guess you could say this comment was. funny

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.

your statement is provably funny.

*puts on sunglasses* YYYYYYYYEAAAAAAAAAAAAAAHHHHHHHHHHH

bahahahahahaha

Wolfgang Ambrus His books are excellent.

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

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.

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

You haven't understood him.

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

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.

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

@@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.

@@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

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

Hah! exactly my thought.

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

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.

Too soon

I'm from German too. Right! :)

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).

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."

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.

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)

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.

by the same logic so does he ...

Gunhaver you can watch anything later

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

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.

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

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.

True

Lol!

@@maxonmendel5757 but not provable!

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).

@@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.

@@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.

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.

Enlighten me.

@@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.

@@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?

@@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.

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.

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.

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 KZfaq 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.

that's what i thought :D

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.

xXUxCXx Nice one! I hope someone answers that!

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.

"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.

Wut 1/2 is the only single prime

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

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

Not quite, Godel talks about the principles themselves

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

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.

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

haha

What led you to that conclusion?

...no...

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.

Maybe

_"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.

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.

Oh come on you’re a great physicist, out of all people surely you should be able to understand

Like me who can't understand your equations XD