Gödel's Incompleteness (extra footage 1) - Numberphile

[@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.

[@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

[@vanderburg.M]

bahahahahahaha

[@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

[@sevrjukov]

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

[@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.

[@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.

[@cassandraxiv5223]

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]

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! :)

[@oybekoyhonim]

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

[@elliott8175]

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

[@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.

[@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.

[@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.

[@patrickwienhoft7987]

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

[@zorrozalai]

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

[@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.

[@Giantcrabz]

prove it

[@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.

[@yorkeR177]

Thanks for uploading extra footage :)

[@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.

[@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.

[@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.

[@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

[@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...

[@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...)

[@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.

[@tothm129]

Does anyone know the paradox in the constitution that this guy mentioned

[@cougar2013]

Mike Toth too late, the election is over

[@jeffirwin7862]

To summarize orochimarujes's comment, you can add amendments that make it easier to add more amendments, eventually leading to a complete breakdown of the system of checks and balances and democracy.

[@circlemouth7567]

Yeah. They didn't account for a foreign country using the internet to interfere with the election by hacking email servers, using an army of bots, spreading fake news, and using complex algorithms to psychologically profile and influence potential voters.

[@Nukepositive]

Exactly what @Jeff Irwin said, but it turned out not to matter. We never make new amendments now; we just selectively enforce the Constitution.

[@ChristopherKing288]

No one actually knows, since Godel died before he published it. Guesses are that either it's the fact that it can be amended in possible way (including to remove amendments), or that judges have complete freedom to interpret them (including other judge's interpretations).

[@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.

[@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.

[@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

[@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.

[@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

[@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?

[@mechtheist]

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

[@xizar0rg]

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

[@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?

[@xXUxCXx]

But if we prove that mathematical consistency is unprovable, doesn't that by the same logic imply that there are no inconsistencies? Or at least that you cannot come across them? Because if you come across an inconsistency it would disprove mathematical consistency, but that is impossible as we proved.

[@shimassi9961]

that's what i thought :D

[@DustinRodriguez1_0]

The problem is that you would never be able to prove that your true statement leads to an inconsistency because you have to prove it for it to be true, and a proof which contains an inconsistency or which causes one in another part of mathematics is not a proof. The statement would have to remain unproved which means you do not know if it is true or not.

[@cougar2013]

xXUxCXx Nice one! I hope someone answers that!

[@SwissBarracuda]

As soon as there is just one inconsistency in a theory, you can prove ANY statement, so also it's consistency (if you're able to formulate it). Gödel proved (from the outside, not within the theory) that if maths (or arithmetic to be more precise) is consistent, it cannot prove that within.

[@cavalrycome]

"if we prove that mathematical consistency is unprovable, doesn't that by the same logic imply that there are no inconsistencies?" If we found an inconsistency, we would have proven that mathematical consistency is unprovable, so no, proving that doesn't imply that there are no inconsistencies.

[@SeanStoneburner]

The limitations of any given formal system are intentional and purposeful, designed to focus on specific domains. While no single system can prove every truth, any true statement can be proven within a stronger, consistent framework. Even the unprovable statements within a system arise because that system is deliberately not designed to handle its own self-analysis, a limitation that can be addressed by stepping outside it.

[@ponchout3982]

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

[@sakshamsingh4378]

One of the best video

[@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?

[@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?

[@adlsfreund]

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

[@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.

[@tiagotiagot]

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

[@Helllllllsing]

Are all undecidable theorem true?

[@paulh.9526]

All undecidable theorems, that only need a simple counter-example to be proven false, are true. If they are undecidable with the axioms we have, then no counter-example can ever be found, thus it is true.

[@Blacksun88marco]

Paul H. But if you can prove them true in this way, doesn't this mean that they are not actually undecidable?

[@martingutlbauer4529]

I like that. My mind is making loops right now - can someone clearify this? ;)

[@RizkyMaulanaNugraha]

@@martingutlbauer4529 Well, it is still undecideable since in order to prove it (by finding counter-example), we can't prove it in a finite statements. The real problem is we don't have a way to finitely decide that undecideable problem is a member of the set of undecideable problem. Since it's not currently provable that the problem is undecided, that problem itself is still can't be proven true or false. To put it simply, let's see Riemann Hypothesis. Let's us suppose that RH is undecideable, then by Incompleteness, RH must be true (provable from outside the system). However if we found zeroes counter-example, then RH is proven false (proven directly from within the system of axioms). The trouble is, to categorize RH as undecideable problem in the first place, you had to prove (within the system) that RH can be categorized as such problem. But by Incompleteness, you can't do that. Thus RH will still can't be categorized until a counter-example is found, or you have already proved it with infinite steps of statements.

[@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.

[@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!

[@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.

[@tacky187]

0:58 that's my new ring tone

[@maxonmendel5757]

True

[@toferg.8264]

Lol!

[@TimothyReeves]

@@maxonmendel5757 but not provable!

[@arunb8841]

@@TimothyReeves LoL..Underrated reply :)

[@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.

[@georgeferris6204]

If you can go outside of the system to make up a new axiom, how come that axiom couldn't be, "If the statement is dimorphic, then we can assume that it is true." Wouldn't that disprove Gödel's Incompleteness Theorem? What am I missing?

[@MuffinsAPlenty]

Gödel's theorem here has a lot of hypotheses. This theorem states that you cannot have a complete, consistent, recursive theory which can fully describe the arithmetic of the natural numbers. Let's break down these words. A theory is a collection of sentences in first order logic. We generally refer to these sentences as axioms. When working with a theory, we assume that all of the axioms are "true". A consistent theory is a collection of axioms which are incapable of proving a contradiction. This basically means that the theory cannot prove a false statement. A complete theory is a theory in which every true statement can be proven from the axioms. A recursive theory is a theory is one in which you are able to identify whether or not a statement is an axiom. In other words, there is some algorithm which, given any statement in first-order logic, this algorithm can determine in a finite amount of time whether or not the statement is an axiom of your theory. So Gödel's theorem states that any theory which is capable of describing the arithmetic of the natural numbers cannot have all three of the properties at the same time. Obviously, we demand consistency. It would be worthless if our axioms could prove false statements. Additionally, we wouldn't be able to work with a non-recursive theory at all. So we have to give up the possibility that mathematics can be complete. If you added the axiom that forces completeness, that would make one of the other two things false. Most likely, it would make the theory inconsistent.

[@georgeferris6204]

Ah! Thank you! Makes sense.

[@georgeferris6204]

It would be pretty cool to see how this model would hold up in a trinary system where true, false, and true||false would all be valid outputs. I wonder how the theorem would hold up then.

[@benjaminbaron3209]

Guy from Germany goes to America: "Wtf is this stuff? Is this supposed to be food? Omg they're trying to poison me!" True Story.

[@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 ☺

[@GSandSDS]

The translation of: "Wir müssen werden, wir werden werden.", at 10:03 would be: "We must become, we will [or shall] become." The correct german translation for: "Wir must know, we will know.", would be: "Wir müssen wissen, wir werden wissen."

[@andreascompagnoni]

Have a look to Gentzen’s consistency proof

[@yodo9000]

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.

[@muhamadhamdy6576]

If axioms are unprovable then on what basis are they chosen? Intuition?

[@soumyajit161]

you can say that.

[@paultikotin]

Yes. The axioms of Euclid were pretty well accepted for hundreds of years since they seem 'right'. Only one, the one about what we mean by 'parallel' caused some doubt. Finally, quite recently some geometers started to think about what happened if the parallel axiom was dropped or altered. The results have been productive. But if you are an engineer, keep that axiom!!! If general relativity concerns you, try the alternatives!

[@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?

[@Giantcrabz]

You have inviolable rights which are violated on a daily basis lol

[@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.

[@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

[@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.

[@borg972]

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

[@Nanaquistillalive]

one question. If there are infinite mathematical statements that cannot be proved, and we can associate a unique prime number to each of the statement,there must also be infinite prime numbers... isn't it?

[@RizkyMaulanaNugraha]

The idea is to assign natural numbers for every provable statements generated within that set of axioms. All provable statements (be it true or false) will have unique numbers (non prime numbers). The axioms will have a finite unique prime numbers. If there is a statement that is not provable (you can't say it is true or false), in the sense that there are no series of logical statements that will lead to that conclusions, you had to include additional axioms if you want that statement to be true. In other words, if your axioms contains #2 and #3 and #5, statements number #6 is provable (you don't care if this is true or false, but is it provable or not) since 6=2x3. statements number #120 is also provable, since you can't factor out to a prime factor of 2, 3, and 5. However statements number #35 is not provable unless you add #7 as the axioms. This is just the illustrations. In practicality you work with the statements first then, if there are no steps that can produce that statements, then you had to introduce another axiom. Since number 2, 3, 5 were taken you put number 7 (prime) for this axiom. Then you prove that previous statements, and assign it the new Godel's numbers, which might be turn out to be 35.

[@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.")

[@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.

[@yoloswaggins2161]

If they are like primes can't you just have an infinite set of axioms and then for it to be complete (you can factor every statement into axioms)? Seems to me that the argument is relying on the fact that you only have a finite set of axioms.

[@paulh.9526]

Indeed. The question is, how do you generate an infinite set of axioms?

[@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

[@MushroomManToad]

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

[@legandable]

since when does labelling an unprovable statement as false mean the original statement is actually right only because we correctly labelled it? The new statement might be right but the new statement only says that the old one is WRONG. they are mutually exclusive.

[@Giantcrabz]

no

[@BatteryAcid1103]

So, I see one inconsistency with Gödel's Incompleteness Theorem, how do we know that any of the improvable statements are coherent? I mean, based on how it was explained in this video, there's no method in place to check whether or not a statement is coherent or not, so obviously statements which make no sense are able to be encoded. (For example "(+79=)^3/+=5" which could be coded by Gödel's coding, but makes no mathematical or logical sense) Obviously a statement like that couldn't be proven, because it makes no logical sense. How do we know that the infinitely many statements which cannot be proven aren't incoherent statements, and the remaining, provable statements are the coherent statements?

[@chaosme1ster]

There's a mathematical way to express "X is a proper formula" -- a kind of Formula(X) predicate. And that can be encoded using Gödel's code. So yes, you can encode rubbish, but also you can distinguish between encoded rubbish and encoded proper formulas.

[@BatteryAcid1103]

Ah, okay. I guess that was just beyond the scope of these videos. Needless to say, I didn't do any research outside of this video, cause frankly I don't have the time. Anyway, I assumed something like that was the case, so thanks for the clarification. :)

[@mitesh8utube]

I don't understand what does it mean to assign a number to axioms. May be because it can't be done.

[@MuffinsAPlenty]

I'm going to paste an old answer I've given someone before, and edit it to address your question. I hope it helps: It's important to note that Göddel didn't encode a "paradox in plain English" into numbers. In logic, we always start with a collection of symbols: logical symbols, constant symbols, variable symbols, function symbols, and relation symbols. First, Gödel encoded all of the symbols in the language of arithmetic into natural numbers - for each symbol, there is a corresponding natural number. Then he encoded every string of symbols into natural numbers - for each string of symbols, there is a corresponding natural number, and this natural number is different from any symbol's natural number. Then he encoded strings of strings of symbols into the natural numbers. And again, each string of strings has a unique natural number that nothing else has. These numbers are the Gödel numbers for the symbols, strings, and strings of strings, respectively. And all of these encodings are done in such a way that if you were given a natural number and had no physical limitations, you would be able to determine whether that natural number is a Gödel number at all, and if it is, which symbol, string, or string of strings it represents. So you are mathematically capable of going backwards too (even if going backwards would take longer than the lifespan of the universe, which, for most of the strings of strings, it would take longer than the lifespan of the universe). It is important to notice that any encoding of these symbols, strings of symbols, and strings of strings of symbols would do provided that it is mathematically possible to decode. So it doesn't depend on the encoding itself. However, I will provide one such an encoding: Send the symbol "(" to the number 3, ")" to 5, "," to 7, "¬" to 9, "⇒" to 11, and "∀" to 13. This takes care of the logical symbols. Then, the kth variable is sent to 13+8k. The kth constant symbol is sent to 7+8k. The kth n-ary function symbol is sent to 1+8(2^n*3^k). The kth n-ary relation symbol is sent to 3+8(2^n*3^k). That takes care of all of the symbols. If you notice, every symbol has an odd Gödel number, and with a little bit of work, you can prove that no two symbols have the same Gödel number. Then, you take a string of symbols, and encode the string as 2^(first symbol number)·3^(second symbol number)·5^(third symbol number) etc., where the kth symbol in your string corresponds to the kth prime number. Notice that every string of symbols has an even Gödel number where the power of 2 is odd. Then, you encode strings of strings of symbols by taking 2^(first string number)·3^(second string number)·5^(third string number) etc., where the kth string corresponds to the kth prime number. If you notice, every string of strings of symbols has an even Gödel number where the power of 2 is even. So we can conclude that every symbol, every string of symbols, and every string of strings of symbols has its own unique Gödel number. So, why did he do this? He wanted to be able to use an axiom set S of the natural numbers to talk about sentences in the language of arithmetic and proofs in those same axioms. A sentence is a string of symbols following certain syntactical rules, and a proof is a string of sentences following certain logical rules. So every sentence has a corresponding Gödel number, and every proof has a corresponding Gödel number. Gödel then did a lot of work proving that a certain relation is what is called "recursive." In particular, Pf(m,n) is a recursive relation provided that S is a recursive axiom set. Pf(m,n) is the relation that m is the Gödel number of a proof using the axiom set S of the sentence with Gödel number n. Now, what it means for Pf(m,n) to be recursive is that, more or less, given two natural numbers m and n, if you were given enough time (again, forget about physical limitations), you would be capable of determining whether Pf(m,n) was true for any two fixed natural numbers m and n, and that process would halt (the process essentially boils down to finding prime factorizations). This means that Pf(m,n) is actually a meaningful relation in the language of arithmetic that you can actually talk about in the language of arithmetic. He then did a lot of work showing that there must exist a sentence φ in the language of arithmetic so that the recursive axiom set S proves the following sentence: φ⇔(∀x)¬Pf(x,[φ]), where [φ] denotes the Gödel number of the sentence φ. Gödel's proof of the existence of φ is non-constructive. In other words, we merely know that such a φ exists, but we are unable to determine what φ actually is. φ ends up being the sentence that is independent from the axiom set S - it can neither be proven nor disproven (as I will explain below). But φ is not the encoding of the Modified Liar's Paradox. Rather, the Modified Liar's Paradox is φ⇔(∀x)¬Pf(x,[φ]). If you were to translate φ⇔(∀x)¬Pf(x,[φ]) into English, it would read "φ if and only if for all natural numbers x, x is not the Gödel number for a proof of φ." This sentence in the formal language (φ⇔(∀x)¬Pf(x,[φ])) basically is an assertion that φ is not provable from the axiom set S. Why? Well, suppose that S proves φ. Then there is an actual proof of it. Then you can encode this proof as a Gödel number, say it is m. Then the statement Pf(m,[φ]) is true. Therefore, the statement (∀x)¬Pf(x,[φ]) is false. But since S proves that (∀x)¬Pf(x,[φ]) is equivalent to φ, it follows that φ must also be false then. Therefore, if S proves φ, then φ is a false statement. However, if S is consistent (meaning that S doesn't prove any false statements), then this is an impossibility. Therefore, if we assume that S is consistent, we conclude that S does not prove φ. However, we can also conclude that φ is true (in the standard model of the natural numbers). Why? If it were false, then since φ is equivalent to (∀x)¬Pf(x,[φ]) (since S proves they are equivalent), (∀x)¬Pf(x,[φ]) would also be false. Hence, its negation, which is (∃x)Pf(x,[φ]), would be true. In other words, there would exist a (standard) natural number n which is the Gödel number for the proof of φ. In other words, φ would have a proof in S (just decode n, and you have a proof). Again, this implies that S could prove a false statement. So if S is consistent, this is an impossibility, meaning that φ must be true (in the standard model of the natural numbers).

[@smith2luke]

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

[@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

[@robertwilsoniii2048]

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

[@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.

[@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?

[@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?

[@no3339]

Brauggi the bold I guess the video didn’t really explain well. Can you explain why the system would need to express arithmetic to reference itself?

[@no3339]

Brauggi the bold That’s kind of what I figured. But for clarification, Gödel’s first incompleteness theorems could be applied to almost any more complex system that could be broken down into mathematical notation that performs arithmetic on the naturals. In other words, could systems such as branches of science be broken down into axioms and theorems about the naturals and then shown to be incomplete due to an established self-reference? Or is this impossible to do? Sorry if I’m bothering you but I see very conflicting answers to this

[@no3339]

Brauggi the bold Well it would make sense to me to have a set of definitions within physics, for example, to take the place of axioms and then use deductive reasoning to reach a conclusion while following the scientific method. The problem would be describing the “axioms” in terms of logical syntax but I believe it is at least reasonable to assume some science systems are incomplete if they are not already limited due to the “axioms” (definitions) not being as rigorous. I was supposing that even if some science branches could be expressed in terms of logical syntax and was capable of basic arithmetic, then it would be incomplete. But since it cannot be expressed in this manner, then it is even more limited than pure math.

[@drmabeuse1]

How can you have a mathematical statement that's both true but unprovable? If it's not provable, how do you know it's true?

[@uzor123]

Yup, that is what gödel said, that any system capable of arithmetic (including math) can't be complete and conistent at the same time. Mathematicians rather choose incompleteness over inconistency. That means that if maths is consistent there are certainly statements out there that can't be dervied (proven) from the axioms.

[@Giantcrabz]

assume it as an axiom

[@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

[@TheHernanNoguera]

Is there a mathematical difference between “incompleteness” and “inconsistency”? Could someone explain?

[@emurillo256]

Inconsistent= wont work somewhere where it should. Like x+1=3 obviously x=2. If I say x=5 then the equation is inconsistent

[@emurillo256]

Incomplete just means unfinished. In the video's sense This means that all the rules and theorems of math will never be fully discovered. We can try to add more axioms but we will never get there :(

[@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

[@davidcampos1463]

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

[@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...

[@BytebroUK]

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

[@MuffinsAPlenty]

Both the First Incompleteness Theorem and the Halting Problem rely on a diagonalization argument (essentially, Cantor's technique of proving different sizes of infinite sets). So they are both related in that way.

[@RobertWF42]

How do you a mathematical statement is not true based on its Godel number? For example "1 = 0"?

[@MuffinsAPlenty]

You don't tell that from the Gödel number. You essentially have to decode the Gödel number and show it's false.

[@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

[@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.

[@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.

[@crmesson22k]

I love how this video is done with a shirt that says I know nothing.

[@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.

[@jglenister0419]

If I'm get a bit of it, it sounds like any mathematical statement is almost given a value in a hash table?

[@wafikiri_]

No, for a hash table is supposed to be able to contain more than one element per hash index, and the way Gödel indexed things would give a unique value to each statement.

[@codycast]

I'm curious why this guys book has two different titles - one for the US and one for the U.K.?

[@dylanmcdonald5759]

codycast harry potter did it...

[@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

[@steffen5121]

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

[@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?

[@hasch5756]

"Wir müssen werden, wir werden werden" *G E R M A N C R I N G E*

[@buffendene9996]

haha

[@toferg.8264]

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

[@davidwilkie9551]

The explanation of the active role of a verb in a sentence was "connecting word", and the mathematical equivalent was an equal sign, so if the subject of an identity is true then the objective realization is the same equally true. It's normally a reversible action in mathematics, so if either side of the equal action is false, both sides are false or the statement is simply a lie. That there are many systems of lies dressed up as "trueisms" (truthiness!) is inarguable. So researching ways to identity veracity is still the core business of mathematics. yay

[@LaminatedMoth]

Doesn't this require our language of mathematics can be contained in a countable infinite? Shorthand and descriptions only describe a set of 'interesting' uncountable infinites that are relevant to our current understanding of mathematics. Wouldn't it be possible to have infinite axioms unable to be represented as a prime, as all other primes have been occupied? Those axioms could contain the "proof" of the supposed unprovable, right?

[@TwelfthRoot2]

1:40 - 1:51 how can he say that with so much confidence?

[@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".

[@oskarrecon8151]

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