Just a quick, picky point about the logic application. What was said in lecture is that incompleteness result follows from considerations of complexity on the assumption that T is consistent. But the proof given actually assumes something stronger, that T is arithmetically sound (that is, anything T proves is true in the natural numbers). Arithmetical soundness implies consistency, but the converse does not hold (this is itself a corollary of the first incompleteness theorem).
Follow up: Of course one can prove the first incompleteness theorem on the mere assumption of consistency, but this proof method needs more machinery to weaken the assumption from soundness to consistency.
Just a quick, picky point about the logic application. What was said in lecture is that incompleteness result follows from considerations of complexity on the assumption that T is consistent. But the proof given actually assumes something stronger, that T is arithmetically sound (that is, anything T proves is true in the natural numbers). Arithmetical soundness implies consistency, but the converse does not hold (this is itself a corollary of the first incompleteness theorem).
ReplyDeleteFollow up: Of course one can prove the first incompleteness theorem on the mere assumption of consistency, but this proof method needs more machinery to weaken the assumption from soundness to consistency.
ReplyDelete