[CrazyStat]

This is incorrect. It absolutely is possible to prove consistency, what Gödel tells us is that in any consistent logic system there are true but unprovable (in that system) statements.

For this particular list, the statements have been proven to both be consistent with ZFC and for their negations to be consistent with ZFC.

[ithinkso]

This is first incompleteness theorem. What deepsun was referring to is second incompleteness theorem - in a consistent system F the statement 'F is consistent' is in fact unprovable (in F).