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 indepth 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, 20th 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}
BuraliForti Paradox
Cantor's Paradox
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, firstorder 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
ZermeloFraenkel Axioms
NeumannBernaysGödel Set Theory
Von NeumannBernaysGödel 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}
NonEuclidean Geometries
NonEuclidean Geometry  Article from MathWorld
^{intro}
Arithmetic
Peano Axioms  Article from MathWorld
^{intro}
FirstOrder Proof Theory of Arithmetic  by Samuel R. Buss
^{most}
Fragments of arithmetic, incompleteness, witnessing theorems
The DedekindPeano Number System  Lecture Slides by Stanley N. Burris
^{quick}
Arithmetic  by Stanley N. Burris
^{more}
Firstorder 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, metatheory
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,
Gentzenstyle calculi
Deduction Theorem  Article from MathWorld
^{topic}
FirstOrder Predicate Calculus
FirstOrder Logic  Article from MathWorld
^{intro}
FirstOrder Logic  Article from Wikipedia
^{intro}
FirstOrder 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 firstorder predicate logic, Kripke semantics
A Brief Introduction to The Intuitionistic Propositional Calculus  by Stuart A. Kurtz
^{intro}
Intuitionistic proofs, CurryHoward isomorphism, semantics
Kripke Semantics [PS]  by Jan Smith
^{intro}
See also:
Natural Deduction
and
Sequent Calculus
HigherOrder Logic
SecondOrder Logic and Foundations of Mathematics  by Jouko Vaananen
^{more}
Informal reasoning, formal languages of the firstorder and secondorder 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, termrewriting systems
KnuthBendix Completion Algorithm  Article from MathWorld
^{topic}
Logic Systems
Constructive Logics. Part I: A Tutorial on Proof Systems and Typed lambdaCalculi  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 powerset 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,
nonconstructive 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, MartinLof'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}
Modeltheoretic definitions and consequences, expressive strength of languages
FirstOrder Model Theory  Article from Stanford Encyclopedia of Philosophy
^{more}
Firstorder 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, LöowenheimSkolem 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
LöwenheimSkolem Theorem
LöwenheimSkolem 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 ChurchTuring Thesis  Article from MathWorld
^{intro}
The ChurchTuring 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 Lambdacalculus, 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
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

Emil Post
 from The MacTutor History of Mathematics archive

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

Copyright © 2003 Alexander Sakharov
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