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I thought I understood Gödel's Incompleteness Theorem to say:

  • Starting from ZF, there only a countable number of proofs you can write

  • The number of possible conjectures is uncountable.

  • Thus, there is a conjecture S such that you can't write down a proof of S, but you also can't write down a proof of -S.

However, I later realized the number of conjectures you can write down is also countable.

What am I missing?

  • 7
    Nope. There are countably many statements in mathematics, and countably many proofs. Goedel's result has nothing to do with countability.2012-08-09
  • 2
    What Goedel found was a way to encode a statement about number theory proofs in number theory that essentially says, "This statement has no proof in this axiom system."2012-08-09
  • 1
    What are you missing? Perhaps a reading of Godel's paper. Or a good modern treatment. The "popular" discussion is not going to give you understanding of this. For example, Godel's paper does not mention ZF.2012-08-09

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