Consistent: In logic, a set of
statements is said to be consistent iff it does not imply a contradiction.
Clearly, this is a highly desirable property for those who are
interested in argueing logically. The reason is that in standard
logic
any inconsistent set of
statements deductively
implies any statement,
and its contradiction.
It is an interesting fact that Gödel proved a theorem to the effect
that set theory, and many systems like it, that are strong enough to
imply most of classical mathematics, cannot be proved to be consistent
within the same system  that is, the statement "this theory is
consistent" must be unprovable in the system if the system is
consistent.
This means that the only means to establish that such a system is
consistent is to represent it somehow in another system. So far,
set
theory is not known to be consistent, though it is widely believed to
be, and should be if it is to be the basis of mathematics.
