Does Gödel say unprovable statements exist, or that they can exist?

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I don't have a very strong mathematical background, but the question I have is straightforward.

Numberphile's video and Veritasium's video are giving different answers regarding the final conclusion of Gödel's incompleteness theorem.

What does this theorem state ?

  • Unprovables statement exists within an axiom-based system ?
  • Unprovables statements CAN exist within an axiom-based system ?