To fully understand what that meant and means requires an understanding of what Logic and Mathematics were 'on about' from around 1870 to l933. To (greatly) simplify, Frege, Hilbert, Russell, Whitehead, & All That Crew were attempting to construct an intellectual tool that would always, when correctly used, derive a True and Valid answer.
Gödel's Proof = You Can't.
By moving to the Second Order, on the other hand, the True/False OR False/True oscillation is, sometimes, capable of resolution.
But that, very often, takes you outside the original axiomatic system.
That was the sailing-over-the-edge moment.
Now it may be that arithmetic really is consistent, but that is another truth function that is "uncomputable." The Fates are kind.
