Gödel's Incompleteness (extra footage 1) - Numberphile

MAIN VIDEO: • Gödel's Incompleteness...
More links & stuff in full description below ↓↓↓
Extra footage part 2: • Gödel's Incompleteness...
Professor Marcus du Sautoy is Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford.
Professor du Sautoy's book as mentioned...
In the US it is called The Great Unknown - amzn.to/2sfkWpb
In the UK it is called What We Cannot Know - amzn.to/2r5yztp
More of his books: amzn.to/2ryEk4r
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science.
NUMBERPHILE
Website: www.numberphile.com/
Numberphile on Facebook: / numberphile
Numberphile tweets: / numberphile
Subscribe: bit.ly/Numberphile_Sub
Videos by Brady Haran
Patreon: / numberphile
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanblog.com/
Sign up for (occasional) emails: eepurl.com/YdjL9

[@MultiSkidding]

guess you could say the original video was. incomplete

[@mrping2603]

guess you could say this comment was. funny

[@hasch5756]

This is also true of the current one, which is why he made a second one right after that, but that cannot cut it either, so he'll just keep uploading infinitely many videos without ever being able to adequately explain the topic. Sad.

[@Bloodsaberxy]

your statement is provably funny.

[@bp56789]

*puts on sunglasses* YYYYYYYYEAAAAAAAAAAAAAAHHHHHHHHHHH

[@thesealab8947]

bahahahahahaha

[@MMrandomdude12]

This guy is really good at explaining stuff.

[@cycklist]

Wolfgang Ambrus His books are excellent.

[@gknucklez]

I came out of it with more questions lol For example, why does a statement, that can't be proven/disproven by the axioms, become an axiom? That's not very logical, or I didn't understand him right

[@vatvslpr]

G Knucklez An unprovable statement doesn't automatically become an axiom, but you can add it (or its contradiction) to the system as a new axiom if you want to. This is because neither the unprovable statement nor its contradiction can create a new contradiction in the system; if they could, you could use that in a proof by contradiction. So an unprovable statement lets you create multiple new systems which assume different truth values for the unprovable statement. The classic example of this is the parallel postulate in geometry. People tried for millennia to prove it from Euclid's other postulates, but they failed because it's an unprovable statement. Instead, you can assume either that the parallel postulate is true, in which case you get plane geometry, or that it's not true, which (depending on how it's not true) give you either elliptical or hyperbolic geometry.

[@thewiseturtle]

Which is a big reason why he's the Professor for the Public Understanding of Science at Oxford.

[@lesselp]

You haven't understood him.

[@twistedsim]

He died because he couldn't prove that the food was poison or not....

[@markorezic3131]

And his theorem led him to believe that then it must be true

[@Chriib]

Hah! exactly my thought.

[@martijnvanweele6204]

If you can't prove it's not poison, you should assume it is.

[@Madoc_EU]

So in a way, his dying proves that the food was poisoned in a way, as because of it, he died. Even if the food isn't actually poisoned.

[@morgengabe1]

Too soon

[@OlafDoschke]

That last bit got me as astonishingly self-referential. The fear of death by poison causing death by starvation. Kind of feels like a made up legend.

[@z-beeblebrox]

It would've been more self-referential if it was believed much later that someone actually was trying to poison him. A true statement that can't be proved

[@OlafDoschke]

Dying by the fear of dying is pretty self-referential, negative feedback loop, perhaps, but how much more self-referential can it get? I can even imagine the fear of getting poisoned being stronger than the fear of starving, up to the point where you're too weak to eat, even if the starvation fear would have become stronger. Being poisoned would make it a paranoia becoming reality, I wouldn't call that self-referential at all.

[@z-beeblebrox]

I mean self-referential to his work, not to itself

[@bmoney1860]

@@z-beeblebrox It's still death by mental illness. Even if you were certain that someone was trying to poison you, you could still find a way to eat. Just go to a grocery store and buy some food. Problem solved. Go to a different grocery store every time. Throw in a few restaurants as well.

[@z-beeblebrox]

@@bmoney1860 I don't know why you're trying to refute a comment I wrote 3 years ago, but I'm pretty confident none of what you said had anything to do with my observational joke about self-referencing

[@OlafDoschke]

The german word you didn't remember at 10:02 was "wissen": "Wir müssen wissen, wir werden wissen."

[@hexagon5610]

I'm from German too. Right! :)

[@unvergebeneid]

Speaking of infinity and Gödelization, it is also noteworthy that every mathematical statement will map to a natural number and therefore the entirety of mathematics is countable. So whenever one encounters an uncountable set, mathematics can't describe every individual member, only the set itself.

[@MisterUnlikely]

I don't know. It strikes me that it should be possible to diagonalize (a la Cantor) and show that the number of mathematical statements is uncountable, and that as a result, the assertion of this mapping has some hidden flaw. Part of the problem, I think, might come from the two-value (true/false) logic systems, or just from a lack of rigor in the creation of language (or both).

[@unvergebeneid]

Well, the statement that all of mathematics is countable is tied to the Church-Turing thesis, which, informally, says that everything a human can compute, a computer can also compute. Since every computer program is just a very long number in that computer's memory, the number of possible computer programs is definitely countable. OTOH, the number of thought a human brain might ever produce is at its very least bound by the number of configurations in the volume of that brain, which is finite. And even if you'd allow for an infinite brain (as idealized computer memory is also infinite, so that's fair enough), you'd still have a countable number of states AFAICS. Neither computers nor brains are bound by binary logic, although I'm not sure what you mean by "lack of rigor in the creation of language."

[@OnTheThirdDay]

Well, saying that all mathematics is countable is just the result of it being a language, since what is meant by mathematics is the valid sentences made in the math we are doing. There are only finitely many symbols able to be used and every sentence has finitely many symbols. One can also say that the English (or any other language) is countable. I do think that the countableness only refers to the formal symbols and stuff, i.e. on the logic/axiom level, since it is clear that all of math is not countable since there are things with uncountably many elements and they are understandable. (I.e. the Cantor set) It just turns out that all the things that we can describe turn out to be countable, because describing them uses finitely many symbols. E.g. there are uncountably many real numbers, but of the real numbers that we can describe in a mathematical sentence, there are countably many because in describing them we are using a language with finitely many letters and sentences of finite length. E.g. 1 is not equal to 2. 1 is not equal to pi. 1 is not equal to e. ... is a way of talking about all the numbers that we can talk about, so in no way can we talk about all real numbers because of the limitation of language. When you invoke an axiom that brings infinity into the picture, like the power-set axiom applied to the axiom of infinity, then you can get uncountably many things and you can say things like: For all x in the real numbers, x^2 >= 0. There are uncountably many real numbers, but we are not counting real numbers, we are counting the sentences.

[@ch4r1z4u0153]

I haven't done any formal logic but I think part of the issue with trying to diagonalise is that statements must necessarily be made of up a finite number of symbols - if you tried a diagonalisable argument, you'd create a statement with a symbol for every natural number. It's the same reason why you can't use Cantor's diagonal argument on the natural numbers (like you do on the real numbers, but right-to-left)

[@MisterUnlikely]

And admittedly, with "words" (at least in the sense we normally think of them), yes, you could even give every letter an ASCII value, just concatenate them all, and come up with a single, unique numeric value. Mind you, statements about mathematics *do* include all irrationals, so that would mean that you have strings of *infinite* length. For example: "The ratio of a circle's circumference to its diameter is 3.1415926535...." would go on forever. Computers only have finite precision and memory; statements do not. We do know that all possible strings of infinite length are, in fact, uncountable.

[@sevrjukov]

Remarkable series on Gödel! Thank you again, Numberphile, for crunching very hard math topics and making them accessible to regular people.

[@PersimmonHurmo]

Godel seemed like a maverick, proving paradoxes and shaking the very fundamentals. Even his death was extraordinary!

[@yorkeR177]

Thanks for uploading extra footage :)

[@combodemo]

Kind of surprised the halting problem wasn't mentioned when he talked about going to other fields to see if they had acknowledged "limitations on what they could possibly know." That's essentially what the halting problem amounts to in terms of computation theory(I don't want to stretch too far and say CS) and is something any intro to CS course would at least mention, I think, and probably the easiest example of it in another field as an example of a fundamentally unanswerable question.

[@elliott8175]

Prof. Sautoy's responses are invigorating, but may I say that the questioner asked really intuitive questions. 'Great interview!!!

[@patrickwienhoft7987]

I'd love a full video of the Axiom of Choice with him!

[@tolek108]

Thank you for this great presentation! Hats off! I'm very thankful!

[@Bladavia]

I really like that last statement about Gödel's death. Reminds me of Nietzsche who ended up completely psychotic, John Nash, etc. There's definitely a fine line between madness and genius.

[@zorrozalai]

Nice video. I have a feeling, that it's a little bit incomplete, that something is missing, but I can't prove it.

[@s4archie]

The question about what happens if your theorem is undecidable, or how will you know has already been covered to some extent. Euclid's 5th Postulate and the Continuum Hypothesis are both formally undecidable within the mathematical system. It has been proved in each case that they are independent of the remainder of the axiom system. These undecidable propositions then give us options in terms of how we progress (as alluded to in the video). In the case of the 5th Postulate we proceed in one of Euclidean geometry, spherical geometry or hyperbolic geometry. I'm not aware of any work having been done based on different options related to the continuum hypothesis, but there are surely choices that can be made and there must be consequences of those choices.

[@adlsfreund]

Gödel starved himself to death? Wow. What a story, Marc!

[@angelmendez-rivera351]

This is all very interesting. One idea I do want to present is the fact that because no set of propositions can prove itself consistent due to incompleteness, it follows that whichever set of propositions Gödel used to deduce and conclude the Gödel Incompleteness Theorem, such set is by his own Theorem unprovable, implying that the Theorem itself is unprovable. Therefore, the very truthiness of the Theorem renders the Theorem as not decidably true since it cannot be proven, and this creates another infinite loop/contradiction.

[@rabidcentrist]

"...pull ourselves outside a system". This right here is an absolutely crucial part of understanding emergent behaviour. As soon as we are cognizant of a structure, we go about figuring out the shape of that system. As soon as we understand the shape of a system, we can envisage the exterior of that shape/structure.

[@Gunbudder]

stop posting these in the morning brady! i have to go to work!

[@htmlguy88]

by the same logic so does he ...

[@Cellkist]

Gunhaver you can watch anything later

[@cycklist]

Gunhaver It's not morning everywhere. I bet you're American.

[@cpawel]

PompeyDB, how very presumptuous of you. If you had done any digging regarding the information of the time of when this video came out in correspondence to the implicit time given by the comment, you would have realized that the person is, in fact, not American. The American East Coast would have had this video available during the middle of the night - around 4 AM by their standards. I am doubtful that anyone would call that the morning, rather than the middle of the night. No, judging by the comment, I would say the person is located in Europe.

[@cpawel]

Hmm, then my apologies to PompeyDB. It is rather unusual for someone to call 4 AM "morning" rather than the middle of the night.

[@sakshamsingh4378]

One of the best video

[@sebastiaanvantil8933]

How can we know if the coding that Gödel came up with is correct? I mean, in the original footage we saw that the number 3 was assigned to the Then statement, but how do we know that it should be 3 and not any other number? If it was any other number, then the outcome of any number assigned to an axiom would be different. Would the system still work if every number assigned to axioms was different? I have so many questions

[@TiagoTiagoT]

Could this be a problem with our languages and how we define the operators and such?

[@rentzepopoulos]

I have not read Gödel's work and probably I am not in position to do so, but while viewing this video, a question came to my mind: Could Gödle's coding actually be introducing incompletness? I mean, could the outcome of his work is exactly the result of some characteristics of this coding? On the other hand, if this is the case, then the inability of mathematics to describe itself may be a proof of its incompleteness in the first place...

[@paultikotin]

Panagiotis Rentzepopoulos It is conceivable that the terms in which Gödel couched his proof necessitated its truth. That is always a potential problem with a proof - it could be circular. I think it's fair to say that this proof has been scrutinised carefully. No one (who has mastered the subject) doubts its truth. It is ironic that this subject was triggered by an attempt to solve that very problem. The hope was that we could reduce all proofs to a series of simple manipulations of the axioms that would guarantee correctness if rigorously followed. Gödel proved that this could not be done.

[@pallavlearn5348]

Can we have some video about - sets, classes and collections as in set theory including the associated paradoxes...

[@chizhang2765]

OK, if the value assigned to a correct, provable statement is the product of those related to all the proven statements used to achieve that one particular proof, what if I can find multiple deduction pathways to the said statement? Does that mean that particular statement can take multiple values or do we arbitrarily select the smallest number to encode that statement? Also, if some statements are used more than once in that particular proof, do we multiply the code number of those statements twice or do we just count them once? I know I have also asked (quite impertinently I regret) a list of questions in the other video, but please do excuse my burning desire to get to the bottom of this! (Really this theorem is so mind-blowing I can't even speak right now!)

[@Cindsardella]

Great epsode on Gödel. I also love the GOT T-Shirt. :-))

[@mechtheist]

If there are two mathematical statements that are fundamentally equivalent, is there some relationship between the numbers Gödel's mechanism produces?

[@Pr1est0fDoom]

The link to part 2 is missing from the description of this video.

[@xizar0rg]

So... does godel theorem imply anything about the magnitudes of the sets of provable and unprovable statements?

[@topilinkala1594]

Best thing IMHO about Continuum Hypothesis is that both parts of derived mathematics eg. infinities galore or organised have shown to be usable by scientists. So it's still usefull mathematics.

[@code-dredd]

BTW, *what* was the logical inconsistency that was said to be in the US Constitution, and was a solution to it ever mentioned/discussed?

[@michaeloneill9020]

Could you prove a math theorem by reference to experience or by experiment?

[@antoniozumpano826]

How we know if the system that Godel prove incompleteness is consistent?

[@garyknight8966]

There are several answers depending how I parse this question. Goedel's are called Incompleteness theorems a little misleadingly, since they don't establish that there are truths that cannot be expressed in terms of consistent logic, only that they cannot all be proven in those terms. If you are asking 'How we know the system that Goedel proves incomplete is consistent' you seem to beg your own answer since Goedel proved that a consistent system cannot be proved consistent within the system, and proved that by stepping outside the system by coding it. However, if you ask 'How we know the [coding] system is consistent in which Godel proved incompleteness [of the lower, coded system]' it is a fun question, because if this higher (coding) system is consistent we can't know it by proof. However, not being certain of the consistency of this system does not negate any proofs that are possible in the system, any more than attainable proofs in the first system are undermined by not knowing it is irrefutably consistent. It is always a Goedel sentence that undermines any effort to prove that the system in which it is expressed is consistent, because it has an undecidable truth. What Goedel proved was that any consistent system is limited (incomplete) in the sense that it will contain unprovable truths: i.e. will have postulates with lines of proof and disproof that cannot be shown to be false in that system. Goedel sentences (truths) in that consistent system will appear to be as likely true as false. In the system's terms, the conjecture (eg. Goldbach's conjecture) and its contrary will for ever appear to be plausibly true and plausibly false. This means the conjecture that (eg. Goldbach) is "true and false" has a certain permanence - even here where consistency entails the law of non-contradiction. It is the self-consistency of a logical system that forces it to be unable to prove every well-formed-formula (expression) in that system as plausible, or likely true. Like the uncertainty principle, there's a great 'cost' to having the 'certainty' of consistency. This leads to the inescapable conclusion that to have confidence at being able to assert the truth of everything conceivably true requires not consistent logic, but some form of modal logic which allows intermediate truth values "yes and no" (reminiscent of entangled states in physics) -- such as we see in human language all the time. For man to have come to such an uber basis of reason is already proof against natural selection (a system presumed to operate on consistent survival rules), and it's the reason that Goedel was able to posit his theorems in the first place. However, it's not the reason he was able to prove them once he'd divined them, as all he had to do was adopt the economy of stepping out of the arithmetical axiomatic scope to its meta-level of coding arithmetical axioms: a level that is itself also pursued using consistent logic. That economy was wise, since the only way to convince [non-contradiction]-bound mathematicians of its truth was to do so in a consistent system where Goedel theorems do not happen to be Goedel sentences (i.e. unprovable there). That system of coding will have other truths that are not provable in that system. Now, a most interesting sentence would be one that is a Goedel-sentence at every meta-level of 'coding' or 'representation.' Possibly Anselm's declaration of "that than which nothing greater can be conceived" is just that. Turing found that within a modal multivalued logic of his devising, this sentence was sound (therefore true, if the 'atomic truths' or axioms are true). It convinced him to be a theist (well, at least a deist).

[@antoniozumpano826]

@@garyknight8966 I do not see any reason for such a long explanation. Things are so much more simple. Let's see arithmetics for instance: a) There are sentence that are not true and also are not false. b) there are so, three kinds of sentence - true; false; not true and not false. Abou consistence, we don't know - if some contradiction appears we do not use this sistem any more. Definition: a sentence is true if it can be proved. Definition: a sentence is false when its negation is true. Definition: we always assume that the system is consistent and if a negation leads to a contradiction so the sentence is true by definition. Also, there is no such thing as conjecture in mathematics, because you can not ask: is this affirmation true? In tha question you suppose that or it is true or it is false. But we agreed that there is a third option, that is, it may be not true and not false. This third option is fantastic because it permits to construct two mathematics from that duality. This happened with the fifth axiom in plane geometry and always happens with the axiom of choice. (You see that there is no sense to say that a sentence is true but can not be proved.) I think that Godel's theorems are super estimated.

[@antoniozumpano826]

@@garyknight8966 Also, it is obvious that a sentence is true if and only if the axioms are true. Indeed, we are not allowed to ask if an axiom is true or not, since an axiom is BY DEFINITION) true. All rational thoughts are based in that situation, veracities are something subjunctive, that is, they depend on veracity os the axioms, or the creeds.

[@Rockyzach88]

Their little animation in the beginning reminds me of the shoot off highway of Langton's ant.

[@HanYang2023]

great video! nice rolex btw!

[@diegovasquez840]

1:00 - A corollary of Turing’s solution to the Entscheidungsproblem (literally: decision problem) says that this question is undecideable, which I suppose is just another way of the universe giving mathematicians the middle finger.

[@ucantSQ]

There's a wonderful book that elaborates on this, The Eternal Golden Braid, by Douglas Hofstadter. It touches upon music, computer science, the visual arts, and formal mathematics. Highly recommended, if you find this video interesting

[@steffen5121]

Help me out a bit. Does this also has consequences on the veracity of statements like 1+1=2?

[@jamescorr8600]

In the previous video he mentioned it is possible to prove a theorem is undecideable, and therefore true, since no false statements are undecideable. Is it possible we can prove that every Gödel problem can be solved in this way? That would sort of sodestep the entire issue

[@htmlguy88]

but if you take the factorization approach instead of the additive approach you can show using mathematics that there will come a point where all theorems will rely on theorems past a certain point. any composite above n^2 has to have a divisor greater than n for example.

[@htmlguy88]

in fact any y-almost prime has to have at least one prime divisors above the yth root of n otherwise the product is less than n.

[@guillermocasanovaaguilar8180]

I am not an expert, so I apologize for the question, but the following question came to my mind. If given a certain number of axioms, there is a theorem that cannot be proved, but if we add other axioms, the theorem can eventually be proved, Is there a theorem that cannot be proved even by adding infinitely many axioms?

[@kodyamour1686]

Question: To say that axioms and statements can be coded as these primes and composites (simply by listing them) presupposes that the set of all statements is countable. How do we know we can do this?

[@alexanderf8451]

Because the set of symbols used in any given system of mathematical logic is finite and they can only be grouped in to sentences with finitely many symbols. That restricts the number of such statements to be countable.

[@TAHeap]

I don't see that circular time / the grandfather paradox is really a problem: • firstly, it all rather depends on your radius of curvature - it the loop's sufficiently long then there will never be any consequence for a locally-experienced universe • secondly, the range of things that can possibly happen in a curcular time universe may be constrained to resemble something like a collection of fixed-points, but if the complexity on view is sufficiently huge then you would never be able to notice the constraints anyway. (There's also possibly something about stability in there too...)

[@francescoangius4787]

I'd like to ask some questions (sorry for my bad english): Gödel assigns a number to every mathematical sentence. This number is a product of prime numbers which are assigned to the axioms used to demonstrate the sentence. Does Gödel consider the chance that a true mathematical sentence has more than a proof? In that case, shouldn't a sentence have more than a number as a "surname"? We could use the product of those numbers as the new surname, but are we sure that it doesn't create some homonymy cases? We could chose to use both surnames with a comma between them (like 14 , 15 which is a sentence demonstrable thanks to the "axioms" 2 and 7 or thanks to the "axioms" 3 and 5) but this logic takes me to ask an another question. Is possible that two axioms could be used to demonstrare two different sentences? In that case, what should we do? p.s. while writing I realised that we could just put a digit at the end of the number that are assigned to sentences that have the same demonstration(s) of other ones: using the example I used above of [14 , 15], if it's possible that two sentences are both demonstrable with 2 and 7 or with 3 and 5, we could call them [14, 15 - 1] and [14 , 15 - 2]. I decided to post this comment anyway for everyone that has the same doubts I had while watching the original video. If anybody wants to correct me or wants to add something else, he's welcome. Have a nice day

[@brauggithebold7956]

Unfortunately, Mr Du Sautoy enormously misrepresents Gödel coding in this series, most likely for the purpose of simplifying it for a wider audience. Now there is no definitive version of Gödel coding. But I have never heard of one that works like how he described it and I'm fairly sure this would not work. Here's how it's usually done: - The Gödel number of a sentence is determined by its logical syntax, not its proof E.g. The law of commutativity would look like this: "For all x For all y + x y = + y x" Now you give every symbol Z a number g(Z) (It ultimately does not matter which symbol gets which number, just that different symbols get different numbers), and multiply accross them as powers of prime numbers like this: "g(law of commutativity) = (2^g(For all) * 3^g(x) * 5^g(For all) * 7^g(y) * 11^g(+) * 13^g(x) * ... etc" (I hope you get the picture). Note that the prime number merely identifies at which position a symbol appears - Now you can do the same for proofs. A proof can be seen as a chain of rules of inference. For example you have sentences P and Q, then "P, P => Q, therefore by modus ponens Q" would be a correctly used rule of inference. So you can code rules of inference as a gödel number as well. "g(P, P => Q, therefore by modus ponens Q) = 2^g(P) * 3^(P=>Q) * 5^g(modus ponens) * 7^g(Q) - Now as I said, a proof is just a chain of correctly applied rules of inference, so you can Gödel code it as well using the same scheme as before. - Every proveable sentence automatically has infinitely many proves. Hope that helped.

[@andrashorvath5923]

My question is the following if you could kindly answer it: If we use the consequences of an undecidable theorem and assume that we could prove any of those with the help of other part of the system, then would it mean that the root theorem, the undecidable one, was true?

[@paulh.9526]

I'm sorry, I can't understand your question

[@Jodabomb24]

Forcing is crazy; spent half of the last semester trying to really understand it and I'm still not convinced I did.

[@ultome9607]

So the axioms of logic are outside of all axiomatics? Because if you add the proof by contradiction to the axions then the sentence results in a contradiction, a neither true nor false statement?

[@oskarrecon8151]

what was the "inconsistency " in the constitution that he had found?doyouknow?

[@hqs9585]

Using prime numbers as your basis for the Godel coding system could be restrictive, what if one uses some other system (maybe not yet discovered)? How do we know that this basis are extensive or universal enough to make "universal" statements

[@borg972]

what does it mean to "work from outside the system"? why can't you put it in the system as well?

[@souleater9189]

how do we know that each prime-encoded statement (or proof?) is uniquely expressed? i gather it's akin to the way a prime factorization is unique to a number, but can we not have a sequence that has all the same counts of logical operations and procedures, just in a different order to tackle a different problem?

[@paultikotin]

souleater9189 Yes. So we also have to numerically code for the position of a proposition so we can tell the difference between A implies B and B implies A. This is done as a matter of routine in computer programs.

[@yousify]

He said in the beginning: the theorem implies that there will be an infinite number of undecidable sentences. If this is true, then Godel would have the right to prove that halting problem is undecidable, right? I do not understand what is useful of Turing's result if we have undecidable sentences by Godel. Could someone explain this to me.

[@alexandertownsend3291]

Are these prime number assigned arbitraririly or how are they chosen? Are there tables or equations for these?

[@wafikiri_]

There are several categories. Certain prime numbers (you may choose the lowest) to code the alphabet used for statements, i.e., 0-9, +, -, =, logical operators, etc. Parts or the whole of a statement then can be asssigned unique numbers for reference in the same or in other statements or lists thereof. Then ordered lists of statements (each list supposedly [dis]proving a theorem) can be assigned unique numbers.... It is a really smart indexing system.

[@jjkthebest]

"There is no ignorance, there is knowledge" Turns out the jedi code is wrong after all.

[@MushroomManToad]

The parts of Mathematics you never knew you wanted to know: Numberphile

[@IDK_OR_DO_I]

It's challenging to count how many times he said "challenge" is this video.

[@dnickaroo3574]

Godel's Theorem is more about the Incompleteness that results when using the Axiomatic Method: he proved there MUST be Theorems which are true but that they cannot be proven to be true using a given set of axioms. Godel did not believe that this was the final word -- humans may agree to loosen the rules of inference and accept the result as a reasonable proof (here he was drawing on Plato's approach to Mathematics). Goodstein's Theorem illustrates what Godel was talking about. The statement of the Theorem only involves the Integers with Addition and Multiplication. Peano's Axioms gives us this part of mathematics. Goodstein's Theorem was proven about 1944; however it was subsequently shown that it could not be proven using only the Peano Axioms (which does seem to be surprising). To prove Goodstein's Theorem one needs to use Transfinite Numbers (which need additional Axioms to the Peano Axioms). Goodstein's Theorem does have some practical uses: it can show that certain Computer Programs will come to a conclusion, rather than continue for ever. Turing had a theorem concerning Computers: that a Computer itself cannot judge whether a reasonably complex program will come to a conclusion or not.

[@EtzEchad]

The Halting Problem in Computer Science is similar to the incompleteness theorem.

[@matthewstuckenbruck5834]

Enlighten me.

[@willmcpherson2]

@@matthewstuckenbruck5834 The Halting Problem shows that there are decision problems (yes/no questions) for which there are no algorithms that can reliably provide the answer. In other words, there are questions that no algorithm can solve.

[@cyberneticbutterfly8506]

@@willmcpherson2 Ah but can you prove how much effort you ought give to try to find out before you give up on something you don't know wether is solvable or not?

[@willmcpherson2]

@@cyberneticbutterfly8506 In general, no, because creating an algorithm that knows exactly when to give up is equivalent to creating an algorithm that knows whether an algorithm halts (making it equivalent to the Halting Problem). Although with both problems, you can use approximations. “Give up after 1000 steps” for example.

[@stephenhughes1862]

Uncle Petros is an awesome book!

[@ThomasJr]

I love how eloquently he speaks. Btw, does anybody knows the DENSITY of statements that are true but not provable in the usual set of axioms of arithmetic? They are infinite, but how infinite in relation to the set of all statements?

[@efulmer8675]

I am very much not qualified to give a definitive answer but I expect they are extremely dense in the same way that irrational numbers fit between every rational number on the number line, but so many of them don't have a constructive use or are simply down some logic path that people haven't bothered to look at yet. We as people use math and numbers all the time, but the numbers we use are very specific ones, the ones that constructively come off of the other numbers (the 'axiomatic numbers' if you will :) ), like the whole numbers extends off of the naturals, then the rationals and the integers extend off of the whole numbers and then the constructables off of them and then the irrationals as the mathematical logic increases in complexity and power, then the imaginaries and the complex numbers in one direction and the transcendentals in another direction... I suggest looking at the Numberphile video "All the numbers" featuring Matt Parker to get a feel for what I mean by the numbers 'building' off of each other and there being so many more than the numbers humans use under normal circumstances. I certainly don't buy Chaitin's Constant (the halting probability 'set' of numbers as the number itself changes depending on the program being checked if it halts or not) numbers of pairs of shoes at the store, because that doesn't make particularly much sense. It's also possible mathematical logic can't lead to inconsistency because not every number may be 'reachable', or something similar.

[@MuffinsAPlenty]

I'm not sure if you're asking about topology on the collection of sentences within a formal language, or if you're one of those people who dislikes referring to cardinality as "size" and chooses to call it "density" instead (my personal opinion is that size is a significantly better word than density when it comes to cardinality, but there are enough people out there in KZbin comments declaring it should be called density instead of size, so what can you do?). I can't speak to actual density in a topological sense (which is the sense in which the word "density" actually makes sense), however I can talk about cardinality. In a formal language, one always starts with a countably infinite number of symbols. Every sentence consists of _finitely_ many symbols. Hence, there are a countably infinite number of possible sentences. This, in and of itself, gives us only two options: there are finitely many undecidable statements or there are countably infinitely many undecidable statements. As you correctly noted, there are certainly infinitely many, so this gives us countably many undecidable sentences. It's the same size as the number of sentences to begin with as well as the number of decidable sentences.

[@mikecimerian6913]

Both mathematics and poetry share this area. The locus of synchronicity. :-)

[@kylecarter1599]

My main question is if this is a law of reality or just mathematics.

[@BytebroUK]

This is maybe a slightly naive question, but is the Incompleteness Theorem related directly or at all to the Halting Problem?

[@horkowl]

Seems to me that Dr. du Sautoy's explanation boils down to saying that no completely consistent system* of mathematics can be generated from a finite set of Postulates**. *Leaving aside the question as to what, exactly, is the definition of "a system of mathematics." For instance, does "algebra" in the broadest sense overlap with "topology," in the broadest sense? Naïvely, it seems to me that all the branches overlap here and there, in which case there could be just one "system of mathematics." ??? **I know that most (not all) mathematicians disagree with me on this, but I think that great confusion is caused by the use of the word "axiom" when what is really meant are POSTULATES (á la H.S. Plane Geometry, and Birkhoff's and Maclean's nomenclature of the "Peano POSTULATES" when they explain these in their classic text /*A Survey of Modern Algebra"). I'm using "Axioms" and "Postulates" according to this distinction: Properly speaking, Axioms (here, in abstract systems of thought) are always the same. For instance, A=A, or to more-or-less quote Aristotle, "A thing cannot both be and not be in the same sense and at the same time": The Law of Non-contradiction, which is axiomatic in all logical thought (even in consideration of real-world phenomena or issues). In this sense, Axioms are simply the rules of logic that we humans MUST assume if our words or thoughts are to have any meaning, regardless of what we're talking or thinking about. But POSTULATES are assumptions which are made without proof that they are true ("true" within the context of the logical system in question). They, along with definitions, FORMALLY are, along with definitions, the foundation of every logical system (including philosophical systems of metaphysics). I don't mean that the system is generated over time, actively, by thinkers, from these; usually it's not. Usually, the system is developed by humans and as they go along they also figure out postulates that are required if the system is to be consistent (non-contradictory) and fruitful. (If two such postulates turn out to be contradictory, then "something is rotten in Denmark.")

[@robertwilsoniii2048]

2:40 is a great plug on AI. I agree on this point.

[@TimJSwan]

12:00 Incorrect. The grandfather paradox does not imply time travel is inconsistent. It just implies that specific sets of events must be consistent within the universe. Therefore, for us to even time travel in the first place, we may have to lie to the travelers that they can accidentally screw up the past - which will allow them to be able to time travel in the first place.

[@carnsoaks1]

timelike loops occur in black hole physics...? circular light paths in time like "spatial" dimension inside the schwartzchild radius, EH appear to exist. Where Tlike D and Slike D swap coordinates

[@davidcampos1463]

When a tree falls in the forest, dose it make a sound? Is a proof only needed when it is asked for?

[@martinepstein9826]

Fantastic interview, but I was really hoping he would talk about nonstandard models of arithmetic, i.e. what you get when you take as an axiom that Godel's statement is false.

[@David_Last_Name]

I believe that would result in an inconsistent system if you included that as an axiom.

[@lagduck2209]

It's not statement, it's theorem that's proven from axioms. To make it false you need to change those, and that probably break whole lot of maths.

[@paultikotin]

I will say again what others have said, slightly differently. Gödel's theorem is not an axiom of arithmetic. If you were to assume the Theorem is false, you would need to label one of its logical steps false (even though it is thought to be true) and go from there. Alternatively, you could start by assuming that one of the axioms Gödel assumes is false. Now that approach has been used. For example in Geometry. The results are worthwhile. But Gödel's proof was about all such axiomatic systems. Drop or alter one axiom and you get a new system. It's still an axiomatic system and it is precisely such a system that Gödel's proof has as its subject

[@thecompanioncube4211]

So Gödel died by self reference. Damn the man was dedicated

[@tacky187]

0:58 that's my new ring tone

[@maxonmendel5757]

True

[@toferg.8264]

Lol!

[@TimothyReeves]

@@maxonmendel5757 but not provable!

[@paulgillespie542]

This reminds me of the PvnP problem.

[@wg9601]

2 and 3 as the only single gap prime in our number system has always been interesting for me. There's a strange tension point between two and three -- they are the smallest even and odd integers in the natural numbers, if we start counting after one -- that can interestingly exploit things. It's why I think the Collatz conjecture can never be proven false ... but I can't prove that my observation is true, nor can I disprove some number is out there which will never reduce to 1.

[@fofolp1213]

Wut 1/2 is the only single prime

[@portreemathstutor]

Can anyone tell me? When was Godel in hospital in America and when did the Sunshine Project begin. Is there any chance that when Godel thought people where trying to poison him he might have been right or was this way to early for that.

[@azzteke]

His name is Gödel, not Godel!

[@DoomCrystal]

How can you assign an integer to every mathematical idea after exausting them because you need to list all of the real numbers first?

[@michaelsommers2356]

_"How can you assign an integer to every mathematical idea after exausting them because you need to list all of the real numbers first?"_ Ideas aren't assigned numbers, but mathematical expressions are. That is, things that are written down, which are necessarily finite.

[@jackkingers]

Awesome concept that I'd never heard about! But how does this coding work if you are relating each operation as a prime, because when multiplying the primes together, we have the freedom of commuting the primes' order! But obvious there are many operations that do not commute!?

[@MuffinsAPlenty]

It's not just multiplying together the numbers for the symbols. There's a sort of hierarchy of encoding. First, you encode all of the symbols in your language. There are countably many symbols in your language, so you can develop a "list of them". And each of these symbols get assigned a certain number. For example, you could send the nth symbol to 2n+1. Then, you encode (finite) strings of symbols by taking 2^(the symbol number of the first symbol in your string)·3^(the symbol number of the second symbol in your string)·5^(the symbol number of the third symbol in your string), etc., using the nth prime for the nth symbol. Notice that two distinct strings of symbols have distinct numbers due to prime factorization being unique. Additionally, every string of symbols has a number which is different from every symbol's number since every symbol has an odd number, and every sentence has an even number. Then, you finally encode all (finite) strings of strings of symbols by taking 2^(string number of the first string in your string of strings)·3^(string number of the second string in your string of strings)·5^(string number of the third string in your string of strings), etc., using the nth prime for the nth string. Notice that any two distinct strings of strings of symbols have distinct numbers due to prime factorization being unique. Moreover, again, a string of strings can't have the same number as a symbol since the form is even and the latter is odd. And a string of strings can't have the same number as a string, since the former has 2 to an even power, while the latter has 2 to an odd power. The reason why we care about strings and strings of strings is because we want to be able to talk about sentences and proofs using the natural numbers. A sentence is a particular string of symbols satisfying certain grammatical rules, and a proof is a particular string of sentences satisfying certain logical rules. In other words, you can turn a sentence in your formal language into a natural number, and you can turn a proof in your formal language into a natural number as well. So you can now use the axioms of natural number arithmetic to prove things about statements and their proofs.

[@life42theuniverse]

9:50 things we can never know precise simultaneous knowledge of position and momentum.. heisenberg uncertainty

[@mauijttewaal]

Uncertainty is a wrong translation, it should be indeterminacy, so nothing to do with knowledge...

[@walterkipferl6729]

does proof by non-provability apply to goldbach? you should be able to find a even number which cannot be expressed if it were wrong, so it being false, you would be able to proof it

[@davidwuhrer6704]

Sure. Go ahead ☺

[@Roxor128]

One of Gödel's contemporaries also met a tragic end: In contrast to Gödel being paranoid about other people trying to poison him, Alan Turing poisoned himself.

[@paultikotin]

Roxor128 With a lot of encouragement from the government and other authorities at the time.

[@smith2luke]

are there mathematical statements such that we can prove that it is impossible to prove whether it is even undecidable?

[@thewiseturtle]

Regarding consciousness, the human brain is operating on a higher dimension of thinking than the 1 or 2 dimensions of simple mathematical thinking, which is why we can do math, because we're thinking more complexly than simple logic. A normal computer can only do one calculation at a time, while a human adult brain (functioning well) can do multiple calculations at the same time. So we can sort of triangulate answers to complex problems, whereas a normal computer can't. This is why the idea of artificial intelligence is so confusing to many people. A simple computer algorithm won't function the way a human brain can, since it's still looking at only one problem at a time. From what I can tell, the human brain can model up to four logic problems (which I define as the difference between a starting state and a goal state) at a time, at least in a mature (wise) adult brain (over the age of about 40 when the prefrontal cortex starts being able to operate at it's full ability, according to current neuroscience).

[@paulh.9526]

What led you to that conclusion?

[@wompastompa3692]

Gödel was a maverick.

[@raymondrogers3797]

Let me see: Say "+" is assigned the number '2' . Then the number "2" has to be assigned to some 'x1'. But then "x1" would have to be assigned to 'x2' and then "x2" ... so "+" builds a ladder through the coding. So for every theorem, I could find a step in the ladder having a value greater than that?

[@javierborda8684]

Is it really true that we have a gap between truth and proof? Is it not just that we can't have a definite proof by being self-referential? All the Theorem is saying is that we can't prove a true statement from the same system we derived this truth from but we sure can do it from a superior one. In fact Goedel created a higher level system precisely to proof that about mathematics. This would in fact make criteria for truth more consistent. Would this be a correct interpretation?

[@simaomarto6140]

Heisenberg's uncertainty seems like the physics version of this theorem. The presenter talked about whether similar "unknowabilities" exist in other fields in science, and this seems like a good example.

[@paulh.9526]

Not quite, Godel talks about the principles themselves

[@ponchout3982]

its like looking at your eyes with your own eyes (without reflection)

[@andreascompagnoni3350]

Have a look to Gentzen’s consistency proof

[@edomeindertsma6669]

Doesn't that depend on the well-foundedness of ɛ0, which was assumed (as an axiom) for that proof? And I'm not sure if it applies to ZFC.

[@PhilBagels]

3:45 No contradictions in math. Does that whole -1/12 thing count as a contradiction? Just wondering.

[@alexanderf8451]

No, infinite series are assigned "sums" in a different sense than finite series are.

[@toferg.8264]

_The_Ultimate_Proof_of_Creation_ by Dr.Jason Lisle deals with logic like this.

[@kennybentley1161]

can you ask him to talk about circular time and einstein's equations?

[@vitalysarmaev]

Gödel's theorem is a statement of the impossibility of solving a recursive object in the form of a direct solution. In arithmetic where there is no recursion, such as arithmetic without multiplication, all propositions are provable. From here, one more small step and you can prove Fermat's theorem in the way that he kept silent about in his diary: most likely Fermat understood the meaning of recursion - this is a function whose domain of definition is the product of the set of values ​​​​of the function itself by the set of the domain of definition of its predecessors. And after a couple of manipulations, you can come to Fermat's conclusion. But the boundaries of KZbin comments are too narrow for me to provide this proof here 😜 p.s. No, I'm not a crazy Fermatist

[@andrewmole3355]

The paradox of Gödel’s death - the fear of being killed, killed him…

[@carnsoaks1]

NETWORKED MINDS, being used already in polymath think-tanks. Yet, this is where the next growth will come from (before Quantum Computing AI takes hold altogether) as with James Maynard & Terry Tao s work together. Their groups each got so far, then their amalgam took them STEPS farther.

[@davidbardvall]

can any two of the "nifinite number of undecidable sentences, sentences that are TRUE but cannot be proved" contradict eachother? if so, has such a construction been made?

[@no3339]

Does gödel's theorems apply to all logical systems? Because I saw that it only works for logical systems that can say something about the basic arithmetic of natural numbers.

[@brauggithebold7956]

Yes, that is correct. The theorems only apply to formal systems that are powerful enough to express addition and multiplication on natural numbers.

[@no3339]

Brauggi the bold So then it would apply to, for example, Euclid’s axioms then? Because, by themselves, they are not powerful enough to express addition, etc. on the naturals. Am I mistaken?

[@brauggithebold7956]

@@no3339 They certainly don't apply to Euclid's axioms. Keep in mind however, that this does not necessarily mean, that Euclid's axioms are consistent and complete. It only means that you cannot use the incompleteness theorems to argue either way.

[@no3339]

Brauggi the bold Why would that be he case specifically for Euclid's axioms? Couldn't you still arithmetize the system and still have the statement: "This statement is not provable from the axioms"? I know that's a simplification of the first theorem but shouldn't it still apply?

[@brauggithebold7956]

@@no3339 Of course you could arithmetize the system. But how would you then establish self-reference, if the system cannot express arithmetic?

[@jlopez47]

About the inconsistency of the US Constitution, does anyone know if Godel referred to the US Constitution in its original form, or in its present state at the time Godel was granted citizenship. This is a very interesting topic.

[@Akiak7]

the set of all sets does not contain itself if you state that any set "contains itself" then... ...the set of all sets with even members may have odd members...

[@gabrielbrunser6401]

Does this raise the question of whether or not there is another or multiple options besides right and wrong, true or false. Maybe there is more than that. We are conditioned from birth that something is either true or false but what about both. Kind of like shrodingers cat, the more we know about some problems the less we know about wether or not it’s true. Maybe...

[@MuffinsAPlenty]

I think you would be interested in looking into things like many-valued logic and paraconsistent logic.

[@johnrembaylo29]

Based on my expierence following statement is expected to be true: "If equations that refer to lows of phycisc can be solved means those solutions aplly to the real world in a certain sense".