This is the (late) weekly open thread. See the tag. You'd think we could automate this. The traditional boilerplate follows.
If it's worth saying, but not worth its own post, then it goes here.
Notes for future OT posters:
1. Please add the 'open_thread' tag.
2. Check if there is an active Open Thread before posting a new one. (Immediately before; refresh the list-of-threads page before posting.)
3. Open Threads should start on Monday, and end on Sunday.
4. Unflag the two options "Notify me of new top level comments on this article" and "
He proved, that there is ALWAYS either at least one "A & ~A" (and therefore many) - either an unprovable theorem exists. Inside all those systems, which contain the "standard calculus"!
He didn't prove an actual "A & ~A", but that one always exists, if there are no unprovable theorems in those "standard calculus systems".
That isn't actually grammatical English, and unfortunately some plausible guesses at reconstructing it produce things that are completely false. So here's what Goedel actually he did: he proved that for any powerful enough system either there is at least one (hence many) A & ~A, or there are A for which neither A nor ~A can be proved by the system.