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He proved, that every system rich enough to contain "infinite arithmetics" is EITHER inconsistent (have paradoxes) EITHER have some non provable sentences.
Almost everybody thought - "Okay, okay, so we'll always have nonprovables. It's a shame, but what can we do?"
But this was not the only one explanation. The "Okay, okay so we can't make the complete "infinite arithmetic" without a paradox. We must cease to even try that." - flies at least as well.
I don't know about that. They key word is "useful". I'm not quite ready to discard and forget Peano arithmetic.