Infinities, impossibilities, and the man in the white linen suit • Hacker News Board

Re: Infinities, impossibilities, and the man in the white linen suit

#61

Post by My_Name » Mon, Jul 20, 2026, 7:55 PM UTC

sorokod wrote: post #19

iainharper wrote: post #2

I wrote this partly because nobody seems to know about Gödel who is arguably a more important figure than Turing. The piece follows four threads where formal limits show up in AI practice today: learnable problems that mathematics cannot sort, neural networks that exist but cannot be trained, self-improving systems that dropped their safety proofs for benchmark scores, and the fact that “will this AI cause harm” is,…

Gödel, Escher, Bach (1980, Pulitzer prize) was how I first got exposed to Gödel. https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach

Ah yes, the eternal golden triangle

Re: Infinities, impossibilities, and the man in the white linen suit

#62

Post by jibal » Tue, Jul 21, 2026, 2:08 AM UTC

My_Name wrote: post #60

"the sentence...is true but unprovable" Personally, I would say that if it cannot be shown to be true or false, it cannot be said to be true (or false). Asking that about that sentence is, to my mind, a category error, like asking what sound yellow makes. I'm reminded of the episode of Star Trek (TOS) where they make a bunch of robots explode by saying "This sentence is false". Any reasonably written software deals w…

What you think "personally" isn't relevant. Turing proved that there are statements that are true in a consistent axiomatic system (of sufficient power for arithmetic) that cannot be proven within that axiomatic system.

That episode of TOS is a silly show and is completely irrelevant.

Re: Infinities, impossibilities, and the man in the white linen suit

#63

Post by hardbass » Tue, Jul 21, 2026, 5:14 AM UTC

Diogenesian wrote: post #59

hardbass wrote: post #57

Earlier quoted context omitted.

Some problem being too chaotic or computationally expensive to perform is a completely different thing that claiming something supernatural. And there is hardly any reason for computers to be confined to silicon digital logic, using coprocessors is a standard procedure, so whatever element if any which exhibits hypercomputation (which is highly doubtful as the universe barely even reaches even a small fraction of the…

To be clear the limitation here isn't silicon logic, it's theoretical logic (specifically general recursive functions). It sure seems like any possible computation can be expressed as a general recursive function, but that's a scientific thesis, not a mathematical theorem. As we have yet to formalize "define a physical system" it is possible that this task isn't actually expressible in 21st century mathematical logic…

Yes but as of yet physical systems have barely scratched even more than 0% of simple turing machines, so presently I'd consider that possibility a wild speculation.

Re: Infinities, impossibilities, and the man in the white linen suit

#64

Post by cyberax » Tue, Jul 21, 2026, 6:31 AM UTC

tsimionescu wrote: post #49

cyberax wrote: post #45

Earlier quoted context omitted.

That's why I added the word "arbitrary" :) > However, this doesn't mean that TMs can't operate on such numbers. Well, it kinda does. For example, you can't compute the sum of two arbitrary real values in a finite number of steps. That being said, it's possible that the universe actually is not "running" on real numbers. This would also mean that our physics built on top of calculus is just an approximation. This has…

I find this topic of precision and of the computation that happens in the universe to be pretty interesting, but also very hard to pin down. For example, let's take two arbitrary real numbers, call them x and y. They're almost certainly irrational uncomputable numbers, of course. Now, what is their sum? It's x + y, of course - I can tell you that without doing an infinity of computational work. Now, is this a useful…

But you have a problem when you want to compare two numbers (to see if the particles overlap). Or if you want to divide one number by another.

In theory, in classical physics if you start a simulation on a Turing machine, then all the initial states must be computable. And you can carry on using symbolic representation of numbers, even with transcendental functions (as long as you compute them to a precision necessary to carry out the next step of the simulation).

But with quantum mechanics, we might have true randomness, so precise classic simulation can become impossible after the first interaction.

Re: Infinities, impossibilities, and the man in the white linen suit

#65

Post by jibal » Tue, Jul 21, 2026, 6:47 AM UTC

c2c2c wrote: post #11

skippyfish wrote: post #5

Aside from seemingly being 100% AI-generated, this article makes a really spurious use of Godel's argument, which I think is best exemplified by the following passage near the end: "These four threads share a common ancestor in what Gödel proved in 1931, and Turing sharpened in 1936. Rule-based systems cannot fully account for themselves. A system cannot certify its own trustworthiness. A learning framework cannot de…

I think this is selling Gödel a little bit short. He proved for the first time that in any system of mathematics with enough complexity to be interesting, there are statements that cannot be proven true or false. The Gödel sentence might seem like a trivial example, but other examples of such things have been discovered since, such as the continuum hypothesis which the article mentions.

The Gödel sentence (which is not otherwise of any interest) is true but unprovable within that axiomatic system. The Continuum hypothesis (which is of great interest) is true or false only by stipulation.

The fact is that neither Gödel's theorems nor the Halting Problem have any actual real world consequences (outside of people talking about them). People say "oh, we can't write a verifier because of the Halting Problem", but that's simply not true since all of our programs are actually FSMs with physically limited data, and the HP is solvable for that subset of TMs. The real limitations are time and space, so this is an engineering problem -- and people who aren't suckered by Halting Problem Hysteria find engineering solutions that work on real programs, bailing if memory or time thresholds are exceeded.