On Mon, Jan 13, 2025 at 5:58 PM Jesse Mazer <laserma...@gmail.com> wrote:
*> Doesn't Godel's theorem only apply to systems whose output can be mapped > to judgments about the truth-value of propositions in first-order > arithmetic?* *That's Godel's Completeness Theorem which concerns First Order Logic (FOL), not to be confused with his better-known Incompleteness Theorems which are concerned with Second Order Logic (SOL). FOL can do stuff like; Socrates is a man, all men are mortal, therefore Socrates is mortal. Unfortunately FOL is pretty weak, if something can be proven in FOL then it can also be proven in SOL, but the reverse is not necessarily true. In FOL you can not define things like "finite" or "continuous", you can't use mathematical induction and you can't even do arithmetic. For those things you need SOL, but it's incomplete, and it can't prove its own consistency.* *John K Clark See what's on my new list at Extropolis <https://groups.google.com/g/extropolis>* 3e4 > > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To view this discussion visit https://groups.google.com/d/msgid/everything-list/CAJPayv3NTTq1FPPz4yjsnfV8KPCs6V7QM-CF1Z0Ors9QRYvmuA%40mail.gmail.com.
