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Search Results - Proof theory

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Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Together with model theory, axiomatic set theory, and recursion theory, proof theory is one of the so-called four pillars of the foundations of mathematics.[1]

Proof theory is important in philosophical logic, where the primary interest is in the idea of a proof-theoretic semantics, an idea which depends upon technical ideas in structural proof theory to be feasible.

Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Giuseppe Peano, Bertrand Russell, and Richard Dedekind, the story of modern proof theory is often seen as being established by David Hilbert, who initiated what is called Hilbert's program in the foundations of mathematics. Kurt Gödel's seminal work on proof theory first advanced, then refuted this program his completeness theorem initially seemed to bode well for Hilbert's aim of reducing all mathematics to a finitist formal system; then his incompleteness theorems showed that this is unattainable. All of this work was carried out with the proof calculi called the Hilbert systems.

In parallel, the foundations of structural proof theory were being founded. Jan Lukasiewicz suggested in 1926 that one could improve on Hilbert systems as a basis for the axiomatic presentation of logic if one allowed the drawing of conclusions from assumptions in the inference rules of the logic. In response to this Stanislaw Jaskowski (1929) and Gerhard Gentzen (1934) independently provided such systems, called calculi of natural deduction, with Gentzen's approach introducing the idea of symmetry between the grounds for asserting propositions, expressed in introduction rules, and the consequences of accepting propositions in the elimination rules, an idea that has proved very important in proof theory[2]. Gentzen (1934) further introduced the idea of the sequent calculus, a calculus advanced in a similar spirit that better expressed the duality of the logical connectives[3], and went on to make fundamental advances in the formalisation of intuitionistic logic, and provide the first combinatorial proof of the consistency of Peano arithmetic. Together, the presentation of natural deduction and the sequent calculus introduced the fundamental idea of analytic proof to proof theory,

Showing 1 to 2 of 2 Articles matching 'Proof theory' in related articles.
Pages: 1

1. How Much Is My Gold Worth?
November 16, 2009

Your gold is worth probably more today than yesterday. There’s a good reason for that. Usually when the dollar is flat or losing value, people feel insecure. They worry about their money. This insecurity drives the gold market higher. It’s called supply and demand. Short supply with a large demand creates higher prices. Today’s economy is living proof of this theory. People believe, and rightfully so, that gold is more stable than paper. You can print as much money as you want, you can’t mine gold if there is a shortage of it in the world. I’ve been in the dental industry for over ... (read more)

Author: Michael Berman

2. Are Right-Handed People Right-Eared As Well?
November 13, 2009

It never occurred to many of us that there is a difference between our left and right ears as far as hearing capabilities are concerned. We always thought that the ears have equal distribution of hearing capabilities but results of studies show that there is indeed a difference. So, which ear can people hear best with? In the results of most studies, they indicate that if a person is right handed, the ear that can hear best is the right one too. Same theory of course applies for left-handed people. Interesting, don’t we think? One proof that right-handed people are “right-eared” is when w... (read more)

Author: Kenny Leones

Pages: 1


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