What Do Gödel's Incompleteness Theorems Mean?

https://www.quantamagazine.org/what-do-godels-incompleteness-theorems-truly-mean-20260518/

by baruchel • 2 months ago

163 65 2 months ago

marojejian• 2 months ago

Interesting points in here.

e.g. that Godel didn't think this scrapped Hilbert's project totally:

Gödel believed that it was possible to redefine what we mean by a formal mathematical framework, or allow for alternative frameworks. He often discussed an infinite sequence of acceptable logical systems, each more powerful than the last. Every well-formulated mathematical question might be answerable within one of them.

lioeters2• 2 months ago

That part you quoted was interesting to me too. I remember once re-reading the incompleteness theorems - where it talks about a "finite set of axioms", it seemed there may be a loophole if we can imagine a theoretically infinite set of axioms, as a way to approach completeness.

Overall I really enjoyed this article, short interviews with mathematicians and philosophers on a topic I've often thought about.

AndrewOMartin2• 2 months ago

Some people, when confronted with a Gödel's Incompleteness Theorems, think "I know, I’ll use a theoretically infinite set of axioms." Now they have aleph-nought problems.

joe_the_user2• 2 months ago

I think the combination of Godel's completeness theorem and Godel's incompleteness theorem stakes out a position in between "truth absolutism" (everything certainly knowable etc) and "truth nihilism" (nothing is truly knowable with any certainty). Which I think is great. Thing, however, is that a lot of philosophers and mathematicians fall into one of these views of truth and so you see people constantly fighting, chaffing at the bit against, this middle ground, claiming it "satisfies nobody" etc. Well, it satisfies me quite a bit.

whattheheckheck• 2 months ago

I wonder if thats why agile is best. We can never fully "prove" the program as theory building

brookst• 2 months ago

I don’t think we’ll ever entirely know what they mean.

watershawl• 2 months ago

It hints at something fundamental to how the universe works, in that there is always an adjacent possible.

MrDrDr• 2 months ago

“incompleteness theorems” established that no formal system of mathematics — no finite set of rules, or axioms, from which everything is supposed to follow — can ever be complete.'

There is usually a 'not sufficiently complex' clause in that definition. Presburger arithmetic is complete: https://en.wikipedia.org/wiki/Presburger_arithmetic

__MatrixMan__• 2 months ago

Right, you need to be able to construct numbers for Gödel's proof to apply.

Hilbert's incidence geometry, for instance, is consistent and complete. It's just rather small.

svantana• 2 months ago

Of all the incompleteness-style theorems, I find the Halting problem to be the most approachable and also the most interesting. Maybe it's because I'm a software dev that dabbles in math rather than the other way around. But that makes me wonder if all of Gödel's theorems can be stated if 'software form', so to speak.

lacewing• 2 months ago

Right, if you're a software engineer, the realization that the two theorems are nearly-equivalent really takes the air out of a lot of the existential philosophizing around Gödel's incompleteness.

Gödel's argument basically says that any system of mathematics powerful enough to implement basic arithmetic is a computer. This shouldn't be surprising to software engineers because the equivalency between Boolean logic and arithmetic is easy to show. And if you have a computer, you can build algorithms whose outcome can't be programmatically decided by other algorithms.

cobbal• 2 months ago

I think that's selling the theorems a little short. A math system with arithmetic is equal to, or more powerful than, a computer. For an example, even classical logic comes with the law of excluded middle that can say (internally) if a program halts or not. Incompleteness applies to all the stronger systems as well.

olmo23• 2 months ago

Also check out https://en.wikipedia.org/wiki/Rice%27s_theorem

basically generalized the halting problem to arbitrary semantic properties.

tialaramex• 2 months ago

It's convenient that Henry Rice lived long before the age of language cults. I don't even think Rice wrote software, he's just a mathematician, he proved this nice property in mathematics. Stuff like FORTRAN and ALGOL happens later.

Also though, just as for the Halting Problem, we are always allowed a three-way split. Rice proves that "Has property" vs "Does not have property" can't be done, but "Has property" vs "Does not have property" vs "Shrug - I dunno, seems hard" is possible, and indeed easy if you're OK with lots of machines landing in the "Shrug" pile. You can expend as much work as you like to shrink that pile, Rice just proved it would need infinite work to empty it completely.

oulipo2• 2 months ago

Halting problem concerns decidability, not completeness

Maxatar• 2 months ago

Sure but that's fairly pedantic. You can derive Godel's first incompleteness theorems strictly as a consequence of undecidability of the halting problem.

bananaflag2• 2 months ago

The undecidability of the halting problem yields an easy proof of Gödel's "zeroth" incompleteness theorem:

Statement: Every sound (i.e. not just consistent, but sound) recursive theory of arithmetic is incomplete.

Proof: Assume it is complete. List all its theorems by a program. Then one can decide the halting problem as follows: for any instance, look whether "the program halts" or "the program does not halt" shows up in the list of theorems (since the theory is complete, one of them must show up; and since the theory is sound, the theorem is true).

jojomodding• 2 months ago

This also makes it obvious that at some point, the halting problem becomes "unprovably hard." There must be Turing Machines for which it is independent of the accepted axioms of mathematics whether or not they halt. And indeed, constructing such machines is not too difficult.

There is current research into finding the smallest such Turing Machine. Here is one with 748 states: https://www.ingo-blechschmidt.eu/assets/bachelor-thesis-unde...

cyberax• 2 months ago

The proof of the first incompleteness theorem is a very technical way of constructing the statement: "The truth of this statement is unproveable".

In the software form, it can be restated as: "there is no finite program that outputs the sequence with this property".

hybrid_study• 2 months ago

It may mean our brains are not currently equipped to understand the universe.

seanhunter• 2 months ago

As far as I can see people always radically exaggerate the effect of the incompleteness theorems. It seems interesting that any nontrivial axiomatic system has statements which are true but unprovable but to say that derails Hilbert’s project seems just obviously untrue when you can for example join math postgrad programs now which are focused on formalisation. [1] So formalisation is very much still going on, probably more so now than ever given advances in theorem provers.

Yes there are undecidable statements (eg the continuum hypothesis) but that doesn’t change the fact that the vast vast majority of statements in any axiomatic system are going to be decidable, and most undecidable statements are going to have “niche” significance like that.

[1] eg https://www.imperial.ac.uk/study/courses/postgraduate-taught...

scott_meyer• 2 months ago

Proof by contradiction requires finding a single contradiction. Practical utility just requires that contradiction be infrequent enough.

random3• 2 months ago

This is more like the popular lay take that are more or less confused about the meaning and implications of the theorems. The fact is that the implications are real yet more nuanced than this - something the article touched on.

Hilbert’s program was to reduce math to a finite set of axioms and was indeed derailed by incompleteness theorems(Gödel) and undecidability (Church, Turing).

Formalizing math with automated theorem provers has little to do with the goal of Hilbert program. Also they aren’t related to the foundational research that it entailed.

Also, as the article mentions, the implications as well as Gödel was largely misunderstood.

jolt42• 2 months ago

We really want to believe that we can understand everything, yet we know we cannot.

dehsge• 2 months ago

At the same time if you imagine a machine that can associate different maths. Would said machine encounter undecidable statements more frequently?

Would the rules of said machine have statements they themselves cannot prove by parameters set in their ‘programmed(by humans, machines, or other machines)’ assumptions?

pdonis5• 2 months ago

any nontrivial axiomatic system has statements which are true but unprovable

Although this is a common way of stating what Godel's incompleteness theorem tells us, it's actually not correct. As was posted upthread, when you combine Godel's first incompleteness theorem with Godel's completeness theorem (all this in first-order logic), you find that, for any sentence that is not provable in a system of first-order logic (such as the Godel sentence for that system), there must be a semantic model of that first-order logic in which the sentence is false. (I gave an example of such a model for the first-order axioms of arithmetic upthread.)

empath75• 2 months ago

the vast vast majority of statements in any axiomatic system are going to be decidable

This is just flatly untrue, in the strictest possible sense, and even in the generous definition of "statements that mathematicians care about", that is going to be very heavily biased towards questions that are decidable, because the questions that are likely to be decidable are the ones that they can even reason about to begin with.

If you're a computer programmer, the CS shadow of Godel comes up _all the time_ in practice, and undecidable situations are quite common and don't really need to be carefully constructed.

This is just the drunk-in-the-streetlight situation. Things seem to be decidable everywhere we look because we look where things are likely to be decidable.

LegionMammal978• 2 months ago

To be fair, in the most literal sense, the vast majority of syntactically-valid statements in a typical FOL encoding will be trivial. One of my side-projects has been trying to find the shortest statements independent of ZF and some of its fragments. In fact, for every slightly nontrivial statement that requires actually building a construction, there are billions more that can be instantly solved via a few simplifications, and statically pruning the search tree a bit is the best we can do.

To actually get independence, we need very rigid statements that don't allow any simple way to fudge a solution or counterexample. If anything, statements that mathematicians (and programmers) care about are biased toward undecidability, simply because they're extremely biased toward nontriviality. We put in the hard work of building towers of rigid definitions to that effect.

dist-epoch• 2 months ago

As far as I can see people always radically exaggerate the effect of the incompleteness theorems

Like people saying Godel theorems "prove" LLMs could never invent new mathematics because being a software system Godel applies to their operation, but not to humans which are not axiomatic systems, and thus humans can go beyond them and beyond the limits of Godel, humans can "know" a result to be true even if Godel says you can't prove it.

eimrine• 2 months ago

Is it releteble to Logic? I have heard that Economy is a subset of Logic, so is this theorem relatable to Economy?

AndrewOMartin2• 2 months ago

Only in the sense that Economists speak entirely in statements that are unprovable.

bo1024• 2 months ago

It's much easier than it seems.

* There are axioms, there are models, and there are theorems.

* A model is a particular structure with objects and relationships. The "standard model of arithmetic" is just the natural numbers 0, 1, 2, ... with normal rules of addition and subtraction and so on. A different model could be the real numbers, or one that includes infinitesimally small numbers, or so on.

* A set of axioms are rules about how a model can work. For example, we can have an axiom for any set of objects called "numbers" with an operation called "addition" that the operation must be commutative (x+y = y+x). The standard model above is one model that satisfies this axiom.

* A theorem is a fact that can be true or false about a given model. For example, "the model has infinitely many objects." If we can prove a theorem from a set of axioms, then that theorem is true for every model that satisfies the axioms. However, there can also be theorems that are true for one model that satisfies the axioms but false for a different model.

Godel's completeness theorem says that if a theorem is true for every model that satisfies a set of axioms, then one can prove that theorem from the axioms.

Godel's first incompleteness theorem says that in any axiomatic system (sufficiently complex) there are theorems that are neither always true nor always false. In other words, there is a theorem that is true for some model of the axioms but false for some other model of the axioms.

danbruc2• 2 months ago

That is interesting, I always thought that the incompleteness theorems says, there are theorems that are true or false in all models but cannot be proved to be so. But if it that is not the case and there always exist models where the theorem is true and false, that makes it sound to me, like the incompleteness theorem is not really about proving things. With that it sounds more like the inability of a sufficiently complex set of axioms to only admit isomorphic models, i.e. have all possible models agree on all expressible theorems. Makes the entire thing sound almost trivial, of course you can not prove what does not follow from the axioms.

pdonis5• 2 months ago

I always thought that the incompleteness theorems says, there are theorems that are true or false in all models but cannot be proved to be so.

As the GP points out, that's not what Godel's incompleteness theorem actually shows. Although it's a common misconception (one which unfortunately is propagated by many sources that should know better).

The key point of the incompleteness theorem is that it shows that (at least in first order logic, which is the logic in which the theorem holds) no set of axioms can ever pin down a single model. For example, no set of first-order axioms can ever pin down "the standard natural numbers" as the only model satisfying the axioms. There will always be other models that also satisfy them. So if you want to pin down a single model, you always have to go beyond just a set of first-order axioms.

Using the natural numbers as an example, consider a model that consists of two "chains" of numbers:

(0, 1, 2, 3, ....)

(..., -3a, -2a, -1a, 0a, 1a, 2a, 3a, ...)

The first chain is, of course, the standard natural numbers, but the second chain also satisfies the standard first-order axioms that we normally take to define natural numbers. So this model, as a whole, satisfies those axioms. And there is no way, within first-order logic, to say "I only want my model to include the first chain". That's what Godel's incompleteness theorem (or more precisely, his first incompleteness theorem combined with his completeness theorem) tells us.

Epa0954• 2 months ago

No.

Godel's completeness theorem can not be understood without bringing in first order logic, because it is a statement of the expressitivity of the language(relative to its semantics). Other more expressive languages, like second order logic (with its usual semantics) is not complete. Trying to explain Godel's completeness theorem without bringing in the language is a path to confusion.

And your explanation of the first incompleteness theorem is also at best confusing. I must preface this with the comment that your definition of a 'theorem' matches what is usually called a sentence or a statement, and a theorem is usually reserved for a sentence which is proven by a axiomatic system. If the axiomatic system is sound, all theorems will be true in all models. The question of completeness is whether or not all truths(aka sentences true in all models) can be proven(aka they are theorems). With this more common usage of the words, Gödel's incompleteness theorems show that every consistent theory containing the natural numbers has true statements on natural numbers that are not theorems of the theory (that is they cannot be proved inside the theory).

Your description of the first incompleteness theorem is also true for complete logics, even for propositional logic (with your definition of 'theorem' as actually meaning statement). It has statements which is true in some models and false in others. This does not make it incomplete.

joe_the_user2• 2 months ago

Actually, I think your statement muddies the waters a bit and the parent gives a clearer picture for those looking for a simple statement of what's going on. The background is the fellow comment: https://news.ycombinator.com/item?id=48224739. Godel's simplest (and roughly original) statement is any system of axioms strong enough to encode arithmetic is either consistent or complete. You can "Or the set of axioms is not enumerable" (as in Second Order Logic and other systems). But when one says that, one has jumped from what would be normally recognized as logic (finite axioms and process) to a mathematical construct with some similarities to naive logic but which "my gran pappy" would not see as logic.

I mean, you can formally construct an axiom system defined to include (via axiom of choice) and assign a truth value to each of the independent propositions of first order arithmetic logic. There, you have consistent and complete system but not one that's a whit closer to being in usable by anyone.

I'm not even a finitist but I think being clear what's going on with these claims is important. It's like saying "the halting problem can't be solved by finite computers but my infinite-foo hypothetical computer can solve it, gotten mention that..."

kerwioru9238492• 2 months ago

My understanding is that for any system of axioms strong enough to encode arithmetic, you can have at most two of these three properties:

1. Complete (for any well formed statement, the axioms can be used to prove either it or its negation)

2. Consistent (can't arrive at contradictory statements ~ arriving at a both a statement and its negation )

3. The set of axioms is enumerable ~ you can write a program that lists them in a defined order (since the workaround for completeness can be just adding an axiom for the cases that are unproven in your original set)

If my understanding is correct, I believe your explanation is missing the third required property.

It's also important to point out that if we cant prove a statement or its negation (one of which must be true) then we know there are true statements that are unprovable. This is a much stronger of a finding than "Godel's first incompleteness theorem says that in any axiomatic system (sufficiently complex) there are theorems that are neither always true nor always false. "

danbruc2• 2 months ago

It's also important to point out that if we cant prove a statement or its negation (one of which must be true) [...]

Is that true, could it not be neither, i.e. independent of the axioms? Or is this assuming completeness which rules out independent statements?

Epa0954• 2 months ago

I just want to be a bit pedantic here (but this is logic after all...), and point out that in point 1 above you are talking about syntactical completeness, and not semantical completeness, which is the kind of completes Gödel proves in his first completeness proof. I think people are often confused because of this overloading of the word. And it is about sentences(a formula withouth free variables), not any well formed formula.

alan-crowe• 2 months ago

As a child, I noticed that the proofs of mathematical theorems were esoteric knowledge, known only to a few adults. I struggled to follow even the simplest proofs, and hoped that one day I might learn to create a proof or two of my own. This was not only a high aspiration, but a dangerous one. I saw no reason why certain knowledge of a true fact would be accessible to humans via proof. Any-one who embarked on the quest to find a proof risked embarking on a doomed quest to find a non-existent proof.

For me Gödel's completeness theorem is the miracle. Every valid statement has a proof. Amazing!

Aim a little higher, every true statement, and there might not be a proof. It is no surprise to me that this is true. It is a big surprise to me that Gödel was able to prove it; ordinary proofs are hard to find, and proofs of the limits of provability presumably even more deeply hidden.

Non-standard models of arithmetic are weird. Theorems that are true of the standard model of arithmetic and false in some non-standard model must surely be convoluted and obscure. The first order version of the Peano axioms nail down the integers, not perfectly, but very well. Restricting one-self to theorems that are true in all models of them, even the weird, non-standard ones, feels like a very minor restriction. Gödel's completeness theorem raises the possibility of writing a computer program to find a proof of every theorem that isn't convoluted and obscure. Gödel completeness theorem is the really big deal.

Except it isn't. That computer program turns out to be one of those wretched tree search ones that soon bogs down. The real problem turns out to be the combinatorial explosion inherent in unstructured search through the Herbrand universe. One needs Unification and one needs a still missing ingredient to give search a sense of direction. The interesting questions are about the "sense of direction" that lets us find some of the deeply hidden proofs that do exist. Will LLM's help? The answer will be interesting, either way.

bananaflag2• 2 months ago

Gödel completeness theorem is the really big deal. Except it isn't. That computer program turns out to be one of those wretched tree search ones that soon bogs down. The real problem turns out to be the combinatorial explosion inherent in unstructured search through the Herbrand universe.

Yup. Incompleteness is sort of a red herring. P≠NP (even though unproven) yields the real, practical, painful incompleteness.

Dylan16807• 2 months ago

P versus NP could be a red herring too.

If P=NP, but the best asymptotic solution is n^7, and it has so much overhead that the best practical solution is n^9, then it doesn't really matter that it isn't exponential. It's still unsolvable for easily accessible problem sizes.

kkylin• 2 months ago

I really do think the incompleteness theorems deserve the attention they get, not just because of what they say about efforts to formalize mathematics and because of the historical context -- remember Gödel numbers came (just) before Turing and the first recognizably modern electronic computers. That numbers can represent things that are not numbers was (IMO) a revolutionary idea.

Having said all that, I'd taken mathematical logic in college to learn about incompletenss, but the most interesting things I got out of it were completeness and compactness. Non-standard models really can be quite interesting.

coldtea2• 2 months ago

That numbers can represent things that are not numbers was (IMO) a revolutionary idea.

When it first appeared, sure. But it wasn't with Godel and Turing that this happened.

renticulous• 2 months ago

Universe is the Godel sentence in the limit of all possible mathematics. Hence it exists and is True.

Fix axioms, we get a Godel sentence G1. Add G1 to axioms, we get G2. So on and so forth. In the limit of this process, we have our Universe.

AxEy• 2 months ago

The real problem turns out to be the combinatorial explosion inherent in unstructured search through the Herbrand universe. One needs Unification and one needs a still missing ingredient to give search a sense of direction.

Shameless plug: I put together a Jupyter notebook walking through the use of Herbrand Universes for a semi-decision procedure for first order logic: https://github.com/aetilley/harrison-rust/blob/main/Herbrand...

alexwwang2• 2 months ago

To put it simple, my understanding to this theorem is that we can not design and implement a program without any bugs.

So it indicates me to think about the boundaries while I design and build a system. These boundaries will protect us from those unsolvable constraints.

coldtea2• 2 months ago

that we can not design and implement a program without any bugs.

If we don't include the compiler and runtime, sure we can.

Godel's theorem is about formal axiomatic mathematical systems with certain properties like the ability to do basic arithmetic.

retrocryptid• 2 months ago

      Natalie mentions  the Newman &  Nagel's text "Gödel's  Proof," a
      (//the//?) 1958 classic on the subject. [[ 1 ]]  Having left IBM
      in December  1990, I spent a  month with the text,  dipping into
      mild insanity, taking to strange  wines to relieve myself of the
      fear that my previous years  long study of Whitehead & Russell's
      "Principia Mathematica" [[ 2 ]] was useless.
   
      I  really  appreciate  the  inclusion of  Alvir's  statement  on
      whether  or not  Gödel  thought he  proved  all logical  systems
      undecidable and incomplete.   About 80% into the  article is her
      quote:
   
      >> Often people will speak as if  the CH is the smoking gun that
      >> shows sometimes  mathematical questions have no  answer.  But
      >> in my  opinion, this situation provides  very little evidence
      >> that   there   are  “absolutely   undecidable”   mathematical
      >> problems, relative to any given permissible framework.
   
      Though  I  would have  added  a  reference to  Infinitary  Logic
      [[ 3 ]]  after dropping  the  reference  to L-omega-1-omega.   I
      suspect most  readers would find discussion  of higher-order and
      modern logic a bit confusing  without a pause for further study.
      But a guide post pointing  in the appropriate direction would be
      good.

      That this is  the only critique I have of  the article speaks to
      Wolchover's  skill  in communicating  complex  ideas  for a  lay
      audience.  I really  liked this article, so  thank you @baruchel
      for posting the reference to it.
   
   :: References
   
      1. https://search.worldcat.org/title/1543160023
   
      2. https://search.worldcat.org/title/933122838
   
      3. https://en.wikipedia.org/wiki/Infinitary_logic

hughw• 2 months ago

From E.T. Jaynes' Probability Theory [1] :

  To understand the above Gödel result, the essential point is
  the principle of elementary logic that a contradiction
  Ā A implies all propositions, true and false. 
  (Given any two propositions A and B, we have A ⇒ (A+B),
  therefore  Ā A ⇒  Ā (A+B) =  Ā A +  Ā B ⇒ B.) 
  Then let A = {A1, A2, ..., An,} be the system of axioms
  underlying a mathematical theory and T any proposition,
  or theorem, deducible from them:
  
  A ⇒ T.

  Now, whatever T may assert, the fact that T can be deduced
  from the axioms cannot prove that there is no contradiction
  in them, since, if there were a contradiction, T could
  certainly be deduced from them!
  
  This is the essence of the Gödel theorem, as it pertains to
  our problems. As noted by Fisher(1956), it shows us the
  intuitive reason why Gödel's result is true. We do not
  suppose that any logician would accept Fisher's simple
  argument as a proof of the full Gödel theorem; yet for most
  of us it is more convincing than Gödel's long and
  complicated proof.

[1] https://bayes.wustl.edu/etj/prob/book.pdf

remywang• 2 months ago

The best way to understand the theorems is to try to understand the proofs, and the short book “Gödel’s proof” by Nagel and Newman is excellent for that. Just like Douglas Hofstadter wrote in the foreword, I found the book an absolute page turner and finished in one afternoon.

kazinator• 2 months ago

Gödel's Incompleteness means that formal systems that are sufficiently expressive can be used to write down new kinds of statements that can neither be derived nor contradicted from their existing axioms. Those statements can be adapted as new axioms.

Unfortunately, Gödel's proof method per se only shows an example that is not so meaningful, involving self-reference. He builds a number-theoretical formal system which can talk about its own formulas, by encoding them as integers (something we do with computers now as a daily matter: all computer program text and other data is arithmetically encoded into binary, which has an interpretation as a number). In the context of Gödel's work, we call this arithmetic encoding "Gödel numbering".

Whether a proposition is true is reformulated as a number-theoretical property: instead of asking, is this proposition or equation true, we ask, is the arithmetic encoding of this proposition an integer which belongs to the set of theorem-integers; is it a theorem-number?

Within this framework, Gödel shows that a proposition can be made which says "G is not a theorem-number", such that this very propostion's own Gödel number is G! In other words, a kind of Quining is going on, whereby the proposition embeds a coded reference to itself. Essentially, Gödel introduces the idea that we can make a statement which says "I am unprovable", in formal, rigorous way. If that statement can be derived from the axioms, then a contradiction results: it was derived, yet it asserts the falsehood that it is not derivable, and so a falsehood was derived from the system's axioms. If it is true, then it points to incompleteness: there is a truth that can be expressed in the syntax of the system, yet cannot be derived.

Thus if we have any system expressive enough to encode the "G is unprovable" statement where that statement itself is G, that system is either inconsistent (allows a falsehood to be derived) or incomplete (allows true statements to be written which cannot be derived).

pdpi• 2 months ago

formal systems that are sufficiently expressive

One of the more interesting bits about this is understanding what "sufficiently expressive" means. The Naturals are incomplete, the Reals aren't.

What Do Gödel's Incompleteness Theorems Mean?

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