Alternative accounts of consistency have, however, been proposed, according to which a system is consistent (1) if no wff consisting of a single variable is a theorem or (2) if not every wff is a theorem.
The first half leads to Gödel's theorem on consistency proofs, which says that if a system is consistent, then the arithmetic sentence expressing the consistency of the system cannot be proved in the system.
In particular, he discovered that, if such a system is ω-consistent i.e., devoid of contradiction in a sense to be explained below—then it is not complete and that, if a system is consistent, then the statement of its consistency, easily expressible in the system, is not provable in it.
The proof of this theorem consists essentially of a formalization in arithmetic of the arithmetized version of the proof of the statement, "If a system is consistent, then p is not provable"; i.e., it consists of a derivation within number theory of p itself from the arithmetic sentence that says that the system is consistent.
The picture that emerges from those overseeing the system is consistent and depressing.
An axiomatic system is consistent if, whenever a wff α is a theorem, ∼α is not a theorem.
"It is very important that we preserve the unity of NATO" and that the new system is "consistent" with the alliance, Robin Cook, the British foreign secretary, said here this week.
More exactly, Gödel showed that, if the system is consistent, then p is not provable; if it is ω-consistent, then ∼p is not provable.
The deduction system is consistent since inference rules do not reveal any inconsistency.
Obviously, if the IMRT planning system is consistent, it is acceptable.
Experiments show that the 3-D system is consistent with the 2-D system.
