The proof of a theorem is a rigorous justification of the veracity of the theorem in such a way that it cannot be disputed by anyone who follows the rules of logic, and who accepts a set of axioms put forth as the basis for the logic system(flt29)
The proof of a theorem is a rigorous justification of the veracity of the theorem in such a way that it cannot be disputed by anyone who follows the rules of logic, and who accepts a set of axioms put forth as the basis for the logic system(flt29)
The proof of the law of infinite conjunction is not what is called a constructive proof. It proceeded by showing the existence of certain suitable objects without saying just which objects they are. Much that is proved in mathematics is insusceptible to constructive proof; much else is proved constructively(mol205)
Where a complete proof procedure differs from a decision procedure, then, is not in being less mechanical. The difference is just that the proof procedure is not a yes-or-no affair; it does not deliver negative answers. Failure to reach an inconsistency proof after a large number of steps, however systematically and mechanically programmed, does not show consistency. A proof procedure is only half of a decision procedure. When there is both a proof procedure and a disproof procedure (for validity or consistency or any other property), there is also a decision procedure(mol214)
A complete proof procedure for validity or consistency or inconsistency of schemata remains far short of a decision procedure, simply because a schema can be consistent without its negation's being inconsistent, and a schema can fail of validity without its negation's being valid(mol217)
When we are concerned with complete proof procedures and decision procedures for the truth of statements, as in elementary number theory, rather than for properties of schemata, the difference between the two disappears- at any rate if our vocabulary includes negation(mol217)
"Metamathematics" is also called proof theory. Its subject matter is mathematical theory itself(mol218)
Proofs of validity are intuitively preferable to proofs of inconsistency. Truth is our game, and direct aim is more straightforward than snaring and trapping(mol226)
Godel's result started a trend of research of mathematics sometimes called proof theory, having to do with recursive functions and related matters, and embracing indeed a general abstract theory of machine computation(wp18)
No analogical argument can amount to a proof. The best it can do is to offer support for a hypothesis(mg201)