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Comment by ahelwer | Hacker News Reader

Honestly this objection always seemed very handwavey to me, especially given how much abuse the incompleteness theorems endure during extension beyond statements about the Natural numbers. Like, what exactly happens if you try to write a bootstrapped theorem prover? Does a deity descend from the heavens to unplug your keyboard? I'd be very interested in reading a paper or blog post documenting a practical attempt here.

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I don't mean to be rude, but this objection only seems handwave-y if you don't understand Gödel's proofs (common), or you have some refutation of them (obviously uncommon, but definitely interesting).

This is a good not-too-technical but not-condescending explanation https://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/081... (it does get into the technical details; it just doesn't go line by line as you would if you wanted to recreate the proofs).

As long as you have the expressive power of Peano arithmetic (which all programming languages ought to, and definitely all proof systems ought to), you can find a Gödel sentence in the system and thus prove that it can't prove, of itself, "x is consistent".

It's worth looking into, if only for the enjoyment.

Have you ever written a compression algorithm?

You do know that Shannon proved that general purpose compression is impossible, right? Why bother if that 75% compression ratio is provably impossible?

It isn't at all obvious that "a theorem prover proving itself correct" and "an axiomatic system proving its own consistency" are the same thing, beyond both being in the category of things which refer to themselves in the domain of mathematical proofs. Software correctness isn't the same as consistency. A program is correct if it refines its specification, and an axiomatic system is consistent if it does not lead to contradiction. Before doing a deep dive into Gödel we need to firmly establish we're actually looking at the right problem.

Even if Gödel is shown to apply in the strict generalized theoretical sense, it might be irrelevant to practice. Theory and practice often differ enough for practice to be very useful; we easily prove halting for most programs we care about, compute solutions to average-case NP-Complete problems without a noticeable rise in CPU temp, and (as the other poster at this level said) write compression algorithms which work well on all relevant inputs. This is poor analogous reasoning, of course, but I include it because demonstration of universal practical pitfall would be very interesting in much the same way specific manifestations of availability loss in fault-tolerant consensus algorithms are interesting.

Really though, I am tired of the above exact exchange playing out in every thread I've seen here on formal verification. Someone raises the obvious concern of bugs in the theorem prover. Someone else raises the bootstrapping solution, cribbed from compilers. Someone else raises the Gödel objection; and there the conversation dies. Dig deeper.

Good write-up. I hear you about the patterns of debate that prop up in these threads. My favorites are the ones where people show up with specific logics, techs, or R&D suggestions that others and I learn from. Such an approach might lead to people solving hard problems over time.

My favorite counter to the Godel thing, esp about termination, was sklogic showing how far from reality these concerns were by illustrating that a while loop counting down toward zero with "stop if zero" condition will guarantee a termination of any algorithm working piece-by-piece in inner loop so long as hardware functions. No magic math at all required to defeat the termination requirement in real-world. I added watchdog timers or even shelf-life of modern parts can do the same.

I did something similar when I was concerned about infinite loops or DOS attacks from malicious input. Just box it up in something I know will work with something checking up on it. Another was algorithm theory telling me QuickSort performed better but choose HeapSort if worst-case is a problem. A smarter programmer told me to just time average for QuickSort, run it with a watchdog, kill it on any instance it takes too long, and use HeapSort for that instance. Those are the kind of tips I appreciate about these terrible problems the theoretical side brings me. Hell, if only theory people would find, generalize, and expand on all the effective cheats in engineering instead of idealistic or abstract stuff. :)

But doesn't the constant Gödel reference push people, exactly, to work toward effective cheats? Imagine if the people putting in effort to make real systems instead spent their time trying to create the self-verifying prover.

No it distracts people by making them think some legend in math showed verification was provably a waste of time. The number that show up to keep saying it on reputable forums like this one leads to increased belief it's true via repetition effect. So, it's overall bad vs mentioning something relevant to practical verifications.

Plus, hardly anyone is making a self-verifying prover. Someone just asked about that. I tried what I hope is the proper response of simply linking to a self-verifying prover that succeeded in that goal up to first-order logic. So, that's already done with us just needing similar ones for checkers of HOL, Coq, ACL2, etc. Contrary to your implication, the Milawa techniques are based on and supporting some of most practical verification work to ever be done. So, one of few times someone totally ignored Godel in what non-experts would consider Godel's domain led to one of best results in formal methods.

We don't need Godel at all in these discussions. We just need to explain what formal methods are, their successes, their limitations, how to properly use them, and ideas for new projects.

I understand the frustration, but the Gödel objection will always be raised as long as someone asks for a prover that can prove itself correct (read "proves false statements or fails to prove true statements," which just is consistency). To put a finer point on it, once you get how "an axiomatic system [can't prove] its own consistency," it actually is rather obvious that "a theorem prover [can't prove] itself correct." How would it? Without axioms or classical logic? What's it doing for the things that aren't it itself?

Notice though, Gödel's incompleteness didn't end formal logic (all the better for Gödel). I know we're in a public forum, but I would note that your "But what about the practical applications!?" is misplaced (though not poor analogous reasoning--I think you're hitting the nail on the head, honestly). Formal verification is not futile because of Gödel's proof. However, a self-verifying formal system is--which is a useful insight, especially for those interested in formal verification.

As an aside, I don't think you need to convince anyone that's actually recreated Gödel's proofs to approve of continued work on formally verified code. We would get excited about that stuff even if it weren't practical.

The incompleteness theorem effectively states that some statements will require an infinite amount of time for this bootstrapping approach to work. The only deity involved is the finite-ness of the natural world.

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