The study of formal proof, mathematical truth, and the axiomatic bedrock the rest of mathematics rests on: set theory, model theory, proof theory and computability. David Hilbert's early twentieth century program sought to place all of mathematics on a finite, provably consistent axiomatic foundation; Kurt Godel's 1931 incompleteness theorems showed that any consistent formal system powerful enough to describe basic arithmetic must contain true statements it cannot prove, reshaping what a foundation for mathematics could even promise.
Facts
Central QuestionWhat are the ultimate foundations of mathematical truth and proof, and can they be made fully rigorous, consistent and complete? 1 Key DebateWhether Godel's incompleteness theorems refuted Hilbert's program outright, or only its most literal, most ambitious form, leaving weaker finitist and relativized foundational projects, and later developments such as reverse mathematics, genuinely viable in its place. 1 Partially Attested
Pure or AppliedThe source ties the field to the foundations of mathematics; it also has applications in computer science. Connections
Associated With
Source MacTutor History of Mathematics Archive
Source MacTutor History of Mathematics Archive
Includes
Source The Stanford Encyclopedia of Philosophy
Source Wikipedia: Axiom
Source Boolean Algebra (Wikipedia)
Source Wikipedia: Cantor's Diagonal Argument
Source Cantor's Theorem (Wikipedia)
Source Continuum Hypothesis (Wikipedia)
Source The Stanford Encyclopedia of Philosophy
Source The Stanford Encyclopedia of Philosophy
Source The Stanford Encyclopedia of Philosophy
Additional Source Wikipedia: Godel's Incompleteness TheoremsLead section
Source MacTutor History of Mathematics Archive
Source The Stanford Encyclopedia of Philosophy
Source The Stanford Encyclopedia of Philosophy
Source Leonid Levin, Faculty Homepage, Boston University
Source The Stanford Encyclopedia of Philosophy
Source Clay Mathematics Institute
Additional Source Wikipedia: P versus NP ProblemLogical characterizations section
Sets, Concepts Source The Stanford Encyclopedia of Philosophy
Source Stephen Cook, Faculty Homepage, University of Toronto
In the Other Atlases
- Also in Philosophy Atlas: Logic, influenced this subject there.
Sources
1. The Stanford Encyclopedia of Philosophy
Center for the Study of Language and Information, Stanford Universityhttps://plato.stanford.edu/entries/logic-classical/Quote, https://plato.stanford.edu/entries/logic-classical/
Typically, a logic consists of a formal or informal language together with a deductive system and/or a model-theoretic semantics.
2. Mathematical Logic (Wikipedia)
Wikimedia FoundationHistory section
also called logistic, symbolic logic, the algebra of logic, and, more recently, simply formal logic
Lead section
Mathematical logic is the study of formal logic within mathematics.
Lead section, foundations
Since its inception, mathematical logic has both contributed to and been motivated by the study of the foundations of mathematics.
View the Source Cantor's Theorem (Wikipedia)
Wikimedia FoundationIncludes: Cantor's Theorem, leadQuote, Includes: Cantor's Theorem, lead
Every set is smaller than its power set
MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and StatisticsAssociated With: Kurt Godel, https://mathshistory.st-andrews.ac.uk/Biographies/Godel/
He proved fundamental results about axiomatic systems, showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system.
Associated With: David Hilbert, https://mathshistory.st-andrews.ac.uk/Biographies/Hilbert/
A systematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axioms and he analysed their significance.
View the Source Continuum Hypothesis (Wikipedia)
Wikimedia FoundationIncludes: Continuum Hypothesis, Opening paragraphQuote, Includes: Continuum Hypothesis, Opening paragraph
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis
Wikipedia: Axiom
Wikimedia FoundationIncludes: Axiom, Mathematical logic sectionQuote, Includes: Axiom, Mathematical logic section
In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical and non-logical.
Wikipedia: Cantor's Diagonal Argument
Wikimedia FoundationIncludes: Cantor's Diagonal Argument, lead paragraph, applications passageQuote, Includes: Cantor's Diagonal Argument, lead paragraph, applications passage
it demonstrates a general technique that has since been used in a wide range of proofs, including the first of Godel's incompleteness theorems and Turing's answer to the Entscheidungsproblem.
Wikipedia: P versus NP Problem
Wikimedia FoundationIncludes: P versus NP, Logical characterizations sectionQuote, Includes: P versus NP, Logical characterizations section
The P = NP problem can also be stated as a question about expressive power in descriptive complexity.
Wikipedia: Godel's Incompleteness Theorems
Wikimedia FoundationIncludes: Godel's Incompleteness Theorems, Lead sectionQuote, Includes: Godel's Incompleteness Theorems, Lead section
Godel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories.
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