Mathematics self-proves its own Consistency (contra Gödel et. al.) • Hacker News Board

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#41

Post by jfarmer » Mon, Jul 29, 2013, 8:10 PM UTC

EGreg wrote: post #34

jfarmer wrote: post #33

Earlier quoted context omitted.

He's just speaking colloquially and I don't know anything about the system of logic he's describing. He means something like "weak relative to all the other ones we'd consider on a day-to-day basis" or "so weak we'd never seriously consider it except as a specialized tool." That's not really the point, though. The point is that "strong" and "weak" are well-defined terms in mathematical logic and he was almost certain…

I think he was saying that their math-fu was weak, too weak for comp sci :)

No, he's almost certainly not. Keep in mind, I know nothing about PL. I'm solely relying on my background as a mathematician to think of the most reasonable interpretation for what he's saying.

He's saying something like, "You can axiomatize Peano Arithmetic in PL and explore the idea of provability therein because that's what PL was invented for. At first glance we might try to use something like PL to do what we're setting out to do, but it's so specialized that it winds up not being worthwhile."

This makes more sense in the context of his paper, anyhow. Gödel's Incompleteness Theorem basically says that Peano Arithmetic can't prove its own consistency within PA unless it's inconsistent

So, we could model PA inside another logical system (e.g., PL) which has notation for certain meta-mathematical statements like "P(X): it is provable that X". You'd then need to demonstrate that a version of PA axiomatized with this logic is both complete and sound WRT PA, etc. etc.

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#42

Post by mietek » Wed, Jul 31, 2013, 1:49 AM UTC

tlb wrote: post #19

I know Carl a little from the late 90s. I figured there was a 50% chance he would come up with something historically important, and a 60% chance he was crazy. (The two options are not quite mutually exclusive.) Indeed, most examples illustrating the Incompleteness Theorem involve self-reference. But, it seems hard to prove anything interesting in a system that precludes self-reference. BTW, the Y Combinator is a way…

Do you consider Planner and the actor model not to be historically important?

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#43

Post by runarberg » Thu, Aug 01, 2013, 10:57 AM UTC

gizmo686 wrote: post #37

runarberg wrote: post #26

I've always been fascinated with naïve set-theory, and how it was refuted after it's apparent inconsistency, the famous Russell's paradox of the set of all sets that don't contain it self, and only contains it self if it doesn't etc. So supposing it's False, yields True, and supposing it's True yields False. But what fascinated me was not the Russell's set, but it's inverse, the set of all sets that contain it self,…

S doesn't seem that interesting. Assume that S does not contain itself. Consider the set P, which can be constructed by adding P as an element to S. Note that P contains itself. Therefore, P is in S. Also note that every other element in P is also in S, and that every element in S is in P. Therefore P and S are the same set. Therefore S contains itself, contradicting our assumption that S does not contain itself. If…

hmm, you are correct. S really isn't that interesting. And my scepticism towards justification through non-contradiction because of possible super-consistancy is therefor based on false premises.