Comment by Chance-Device | Hacker News Reader

Comment by Chance-Device | Hacker News Reader

Parent Full thread

I swear this was on the front page 2 minutes ago and now it’s halfway down page 2.

Anyway, I’m not really sure where Penrose is going with this. As a summary, incompleteness theorem is basically a mathematical reformulation of the paradox of the liar - let’s state this here for simplicity as “This statement is a lie” which is a bit easier than talking about “ All Cretans are liars”, which is the way I first heard it.

So what’s the truth value of “This statement is a lie”? It doesn’t have one. If it’s false, then it’s true. But if it’s true, then it must be false. The reason for this paradox is that it’s a self-referential statement: it refers to its own truth value in the construction of its own truth value, so it never actually gets constructed in the first place.

You can formulate the same sort of idea mathematically using sets, which is what Gödel did.

Now, the thing about this is that as far as I am aware (and I’m open to be corrected on this) this never actually happens in reality in any physical system. It seems to be an artefact of symbolic representation. We can construct a series of symbols that reference themselves in this way, but not an actual system. This is much the same way as I can write “5 + 5 = 11” but it doesn’t actually mean anything physically.

The closest thing we might get to would be something that oscillates between two states.

We also ourselves, don’t have a good answer to this problem as phrased. What is the truth value of “This statement is a lie”? I have to say “I don’t know” or “there isn’t one” which is a bit like cheating. Am I incapable of consciousness as a result? And if I am indeed conscious instead because I can make such a statement instead of simply ”True” or “False”, well I’m sure that an AI can be made to do likewise.

So I really don’t think this has anything to do with intelligence, or consciousness, or any limits on AI.

(for the record, I think the Penrose take on Gödel and consciousness is mostly silly and or confused)

I think your understanding of the incompleteness theorem is a little, well, incomplete. The proof of the theorem does involve, essentially, figuring out how to write down "this statement is not provable" and using liar-paradox-type-reasoning to show that it is neither provable nor disprovable.

But the incompleteness theorem itself is not the liar paradox. Rather, it shows that any (consistent) system rich enough to express arithmetic cannot prove or disprove all statements. There are things in the gaps. Gödel's proof gives one example ("this statement is not provable") but there are others of very different flavors. The standard one is consistency (e.g. Peano arithemtic alone cannot prove the consistency of Peano arithmetic, you need more, like much stronger induction; ZFC cannot prove the consistency of ZFC, you need more, like a large cardinal).

And this very much does come up for real systems, in the following way. If we could prove or disprove each statement in PA, then we could also solve the halting problem! For the same reason there's no general way to tell whether each statement of PA has a proof, there's no general way to tell whether each program will halt on a given input.

Nice reply. I don’t know anything about Peano arithmetic, or how it applies to the halting problem, so I can’t really evaluate this. All I know is the description of the proof that I read some time ago. Maybe there’s more to dig into on it, but as you say at the start of your post, likely none of it has anything to do with what Penrose is arguing for.

Penrose is saying a computer cannot understand / has no intelligence, and the incompleteness theorem proves this. Think in terms of the semantics (the meaning) of what the AI is outputting. The incompleteness theorem proves it does not understand and can never understand the semantics. It operates at the syntax level. How can something operating at the syntax level be intelligent? All it can do is look at symbols and rearrange them. There are algorithms to categorize and give these symbols (tokens/words) weights etc but all it can do is rearrange symbols. Do humans re-arrange symbols, no, they understand. They experience the color red, but to a computer the color red is a number. What is the connection between the number for red and the color red? AI pretty much collects every instance of red and in some sense approximates what red actually is.

> I swear this was on the front page 2 minutes ago and now it’s halfway down page 2.

It set off the flamewar detector. I've turned that off now.

Thanks!

Reply on news.ycombinator.com