theorem - Article from MathWorld
intro
Gödel's Incompleteness Theorem - 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
Gödel's Theorems by Peter Smith
most
Axiomitization of formal theories, arithmetics, recursive functions,
Gödel's first and second incompleteness theorems
Incompleteness Theorems - by Karlis Podnieks
topic
Liar's paradox, self-reference lemma, Godel's two incompleteness theorems
Gödel to Rosser - by Alexander Milowski
topic
Godel numbers and diagonalization, incompleteness of arithmetic systems
Intuitionism
Intuitionism - Article from Wikipedia
intro
Intuitionism - from Britannica Concise
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
Constructive Mathematics - by Fred Richman
intro
Notes on the Foundations of Constructive Mathematics - by Joan Moschovakis
most
Background, formal systems and semantics for intuitionistic logic,
intuitionistic logic in mathematics
Introduction to Constructive Logic and Mathematics [PS] - by Thomas Streicher
most
Semantics of constructive logic, embedding, constructive arithmetic and analysis,
Kleene's realizability, Kreisel's realizability, higher-type arithmetic, Markov's principle
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
Classical and Constructive Logic - by Jeremy Avigad
more
Classical and constructive viewpoints, number theory and other examples,
historical and philosophical issues, symbolic logic, double negation translation,
relationship between classical and constructive logic, mathematics,
logic and computer science
Constructivism and Proof Theory - by A.S. Troelstra
most
Intuitionistic logic, its semantics, Heyting arithmetic, constructive mathematics,
proof theory of first-order logic and mathematical theories
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
Model Theory - by Stephen G. Simpson
most
Complete theories, compactness theorem, decidability, closed fields,
saturated models, quantifier elimination, prime models
Fundamentals of Model Theory - by William Weiss and Cherie D'Mello
most
Compactness, Löowenheim-Skolem theorems, diagrams, embedding, model completeness,
model completions
About Model Theory - by Karlis Podnieks
topic
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
Löwenheim-Skolem Theorem - Article by Peter Suber
more
Skolem's paradox, incurable ambiguities
Skolem Paradox
Skolem Paradox - Article from MathWorld
intro
Computability
Introduction to Logic and Recursion Theory - by Edward Boyden, from The Babbage Group web site
intro
Propositional and first-order logic, completeness and consistency,
recursion theory, Post's problem
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
Enumerability - Article from MathWorld
intro
Creative Sets - Article from MathWorld
topic
Lambda Calculus
Lambda Calculus - Article from Wikipedia
intro
Lambda Calculus and Combinators - by Anthony Aaby
quick
The Lambda-calculus, Combinatory Logic, and Type Systems - by Roger B. Jones
intro
Lambda Tutorial - by Chris Barker
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
Category Theory: The Language of Mathematics - by Elaine Landry
more
Philosophy of Mathematics
Foundations Study Guide: Philosophy of Mathematics - by David S. Ross
intro
History and overview of logicism, formalism, intuitionism, and objectivism
Logic and Mathematics - by Stephen G. Simpson
intro
Aristotelian logic, predicate calculus, formal theories
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
Philosophical Foundations - by Anthony Aaby
quick
General Information
FOM (Foundations of Mathematics) - Automated email list
for discussing foundations of mathematics
Proof Theory - email list
Metamath Home Page
Metamath is a language for expressing theorems and their proofs
Practical Foundations of Mathematics - online book by Paul Taylor
most
First-order reasoning, types and induction, posets and lattices, Cartesian closed categories,
limits and colimits, structural recursion, adjunctions, algebra with dependent types, quantifiers
Foundations of Mathematics - by Stephen Simpson
most
Computable functions, undecidability of arithmetic, real number system,
informal and axiomatic set theory
An Introduction to Mathematical Logic - by Wolfram Pohlers & Thomas Glaß
most
First-order logic, model theory, decidability, Peano arithmetic, other logics
Reference
Glossary of First-Order Logic - by Peter Suber
Foundations of Mathematics - list of articles from MathWorld
Stanford Encyclopedia of Philosophy - table of contents
Personalities
|
Alan Turing - from The MacTutor History of Mathematics archive
|
Alan Turing
- from Stanford Encyclopedia of Philosophy
|
|
Emil Post
- from The MacTutor History of Mathematics archive
|
|
Andrey Markov
- from Laboratory of Mathematical Logic of Steklov Institute of Mathematics at St.Petersburg
|
Link Collections
Mathematical Logic Around The World - by Boris Piwinger (University of Bonn, University of Vienna)
Math Topics: Logic/Foundations - from Math Forum @ Drexel
MavicaNET - Logic
Mathematical Logic and Foundations - from MathGuide
Logical Systems - by Peter Suber
Logic - by Denis Roegel
Science > Math > Logic and Foundations - from Open Directory Project (presented by Google)
Science > Mathematics > Foundations of Mathematics - from Yahoo
Copyright © 2003 Alexander Sakharov
Need to relax? Try brain teasers. I would recommend those marked 'cool'.