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I just can't get this dumb question out of my head: How could any "incompleteness theorem" ever be completely true?

• gjm11 3y ago
  In the same way as a theorem that says "the sequence of prime numbers is infinite" can be finitely long, or a theorem that says "the square root of 2 is irrational" can be rational :-). If the theorem just said "everything is incomplete", maybe that would be a problem, but that isn't what it says. Goedel's (first) incompleteness theorem says: if you have a formal system with such-and-such properties, then there have to be statements it can neither prove nor disprove. The theorem itself isn't a formal system with such-and-such properties. It isn't talking about itself. (Amusing though it would be if it were, since one of the neat things about Goedel's proof is the way it gets mathematical statements to kinda-sorta talk about themselves.)
• dandanua 3y ago
  The correct way to understand it is this: "We can't pack (i.e. compress) all the mathematical knowledge in a finite set of axioms and rules." This statement doesn't look paradoxical.