patrick.net

Heraclitusstudent says

So any AI system capable of considering a sentence as simple as "This sentence is false" IS inconsistent.

So you are telling me to remove axioms and I'll be fine? Seriously? Which basic fact about arithmetic do you want to remove from such AI system because of this inconsistency????

No, what I told you is that you aren't going to have inconsistencies. Teaching a computer how to deal with absurdities is simple and beside the point.

I see that you're back to confusing Godel's method of proof with what the proof implies. The proof only implies something very simple and straight forward: that a system can not be both consistent and complete. It does not say that the only way in which such inconsistencies arise as the system becomes more complete (meaning as more axioms and theorems are added to cover the infinite directions and lengths to which you can go starting from the initial axioms) is with such absurd statements.

Maybe the issue is you don't get what complete means in this context ?

A system does not have to be anything close to complete to deal with the set of problems you are going to use it for. Or to prove what you set out to prove.

If you are going to have a computer do something useful such as proving the four color theorem, you are going to have precise definitions and some basic axioms and theorem of geometry that are consistent that were used in coming up with the proof. (actually most of the consistent basics of plane geometry are accepted and not really even addressed in such a proof). Also, I know that in that example the people did the proof simply using a computer under their guidance.

I have to stop. Either I am missing something or you don't really have a point to make that I understand. We're
going in circles, and you want to get your point across without really attempting to understand what I'm saying. I believe I understand what you're saying. We're talking at eachother. Maybe I'm right but you're saying something else ? Or maybe I'm wrong, and you can't explain why ?

marcus says

A system does not have to be anything close to complete to deal with the set of problems you are going to use it for.

That's simply not true. To be useful a system needs to have a complete picture of what geometry is, or what numbers are. You can't leave half the definition out of it.

marcus says

The proof only implies something very simple and straight forward: that a system can not be both consistent and complete.

I 100% agree with that.

marcus says

It does not say that the only way in which such inconsistencies arise as the system becomes more complete (meaning as more axioms and theorems are added to cover the infinite directions and lengths to which you can go starting from the initial axioms) is with such absurd statements.

So you are saying the problem is not necessarily with self referential assertions.
Well, I agree I don't have a formal proof that there is not an other problem outside self-referential assertions.... But....

But first let me return this: there is no formal proof that there is an other problem outside self-referential assertions. In particular everything Godel did in his proof is about a self-referential assertion.

Second I would argue there are very strong reasons to believe there is no other problem. Because to believe there is an other problem you basically need to believe either:
- there is something intrinsic about a complete arithmetic picture that is rotten (inconsistent with itself);
- there is no way to derive in formal logic everything a human can "prove" as true.
Either of these are extremely strong assertions that fly in the face of everything we observe in practice.