[adastra22]

I was talking about Gödel, not Goldbach:

> The proof of Gödel's result's involves very carefully formalizing what statements and proofs mean so that they can be encoded as statements about arithmetic. He then shows there is a statement with encoding G that says "The statement with encoding G cannot be proved" – if it is true, then it cannot be proved.

Sorry I meant to quote this bit at the beginning of my comment. Parent comment which I was replying to talks about both.

[dudeguy3301]

wait, so you are casually dismissing Gödel's work as a logical "gotcha"? don't premise opinions about complex proofs on other peoples woefully impoverished and misrepresented explanations of said proofs

that is disrespectful and very very very short sighted.

also, go read up (...on Gödel, Cantor, Turing, Tarski, etc...)

[gsinclair]

I understand precisely what adastra22 meant from the moment I read it. Unfortunately two people now have rushed to unkind characteristics of what that (perfectly sensible) meaning was.

[dudeguy3301]

sure, what was written was sensible but has no bearing on the proof. speculation isn't critical thinking, as adastra22 says, be careful to what you are responding to. my point is the lack of rigor and understanding because the proof isnt being discussed, hand waving about an unread and very important mathematical work is irresponsible. thats MY point

[adastra22]

Please try to give the people you talk to the benefit of the doubt, and read carefully what you are responding to.

My training is as an applied physicist. We physicists have an interesting relationship with math. Obviously math is essential to the work that we do, but the physical world decides whether the math is right, not the other way around. Our mathematical models technically permit things like negative mass, time flowing backwards, or magnetic monopoles. But that doesn't mean tachyons, time machines, or fundamental magnetic particles exist--they don't, so far as we know. So I'm trained to actively disregard non-physical, not relevant mathematical implications. I'm sorry if this offends a pure mathematicians sensibilities, but pragmatically it is very useful.

Or take a different field: in computational semantics, a branch of formal linguistics, there are many models for inferring a formal logical statement from an example written sentence or spoken utterance, and then determining the validity (truth) of the statement. These models get caught up on stuff like "This sentence is false." What's the truth value for that sentence? If it is true then it must be false, and if it is false then it must be true. Error, validity of this statement can't be determined! But hey, it turns out that in practice this basically never happens unless the speaker is really confused, misspeaks, or deliberately evasive. Real sentences don't have this self-referential, circular logic structure because that's not how people think or communicate.

Now "this sentence is false" goes back to the greeks, IIRC, and Gödel's theorem is slightly different. Gödel's main work is in the formalization of proofs and proof systems, and I don't want to take away from that in any way. But the incompleteness theorem always seems to be explained through these sorts of self-referential examples and I have yet to ever see it reduced to a practical problem with real-world implications. Hence my question. Does Gödel's incompleteness theorem actually constrain a real world application of proof systems, where we tend to be interested in non-cyclical logical arguments?

[thrwycct123]

> Obviously math is essential to the work that we do, but the physical world decides whether the math is right, not the other way arounnd

That was the belief before the 1800s. But with the discovery of non-euclidean geometry, math has been divorced from the physical world. Math is simply a system of axioms and proofs. Math is purely abstract and logical. Whatever math that physicists use just simply happens to align with the physical world.

Also, the physical world doesn't confirm whether the math is "right". Math is deductive, not inductive. As long as the math is derivable from the axioms, it is right. The physical world/experiments determine whether the mathematical model aligns with the physical world. The physical world has no say in math. Not anymore.

> So I'm trained to actively disregard non-physical, not relevant mathematical implications.

In the past, when a mathematical model predicted something ( relativity to quantum physics to elements ), experiments were conducted to determine whether the models aligned with the physical world. If the mathematical models make predictions that physicists currently can't verify with experiments, should we disregard it? Do we need technology to advance to where we can conduct experiments. Do we need physics to become more abstract? Perhaps model the physical world in the virtual world. Would experiments in the virtual world be applicable to physics? Seems like physics is both at a dead-end and on the cusp of a revolution.

[adastra22]

The question “does this have relevant applications?” Is still well formed and answerable. That’s all I’m asking! I find it odd that so many are taking offense.

[randomswede]

If your question is "does Gödel Incompleteness have practical applications", my answer is "probably". Not necessarily directly, but indirectly it has inspired much other work.

One of the things it inspired was Turing's work on uncomputable numbers. It turns out that computability and incompleteness are intertwined, which at least I find interesting. And without the "uncomputable numbers" malarkey, I wonder if the Turing Machine formalism would exist (answer, "probably, but maybe looking different and with another name"). And, well, the Halting Problem is essentially Gödel Incompleteness (imagine handwaving here).

As for proof systems, again, the answer is "probably". Knowing that there are true, unprovable, statements in a formalism is something that informs how you approach it, you need to put a limit on how far to go before you say "I don't know" and taht is in and of itself important.

[dudeguy3301]

still doesn't sound like you have ever read the proof, nor about it. so aren't you engaged in this discussion strictly by speculation? logical fallacy? just because you have some point to make has no bearing on the work you are adjacent to and not actually discussing. making your own points and arguments about logical gotchas is great, has nothing to do with critically understanding a work you haven't read. asking for giving the 'benefit of the doubt' to speculation informed only by a comment section is ridiculous. do your homework before asking for generous responses to your internet quibbling

[adastra22]

And what did your post contribute?

Never mind then, I’m sorry I bothered.

[ColinWright]

I'm late to the party, so I'm not sure you'll see this. Hope you do. Different people will each have their own contexts and their own concepts of "useful" or "interesting". I'm making assumptions about yours ... apologies if I misrepresent you.

The proof of Gödel's result uses the paradoxical statement "This statement is false", but that's being used to prove this very general result about all systems. So the hunt is then on to find "natural" statements that are True but Unprovable.

But the "unprovable" bit should more completely be stated as "unprovable in a specific axiomatic proof system". If we want to prove that statement S is "True but Unprovable" then we must actually prove that it's true. So if we've proved it's true, what does it mean to say it's unprovable? We just proved it! What's going on?

So let's take a specific example.

Peano Arithmetic (PA)[0] is an axiomatic proof system intended to capture Natural Numbers and their behaviour.

The "Goodstein Sequence" G(m) of a number m is a sequence of natural numbers ... you can find the definition here[1]. It's not hard, but it's longer than I want to reproduce here.

Goodstein's Theorem (GT) says that for every integer m greater than 0, G(m) is eventually zero.

It has been proven that GT cannot be proved in PA, but it can be proved in stronger systems, such as second-order arithmetic.

So the statement of GT is not self-referential, along the lines of "This Statement Is False" sort of thing. It's an actual statement about the behaviour of integers, so it's not a self-referential trick.

Your question now is: What's the point? How is this useful or relevant?

Much of modern (pure) mathematics is chasing things because the mathematicians find them interesting. The vast, vast majority will never, of themselves, be useful by (what I expect are) your standards.

But it was once thought that factoring integers was of no practical use, and only pursued or investigated by cranks. Imaginary Numbers were thought to be bizarre, useless, and dangerous. Non-Euclidean Geometry was thought to be utter nonsense, and held up as part of the "proof" that the fifth postulate was unnecessary and was deducible from the other four. All three of these now form critical components in modern technology.

Even more, to the average person on the street, anything to do with algebra is completely pointless.

For you, Gödel's theorem is completely pointless and useless and probably of no interest at all, but it helps us understand the limitations of formal systems. The techniques that have been developed in the time since it was proved have helped us understand more about what computer verification systems might or might not be able to accomplish.

Of itself, Gödel's theorem might not be of direct, immediate, and practical use, but the work it has inspired has tangentially been useful, and may yet be moreso.

But not for everyone. After all, some people don't care about the Mona Lisa, or Beethoven's Fifth Symphony, or Michaelangelo's David, or the fact that people have walked on the Moon, so why should people care about results in Pure Mathematics?

That's the thing about Pure Mathematics. Sometimes it ends up being useful in ways we never expected.

[0] https://en.wikipedia.org/wiki/Peano_axioms

[1] https://en.wikipedia.org/wiki/Goodstein's_theorem#Goodstein_...