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Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also focuses on applications in computer science, linguistics, and philosophy.
Aug 13, 2018 · Proof Theory. First published Mon Aug 13, 2018; substantive revision Wed Feb 21, 2024. Proof theory is not an esoteric technical subject that was invented to support a formalist doctrine in the philosophy of mathematics; rather, it has been developed as an attempt to analyze aspects of mathematical experience and to isolate, possibly overcome ...
Apr 16, 2008 · The development of proof theory can be naturally divided into: the prehistory of the notion of proof in ancient logic and mathematics; the discovery by Frege that mathematical proofs, and not only the propositions of mathematics, can (and should) be represented in a logical system; Hilbert's old axiomatic proof theory ; failure of the aims of ...
- Jan von Plato
- 2008
Proof Theory is concerned almost exclusively with the study of formal proofs: this is justifled, in part, by the close connection between social and formal proofs, and it is necessitated by the fact that only formal proofs are subject to mathematical analysis. The principal tasks of Proof Theory can be summarized as follows.
Proof theory was created early in the 20th century by David Hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics| in arithmetic (number theory), analysis and set theory.
Proof theory is nevertheless not merely a study of different kinds and methods of logical proof. From proof-theoretical results—e.g., from normal forms of proofs—one can hope to extract other kinds of important information.
In this somewhat extended appendix, we are going to discuss the evolution of the formal axiomatic standpoint in Hilbert’s foundational thinking. To begin with we articulate in greater detail than we did in section 1 of the main article Dedekind’s way of defining abstract concepts, like that of a simply infinite system. Dedekind had seen very clearly that such a structural definition raises ...
