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.
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.
Yes, it is useful. Like a buggy program may be useful. But there comes the time of refactoring the whole thing.
Perhaps on an entirely new architecture.