Possibly:
...within any given branch of mathematics, there would always be some
propositions that couldn't be proven either true or false using the
rules and axioms ... of that mathematical branch itself. You might be
able to prove every conceivable statement about numbers within a
system by going outside the system in order to come up with new rules
and axioms, but by doing so you'll only create a larger system with
its own unprovable statements. The implication is that all logical
system of any complexity are, by definition, incomplete; each of them
contains, at any given time, more true statements than it can possibly
prove according to its own defining set of rules.
Gödel's Theorem has been used to argue that a computer can never be as
smart as a human being because the extent of its knowledge is limited
by a fixed set of axioms, whereas people can discover unexpected
truths ... It plays a part in modern linguistic theories, which
emphasize the power of language to come up with new ways to express
ideas. And it has been taken to imply that you'll never entirely
understand yourself, since your mind, like any other closed system,
can only be sure of what it knows about itself by relying on what it
knows about itself...
...within the system, there exist certain clear-cut statements that
can neither be proved or disproved. Hence one cannot, using the usual
methods, be certain that the axioms of arithmetic will not lead to
contradictions ... It appears to foredoom hope of mathematical
certitude through use of the obvious methods. Perhaps doomed also, as
a result, is the ideal of science - to devise a set of axioms from
which all phenomena of the external world can be deduced...
...He proved it impossible to establish the internal logical
consistency of a very large class of deductive systems - elementary
arithmetic, for example - unless one adopts principles of reasoning so
complex that their internal consistency is as open to doubt as that of
the systems themselves ... Second main conclusion is ... Gödel showed
that Principia, or any other system within which arithmetic can be
developed, is essentially incomplete. In other words, given any
consistent set of arithmetical axioms, there are true mathematical
statements that cannot be derived from the set... Even if the axioms
of arithmetic are augmented by an indefinite number of other true
ones, there will always be further mathematical truths that are not
formally derivable from the augmented set...
http://www.miskatonic.org/godel.html
http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorem
My conclusion; I don't think that the idea of proof is appropriately
used here, since certainty about anything's decidability is in doubt,
concepts included. But he may have shown how in order for the case of
complete enough justification for the needs of math, that
incompleteness is necessary, for sufficient warrent of the case.
--------------------------------
Russell's paradox represents either of two interrelated logical
antinomies. The most commonly discussed form is a contradiction
arising in the logic of sets or classes. Some classes (or sets) seem
to be members of themselves, while some do not. The class of all
classes is itself a class, and so it seems to be in itself. The null
or empty class, however, must not be a member of itself. However,
suppose that we can form a class of all classes (or sets) that, like
the null class, are not included in themselves. The paradox arises
from asking the question of whether this class is in itself. It is if
and only if it is not. The other form is a contradiction involving
properties. Some properties seem to apply to themselves, while others
do not. The property of being a property is itself a property, while
the property of being a cat is not itself a cat. Consider the property
that something has just in case it is a property (like that of being a
cat) that does not apply to itself. Does this property apply to
itself? Once again, from either assumption, the opposite follows. The
paradox was named after Bertrand Russell, who discovered it in 1901.
http://www.iep.utm.edu/p/par-russ.htm
http://plato.stanford.edu/entries/russell-paradox/
http://en.wikipedia.org/wiki/Russell's_paradox