Post by timhttp://www.salon.com/2013/07/14/how_does_one_prove_the_unprovable_partner/
There's a USENET group about that.
It's been shown that there will be statements that are unprovable for
any axiomatic system; the temptation is to then take those statements
as additional axioms and extend the initial system.
There are also various logics that can be employed, some more powerful
than others, so that also is an aspect.
And then there are encodings such that proving a theorem corresponds
to a calculation. And we get into computational complexity, and levels of
"hardness". Some calculations will give the right answer, but would take
an infinite length of time to achieve it. Halting problems.
------------------------------
Everything you've learned in school as 'obvious' becomes less and less obvious
as you begin to study the universe. For example, there are no solids in the
universe. There's not even a suggestion of a solid. There are no absolute
continuums. There are no surfaces. There are no straight lines. - R.
Buckminster Fuller
------------------------------