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Well, it's not possible to prove the consistency, thanks to Godel. Maybe one of your new theories would contain a statement, inconsistent with the rest of ZFC.

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When we say that a statement P is independent of ZFC what we're really saying is "if ZFC is consistent, then P is independent of ZFC (hence both ZFC+P and ZFC+not P are consistent)".

This is the only sensible way to interpret claims of independency, since if ZFC is inconsistent it just proves every statement so there's no independent ones

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.

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).

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