Sentence examples for notion of proof from inspiring English sources

Perhaps the most important contribution to the foundations of mathematics made by the ancient Greeks was the axiomatic method and the notion of proof.

Europeans discovered the notion of proof, the power of generalization, and the superhuman cleverness of the Greeks; they hurried to master techniques that would enable them to improve their calendars and horoscopes, fashion better instruments, and raise Christian mathematicians to the level of the infidels.

Gödel came up with the surprising discovery that this was not the case for type theory and related languages adequate for arithmetic, as long as the following assumptions are insisted upon: The set of theorems (provable statements) is effectively enumerable, by virtue of the notion of proof being decidable.

Gödel's incompleteness theorem, generalized likewise, says that, in the usual language of arithmetic, it is not enough to look only at ω-complete models: Assuming that ℒ is consistent and that the theorems of ℒ are recursively enumerable, with the help of a decidable notion of proof, there is a closed formula g in ℒ, which is true in every ω-complete model, yet g is not a theorem in ℒ.

Modus ponens is the only deduction rule; this gives the usual notion of proof and provability of the logic BL.

In other words, through the advent of computer proofs the notion of proof has lost its purely a priori character.

For a definition of the notion of proof-theoretic strength and for surveys on proof theory see for example (Rathjen 1999, 2006b).

Echoing one of Strauss's major themes, Shulsky and Schmitt criticize America's intelligence community for its failure to appreciate the duplicitous nature of the regimes it deals with, its susceptibility to social-science notions of proof, and its inability to cope with deliberate concealment.

Only the consistency of the axioms that define the concept guarantees the legitimacy of C. In a nutshell: paradoxes tell us that we must develop a metamathematical analysis of the notions of proof and of the axiomatic method; their importance is methodological as well as epistemological.

The usual notion of a proof in a Hilbert-style axiomatic system is quite lax, but it can be tidied up to obtain the notion of traversing proofs.

However, Jevons argues that it is impossible to prove the fundamental laws of logic by reasoning, since they are already presupposed by the notion of a proof.