How can a statement that is unprovable be true?
I always had the impression that unprovable means you could add either the statement or its negation as an axiom, and both resulting systems are as consistent as the system you started with
I always had the impression that unprovable means you could add either the statement or its negation as an axiom, and both resulting systems are as consistent as the system you started with
Goldbach's conjecture is that "every even number bigger than 2 is the sum of exactly two prime numbers", so 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, etc.
For this statement to be *true* it just means that every even number there must exist two primes that add to that number. This is a statement about infinitely many integers.
A *proof* of Goldbach's conjecture consists of a finite number of formal reasoning steps that start with some axioms and end up at the statement of the result.
To this day, it seems as though Goldbach's conjecture is true. It holds for every number we've been able to test. However, no one has proved it is true or false yet (or proved that it is unprovable).
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.
It's kind of confusing at first, but there is an easier way to get an intuition why there might be true statements that cannot be proved. Think of each statement about the natural numbers as a subset where each number in the subset makes the statement true. There are uncountably many subsets of the natural numbers (by Cantor's diagonalization argument). Proofs are finite chains of finite statements so there are only countably infinitely many of these. Therefore there must be subsets/statements that are true that do not have a matching proof.
The approach that Gödel's proof takes is not too different to the above argument – it is essentially a diagonalization argument – the complexity is in making the encoding of statements are numbers very precise.
Whether or not a visual demonstration like that is actually a “proof” is a separate question. It definitely wouldn’t satisfy Hilbert, and doesn’t meet this definition:
> A proof of a statement S is a finite sequence of assertions S(1), S(2), … S(n) such that S(n) = S and each S(i) is either an axiom or else follows from one or more of the preceding statements S(1), …, S(i-1) by a direct application of a valid rule of inference.
I also don’t know of any visual “proof” like that which can’t be explained much more rigorously and powerfully with a formal set of assertions. But pulling threads like this and really asking what makes a proof a “proof” are some of the deepest questions I think a person can ask. It’s worth doing if only to appreciate what an incredible accomplishment all of the formal set theory work is in unifying and attempting to define meta concepts like “proof” itself.
For such proofs to be contained in a finite space, the verifying person or machine needs to be able to distinguish between arbitrarily minute differences between proofs.
Again, that doesn’t really count as a “proof” by modern standards, but it’s how the ancient greeks thought (they used more than just visual intuition/they also used more rigorous and formal propositions than that water thing, but they were visual and didn’t involve finite sets)
Of course, this is not an argument that our brains are absolutely finite and classical. Perhaps analog computations is required for an appreciation of beauty, or quantum physics is necessary for us to fall in love. But we seem to be able to check math proofs without them.
A proof (there) is just a finite number of symbols which happens to have a specifix form (A=>B AND A), where A and B are sentences, which are finite sequences of symbols having a slecific form… Wait, I am telling you a half of what Gödel did to prove his result.
But one of those two must be true, you just can't prove it. Of course you can add a new axiom that allows you to prove one or the other (or accept one of those statements as an axiom), but you will still have other statements that you can't prove.
https://en.wikipedia.org/wiki/Continuum_hypothesis#Independe...
Also:
https://en.wikipedia.org/wiki/Axiom_of_choice#Independence
And thanks to Wikipedia for this elaborate list:
https://en.wikipedia.org/wiki/List_of_statements_independent...
Three alternative accounts:
* Both the CH and ¬CH mathematical universes really exist, so we just have to choose which one we're more interested in at a given time. Like one might say there are the "reall numbers" and the "realle numbers", both valid and interesting constructions which humanity was just slow to recognize the distinctions between (because they were initially less relevant to our interests and our day-to-day lives).
* We are actually ultimately thinking about one or the other of them, or are in some sense in one or the other mathematical universe, but we don't know enough about our intuition about the reals to be able to specify or explain which one. (Maybe we need other properties whose obviousness or relevance humanity is not smart enough to notice?)
* Some finitist or ultrafinitist approach is actually right: the real numbers are a formalism that, while reasonably motivated by historical attempts to "complete" mathematics in various ways, doesn't correspond to anything Platonically real or to anything intellectually relevant to humanity. (In this account, there is potentially no answer to the question because the real numbers don't exist at all. Neither CH nor ¬CH refers to a mathematical reality, just to games about formalisms.)
It's also a provable statement (via excluded middle)
I do believe this opinion places you very high on the 'confidence' axis, but not especially far along the 'competence' axis.
> 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.
that is disrespectful and very very very short sighted.
also, go read up (...on Gödel, Cantor, Turing, Tarski, etc...)
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?
Never mind then, I’m sorry I bothered.
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.
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.
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_...
It is fated to be proved as a trivial corollary to some more important mathematics; a corollary that no one would have bothered with if not for the historical importance.
The unsolved Twin Prime Conjecture, of roughly the same age, is expected to lead to much more interesting mathematics if it is proved.
But in general, Gödel applies to all formal systems that satisfy certain properties, it's just that the exact unprovable sentences will be different (since you may just add that sentence as an axiom) that's why the general example is very abstract. It shows that sufficiently rich theories are not only incomplete, but also incompletable.
The CH is undoubtedly meaningful, natural, and of huge interest to (a subset of) mathematicians.
There are huge branches of mathematics that don't find immediate practicality in physics. Often these end up being practical in cryptography or quantum mechanics, but sometimes they don't. But to dismiss the entire field that relates to the cardinality of real numbers as uninteresting if someone can't give you a practical application of it shows a simple disregard for other fields of study.
IMO the most weird thing is the following, intertwining consistency of ZFC with Diophantine equations[2]:
> One can write down a concrete polynomial p ∈ Z[x1, ..., x9] such that the statement "there are integers m1, ..., m9 with p(m1, ..., m9) = 0" can neither be proven nor disproven in ZFC (assuming ZFC is consistent). [...] the polynomial is constructed so that it has an integer root if and only if ZFC is inconsistent.
[1] https://en.wikipedia.org/wiki/List_of_statements_independent...
[2] https://en.wikipedia.org/wiki/List_of_statements_independent...
My understanding comes from Keith Devlin’s wonderful book “Mathematics: A new golden age”.
Goodstein's theorem and the Paris-Harrington theorem are some examples of this for ZFC. There are several more, maybe a logician could chime in
Because if you show that the converse of Goldbach's conjecture is not provable, you have actually proven Goldbach's conjecture (since you have shown that there is no counterexample!).
>Think of each statement about the natural numbers as a subset where each number in the subset makes the statement true. There are uncountably many subsets of the natural numbers (by Cantor's diagonalization argument).
Don't we only care about the countable set of statements that can written down in a given logical system? Say second order logic + ZFC.
By “converse” I think you mean “negation”.
A statement being unprovable means we will never know whether it is this or false. > Because if you show that the converse of Goldbach's conjecture is not provable, you have actually proven Goldbach's conjecture (since you have shown that there is no counterexample!).
Nope: demonstrating that (the opposite of Goldbach’s conjecture) is unprovable is logically equivalent to demonstrating that (Goldbach’s conjecture) is unprovable. It means we’ll never know either way.
I think this is false. If you find a number N that is not the sum of two primes, you did disprove Goldbach's conjecture. Any such number would be smaller than infinity, and so there would only be a finite set of primes smaller it to check for.
So basically, if Goldbach's conjecture turns out to be false it IS going to be PROVABLY false.
Only if Goldbach's conjecture is actually true will it be the case that it is impossible to prove its negation. But if it is ALSO unprovable (but STILL true) you will NOT be able to prove that the negation is unprovable, because that would mean that there doesn't exist ANY number N that disproves the conjecture, so would prove the unprovable original conjecture....
Consider the statement "All swans are white", but you live in a universe with an infinite number of swans.
Lets assume that there exists at least one black swan. Proving the statement false is trivial once you find the first black swan.
However, if all swans in the given universe ARE white, and you have no way of inspecting every one, you can never PROVE that they are all white. Also you can NOT prove that it is impossible to prove the negation.
Yes, you can. Math is not physics. Pythagoras was able to prove his theorem about every triangle in the universe without inspecting every triangle in the universe. The trick is that "the universe" is not the physical universe, it's just a choice of axioms.
My understanding is that statements about members of infinite sets can be decidable if there exists a method to set up use recoursion/induction to cover all of them. For the set of right triangles, this is relatively straight forward (if we ignore the complexity introduced by real numbers).
For statements on infinite sets where it is not possible to reduce a proof to such recoursion/induction, you quickly end up needing an infinite number of steps to cover all cases, meaning the problem is undeciable/uncomputable.
See the Church-Turing thesis: https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis
Lets hypothetize that Goldbach's conjecture is true and unprovable (let's call it S1). In that case, the following statments are both true: C1) It is impossible to prove S1. C2) It is impossible to prove the converse of S1. (since in this case S1 is true, the converse is actually false, so cannot be proven).
Now, if we go into meta-proofs, we may be able to prove C1 (whether or not S1 is true). But we will never be able to prove C2 (if S1 is true and unprovable, C2 will also be true and unprovable).
Now, if S1 is actually false, We may actually at some point find a proof of that. But that's not really so interesting.
The interesting part is that there DOES exist statements like this that ARE true AND unprovable (even if we don't know which statments), we just don't know WHAT statments those are.
At 13:46 it gets into the "is there a way to prove every complete statement?" and furthermore goes into the Godel math for "there is no proof for the statement with Gödel number g" where that statement itself has a number "g".
The entire video is a good watch though.
And the related problem is that with Godel we don't know if math is consistent either.
GEB is a marvellous work that is accessible to anyone with reasonably good school grade maths. I chanced upon it by accident in the school library one day and was hooked after a few pages.
Anyway the crux of the matter is that you can very carefully construct a statement about a system that can't be either proven or disproven by that system! I don't have anything like the formal knowledge to really get to grips with it and the discussions on infinities and so on are pretty mind blowing. However, you feel that DH is imparting glimpses into the sheer beauty of the ideas he covers.
Wait 'til you discover what ricercar and quining is all about - bloody lovely. Just read it but take your time. There is something in there for everyone. You often get told by clever people about the links between maths, music and art. Mr H easily gives the best argument I've ever seen that attests to that being true, whilst giving your brain a right good kicking.
There was a Dutchman, a German and an Austrian who walked into a book ...
Close but not quite as that's an inconsistent statement.
"This statement is unprovable." is the approach Godel takes and eliminates the inconsistency. Either that statement is true, in which case it's unprovable, or it's false in which case there exists a proof of a false statement.
I just picked a classic to start off my prior comment which morphed more into praise for GEB than Goedel's brain breaker.
I have dim memories of a cracking constructive argument starting off with some very basic axioms where one was written as S, and two as SS and so on, then it all went a bit mad but the genius of Hofstadter is to make all that impenetrable palava nigh on accessible to the layman (with a bit of effort from the reader).
The horrendous thing about Goedel is that the final flourish is clearly correct (I assume that responsible adults have filled in the formal bits, I'm sticking to the lies to children version). It is both terrifying and perhaps obvious at the same time. I can imagine the sense of dread when mighty edifices such as Herr Hilbert's suddenly looked a bit shady and then anger, followed by disbelief and finally acceptance as the big hitters really got to grips with it. The world hasn't come to an end because of Goedel but it certainly got a bit more interesting.
I think that it is almost comforting that we have a system that can be complicated enough be to capable of saying things about itself that can't be proven within itself. I think that there is a good chance that our system of mathematics will eventually become complicated enough and no more. Obviously there will always be spherical cows and some absolutely mad numbers and when I say eventually - that will take forever (nearly).
My take on that statement is that it is not saying anything about the world. It only refers to itself, making it a self-contained mini-universe with no relation to the real world.
So, since it's not saying anything, it's neither true nor false. Only something confusing that feels like it should have some meaning.
Now, how do we attach meaning to the actual statement. Do the English words actually mean the same thing to me as they do to you? Probably.
Should any statement be applicable to anything - world or otherwise? It is just a statement and not my best man's speech at my brother's wedding 25 odd years ago, which I'm sure you can appreciate that I consider that being rather more important.
So here we have a statement that is unable to be consistent and as you say, it sounds like it should have meaning but doesn't.
So we have a way of saying things with spoken language that are logically inconsistent that are also grammatically correct and that is a sort of flavour of what Goedel proved with his incompleteness theorem.
> the links between maths, music and art.
Mr. H. is by reputation a very competent violinist; even though he's a mathematician, he can pronounce with some authority on subjects like Bach.
See https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... and https://en.wikipedia.org/wiki/Peano_axioms#Nonstandard_model...
Godel's "completeness" theorem gives a converse. If the statement were true about all possible models, then it would be provable.
Concerning multiple models of ZFC: I'm always confused by such statements about the foundations of set theory itself, they seem weirdly self-referential. ZFC certainly can't prove that it has multiple (or even any) models. Does such a statement need additional axioms, or is there a general theorem like "If a first order theory has any model, then it has multiple ones."?
Löwenheim-Skolem implies the existence of a countable model of ZFC. https://en.m.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skole...
"What I meant to say is that multiple models are not the only reason for something to be true but unprovable, the incompleteness theorem also holds in more general conditions." I can't give a clean rebuttal for this, but I believe this to be profoundly mistaken. It might be technically correct though, depending on how you'd formalize this statement. To formalize math you need a logic that has some properties: it should be decidable whether a proof is correct, you should be able to write it down, it should not be contradictory. If you take these together the only way a statement is unprovable, is if it is independent from the axioms, i.e. there exist multiple models.
Edit: however, consistent second order theories don't always have a model. In first order, if S is an undecideable statement in theory T, then both T+S and T+!S have models, both of which are models for T, so undecideable statements always come from multiple models. But that does not need to be the case in second order, so your claim "it is independent from the axioms, i.e. there exist multiple models" is not neccessarily true, i.e. there may be cases when decideability of some statement fails in a theory with a unique model. Or is there another argument for your claim?
Forgive me for spamming questions, I just try to understand how these things fit together. But maybe we should just stick to first order, since everything else is too weird anyway.
Without biasing for a particular model, we could simply say that every sufficiently expressive proof system admits either zero models or more than one model.
We can define a notion of complexity for any given integer as being the size of the smallest program that returns that integer. Obviously, I'm being imprecise here, but it should hopefully be clear that it is possible to get the details right and the precise nature of those details aren't going to be relevant for what follows.
Now suppose we have a program that can find the complexity of a given integer. Then, we can write the following program:
for (i = 0; ; i++) {
if (complexity(i) > 2*K) {
return i;
}
}
As a result, we have some integer that has a complexity--it's still well-defined (at least if you have the axiom of choice, but I'm not sure if that axiom is necessary)--but we can't necessarily prove that any integer has a given complexity.
However we can write a program that does a brute force search through all proofs from ZF looking for the complexity of i. As you just showed, this program will not establish the complexity of all integers.
But how does it fail? It fails because there are programs which do not halt, that it can't prove don't halt, which if they halted COULD return i.
And therefore the complexity cannot be determined from ZF.
The prevailing philosophy of mathematics says it is well-defined. But there are alternate (entirely consistent!) philosophies under which complexity is not well-defined at all.
If you are a platonist and believe in some preferred model where every statement is decided this makes perfect sense. For the rest of us, this just means that in a sufficiently complex system there will be undecided statements. Which is not such a big surprise – but a rather awesome technical exercise!
I'm somewhat on thin ice on the mathetematical formulation, but I suppose something could be undecidable but true with respect to all standard models of a theory, while not true for some non-standard model.
For instance, there may be statements acting on infinite sets (such as N) that may not be compressed into a recoursive formulation, which means that a proof of the valididity of the statement (if it is true) would require an infinite number of steps.
The term "standard model" is interesting, because it pushes the problem a bit. So, let us take the natural numbers. Sure, we can prove in set theory that some of the undecided statements of Peano arithmetic are true for the standard model. But seen from the outside this means that we have changed our axiomatic system from Peano arithmetic to set theory. But there are still undecided statements – even arithmetic ones – in this new system! These will hold in some models of set theory, while being false in some other.
So, saying “Oh, I mean the standard ℕ, not any of those other ones” does not get us off the hook, because it raises the question: “The standard ℕ... in which model of ZF?”
Kruskal's tree theorem is a more interesting example, you should look into that. You can prove it undecidable, so you can add either it or it's negation like you say, in Peano's arithmetic, but you can prove it ZFC or I think even less expressive axiomatic systems for set theory.
However, Gödel made explicit in the very first statement of his incompleteness theorem that assuming more axioms cannot ever eliminate incompleteness (except by the extremely undesirable event of eliminating consistency). Gödel shows that, if a set of axioms (including additional axioms on top of your original set) doesn't allow you to prove everything (by virtue of being inconsistent, something called the "principle of explosion"), then there are necessarily statements that those axioms can't resolve the truth of.
So you can add more axioms and that will either make your system more powerful or inconsistent (depending on whether you chose inconsistent axioms to add), but you still can't get rid of the incompleteness phenomenon that way!