[kbrkbr]

> is not fully consistent

Not being able to prove the consistency of a system within the system does not entail that the system is inconsistent (or not fully consistent).

[pixl97]

I think it means we could only ever prove it is inconsistent, and never prove that it is consistent.

[tux3]

If we can only prove that an inconsistent system is inconsistent, and we can never prove our (presumably) consistent system is consistent, then it is incomplete.

We can't say it's inconsistent just because we can't prove otherwise. We can say it may be incomplete (or inconsistent, and we haven't noticed yet)