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(1) Assume my reasoning powers are captured by some formal system F (to put this more briefly, "I am F"). Consider the class of statements I can know to be true, given this assumption.
(2) Given that I know that I am F, I know that F is sound (as I know that I am sound). Indeed, I know that the larger system F' is sound, where F' is F supplemented by the further assumption "I am F". (Supplementing a sound system with a true statement yields a sound system.)
(3) So I know that G(F') is true, where this is the Gödel sentence of the system F'.
(4) But F' could not see that G(F') is true (by Gödel's theorem).
(5) By assumption, however, I am now effectively equivalent to F'. After all, I am F supplemented by the knowledge that I am F.
(6) This is a contradiction, so the initial assumption must be false, and F must not have captured my powers of reasoning after all.
(7) The conclusion generalizes: my reasoning powers cannot be captured by any formal system.