Branches of Mathematics
Proof Theory
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Proof theory is the branch of mathematical logic that treats mathematical proofs themselves as formal objects, so that what a proof can and cannot establish becomes a subject for mathematical analysis in its own right.
Facts
Central QuestionWhat can be proved about a formal system's own proofs from outside that system, using methods restrictive enough to count as secure while still powerful enough to say something worth knowing. 1 Key DebateWhether David Hilbert's program of grounding all of mathematics through finitary consistency proofs could succeed. Hilbert's program aimed to certify the sophisticated formal theories mathematicians rely on as consistent by strictly finitary metamathematical arguments, and Kurt Godel's incompleteness theorems demonstrated that the program as originally conceived cannot succeed, since a sufficiently strong consistent theory cannot prove its own consistency. 1 Cross-Tradition Connections
Sources
1. Wikipedia: Proof Theory
Wikimedia FoundationLead sectionQuote, Lead section
proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques.
1. Wikipedia: Proof Theory
Wikimedia FoundationOrdinal analysis sectionQuote, Ordinal analysis section
For a consistent recursively axiomatized theory T, one can prove in finitistic arithmetic that the well-foundedness of a certain transfinite ordinal implies the consistency of T.
1. Wikipedia: Proof Theory
Wikimedia FoundationHistory sectionQuote, History section
The failure of the program was demonstrated by Kurt Godel's incompleteness theorems, which showed that any omega-consistent theory that is sufficiently strong to express certain simple arithmetic truths, cannot prove its own consistency.
Stone-Weierstrass Theorem (Wikipedia)
Wikipedialead paragraphQuote, lead paragraph
Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques.
Stone-Weierstrass Theorem (Wikipedia)
WikipediaHistory sectionQuote, History section
The failure of the program was demonstrated by Kurt Godel's incompleteness theorems, which showed that any omega-consistent theory that is sufficiently strong to express certain simple arithmetic truths, cannot prove its own consistency, which on Godel's formulation is a Pi^0_1 sentence.
Wikipedia: Godel's Incompleteness Theorems
Wikimedia FoundationIncludes: Godel's Incompleteness Theorems, Implications for consistency proofs sectionQuote, Includes: Godel's Incompleteness Theorems, Implications for consistency proofs section
Gentzen's theorem spurred the development of ordinal analysis in proof theory.
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