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. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
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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 Classification
Pure or Applied Proof Theory
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Additional Source Wikipedia: Godel's Incompleteness TheoremsImplications for consistency proofs section
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1. Wikipedia: Proof Theory
Wikimedia FoundationLead section
proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques.
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.
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.
View the Source Stone-Weierstrass Theorem (Wikipedia)
Wikipedialead 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.
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.
View the Source 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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