Mathematics - Russell's Paradox

[@boltfan36]

This thread is really entertaining! Please keep in mind, as has been mentioned several times in the thread, the paradox is based on a mathematical equation. Russell used visual representations such as the "Barber Story" to illustrate the problem. The original theory was presented by Gottlob Frege who tried to develop a foundation for all of mathematics using symbolic logic. He established a correspondence between formal expressions (such as x=2) and mathematical properties (such as even numbers). In Frege's development, one could freely use any property to define further properties. Russell's Paradox (Principles of Mathematics, 1903) discusses that the problem exists when one attempts to use an expression to prove itself. To keep this on a level for people like myself who are not die hard mathematicians, we prove division through multiplication and vice versa. We would not prove division by repeating the process, all you have done in that case is repeated a potentially flawed result. (forgive the oversimplification, but we get the point right?) Russell goes on after doing the math to illustrate with a story or visual aide. Frege saw the paradox and could not solve the issue. Later a mathematician name Zermelo also found the same flaw, he attempts to solve the Russell Paradox with an *Axiom (theory in which you accept an answer that cannot be proven) For those who are interested, you can look up the Zermelo-Fraenkel set theory.. It may make total sense to you, but from what I read, it really breaks right back down to Russell's Paradox.

[@Drumsmoker]

This is a paradox: What if Pinnochio said "My nose will now grow" Remember, his nose only grows when he lies.

[@jimkokko5]

Here's another one: Statement A: "Statement B is true" Statement B: "Statement A is false"

[@tomtomspa]

***** it is the same paradox of the pinocchio's one. It's like saying: i'm lying.

[@LaureanoLuna]

Macho, the Pinocchio paradox suggests that it is impossible that ltelling a lie be equivalent to displaying a particular physical trait, for otherwise you could produce a paradox (a sentence that has no possible truth value) by talking about a physical event. And this seems impossible: sentences about physical events are always meaningful and determinate.

[@ArpanD]

@@jimkokko5 Actually, the Russell paradox is a better one. Because here, we are not specifying what is statement A and/or statement B. We are just relating one statement to the other, but not specifying either statement with reference to anything else.

[@lincolnjohn8227]

No it wouldn't grow, because there are statements that can be neither true nor false. Pinnochio's nose would only grow from a false statement.

[@random_content_generator]

Congratulations, you actually explain it quite clearly in only one minute.

[@awesomedancer78]

Couldn't someone else shave the barber like damn shaving isn't that hard

[@emmanouil2586]

Yes but that would mean that the barber doesnt shave himself, and since the barber shaves all the people who dont shave themselves, he would shave himself

[@michakochanski818]

Thank you for explaining, i had the same problem

[@SH19922x]

This paradox is nonsensical and has been proven so, only under the strict rule of "the barber shaves EVERYONE who doesn't shave themselves. Meaning every person has to see a barber to get shaven if they don't shave themselves. But the point is The beard must be shaved for one way or another for it too work. If i break the rules the beard doesn't get shaved at all.

[@morpheus6749]

Jack of Blades Yeah you're not really getting point. Changing the problem statement isn't a solution to the problem.

[@morgengabe1]

Morpheus, pkb.

[@lishlash3749]

The source of this logical paradox is the hidden assumption of the Law of the Excluded Middle, which dictates that all propositions must be either True or False with no exceptions. Systems of logic which make no such assumption are able to define the "Set of All Sets Which Do Not Include Themselves" as the boundary that divides all sets into Self-Inclusive and Non-Self-Inclusive categories, a boundary which itself belongs to neither of those two mutually exclusive groups.

[@КириллИванов-з7д]

But if the proposition is nor true neither false, then it is not a proposition, isn't it?

[@nathanielhellerstein5871]

@@КириллИванов-з7д Maybe it is anyhow. Grey areas exist; that is the Paradox of the Boundary. If point A is black, and point B is white and here it is day, and there it is night then what do we make of the points in-between? For surely it is clearly seen That somewhere there must be a border, which, though its edge creates this order itself does not commit its troth to either side. So is it both? Or neither? How to read this rhyme? What place to place the time of time? For is the present old or new? And is the boundary false or true?

[@patrickwithee7625]

The law of the excluded middle (LEM) has little to do with Russell’s Paradox. That is, even if you get rid of LEM, you can still prove that the set is a member of itself if and only if it isn’t, which is always contradictory.

[@xmikeydx]

All *and only* those men who do not shave themselves. Otherwise the barber may shave men who shave themselves, too, and this yields no paradox since the barber may shave himself. Similarly for set-theoretic entities: Russell's paradox works ("works") only if the set of all sets that are non-self-membered contains all and only non-self-membered sets. Otherwise the superset in question would contain as possible subsets self-membered sets.

[@paulkomladarku3347]

The barber's case: "I don't shave my own beard so the barber is supposed to shave me, but there's only one barber which is me"

[@JF-py3vq]

he gets a lighter and hairspray.

[@smarvysmarv8735]

If R= set of all sets (not members of themselves but if R is not in the set of all sets, then it is not the set of ALL sets. But if R is in the set of all sets, it’s a member of itself. Pretty simple!!!!

[@dmitryalexandersamoilov]

It seems to me that this problem can be easily solved if you simply make room for the concept of the viewpoint or origin of the mathematical operation not being able to perform the operation on itself. This may be a property of abstract mathematical objects.

[@jimkokko5]

Dmitry Alexander Samoilov the conclusion of the paradox is that a set cannot contain all sets, because it cannot contain itself.

[@dmitryalexandersamoilov]

@@jimkokko5 i see, would it be a good axiom to say: sets that contain themselves are equal to the null set?

[@jimkokko5]

Dmitry Alexander Samoilov I think so. However I’d say that the paradox is a sufficient proof of that, so maybe not an axiom but rather a fact?

[@eddieprasad108]

in my country there are women barbers

[@nomealeatorioxd4639]

Paradox Rekt.

[@NeoHoshi]

+Edd Dee Women who exclusively cater to men?

[@wbx9126]

Hah!!! Bet you didn't see that coming did you Russell

[@sneed2600]

How

[@glennhitzert5981]

If everything you say is a lie, you cannot say that you always lie, for you would be telling the truth. If you would be telling the truth, you wouldn't always be telling a lie, which makes it a lie if you said you always lie. But if that is a lie, you must be telling the truth....

[@travonpough5626]

Glenn Hitzert I like it

[@Celtics-x4w]

Dope

[@AjinkyaMaurya]

Lovely

[@Nicktjohnsonll]

please can someone answer this. imagine yourself on a box, would the shape your observing around you be consider a box shape? or is there another name for this because looking at a cube and being inside of a cube and looking around would be different shapes wouldnt it?

[@Nicktjohnsonll]

imagine your self in a box is what I meant to say

[@dashinblu]

Well why don't you try to find out for yourself?(not trying to be mean) Its more fun and beneficial for you :)

[@Nicktjohnsonll]

Min Khant Htoo I wouldn't even know where to begin. but thank you, it would be beneficial.

[@edwardburroughs1489]

Made Mafia Music Do you have a room? They tend to be rather box shaped.

[@chrisg3030]

@@edwardburroughs1489 I'm in a room, and looking at one wall its entirety I see a square with 4 edges radiating outwards from the corners towards me. If I had eyes in the back of my head as well I would presumably see the same image of the opposite wall simultaneously, completing my view of the room's interior. I can't imagine what this would look like though. Maybe that's what was meant.

[@Bakugantsuvai1]

@Thomas Becker no you are mixing yourself up with the definition of the subset, a subset can be the whole set itself but you cannot have a set of all sets. That is the main subtlety right there.

[@jimkokko5]

HimikoWerckmeister Well said:)

[@Bakugantsuvai1]

***** Thank you, trying to learn this stuff right now and Russell's paradox is a pain in the butt to understand due to the broad definition of axiom of comprehension and extensionality.

[@ThemisTheotokatos]

Hi. I am sorry but I can't understand Russell paradox. Why can't all sets be a subset of a universal set and contain all of them? The story with the barber .. if the barber shaves all that do not shave them selfs and not the ones that do shave them selfs, what is the problem for the barber to belong to one set one day and another to another set?

[@jimkokko5]

So, in what set is he gonna be today?:) Think of it this way: let's say you have a box. And you want it to contain EVERYTHING in the world. Somehow, you manage to do that. However, your box does not contain everything yet. Because, there is still one more item in the world: the box itself! So, to make the box contain everything, you must put the box IN the box itself. That is, more or less, Russell's Paradox.

[@99bits46]

That means the statement "He shaves all those who don't shave themselves" is wrong, there's no paradox

[@Lisnageeragh]

Here , here....like certain other junk floating around these days trying to sound smart.

[@nathanielhellerstein5871]

You can Russell-paradox Plato's Theory of Forms. Let Russell's Form take as examples all Forms, and only those Forms, which are not examples of themselves. The Form of all Forms is a Form, so it's an example of itself, and therefore it's not an example of Russell's Form. The Form of all carrots is not a carrot, so it's not an example of itself, and therefore it is an example of Russell's Form. For any form F, F is an example of Russell's Form just as much as F is not an example of F. Therefore Russell's Form is an example of Russell's Form just as much as Russell's Form is not an example of Russell's Form.

[@colinturner6588]

This is a fantastic video. Great work!!

[@ArtemisLogic]

The barber is shaved by someone out of his country. Paradox Over. Other possible answers with varying validity are: The barber is not a man, he is a child. The barber shaves himself (none of the information here excludes this). The barber is a women. The barber doesn't actually live in the country where every man is shaved. The barber was born with a genetic defect that prevents hair from growing. The barber had facial electrolysis. The barber doesn't shave, he waxes. The barber uses hair removal cream. The barber had his cheeks burnt off in a fire when he was younger. The barber is shaved by a sentient robot, alien, or animal. The barber is a sentient robot, alien, or animal.

[@sandiladhikari6305]

Why couldn't the second answer solve the paradox? I don't undeestand.

[@vivianbshoutyhurani]

At last !! I understand this paradox

[@herbertwells8757]

The narrator sounds very drunk.

[@davidjames1684]

When deciding who will shave each person, apply the rule ONCE per day, and the paradox disappears. For example, suppose there were 12 people total (including the barber), and the barber could shave up to 12 people per day, so he would check 1 person every hour starting at 7am and ending at 6pm. Each of the 12 people were scheduled at the same time each day. If the barber was scheduled at say 12 noon, he would simply observe himself (to see if he had previously shaved himself earlier, such as at 6am), and act according to the rule. If he did not shave himself on a particular day and 12 noon came around, he would observe himself and realize he needed to shave himself. This is applying the rule once per day per person. Similarly, in computer science, if you have a statement such as if (x = 1) then y = 2 endif, meaning if x is equal to 1, then assign y a value of 2. Now let's suppose x = 2, but also y = 2. Did we violate any rule? Will the computer see this and say "hey, if y = 2 then x cannot also be 2)? The answers to those 2 questions are both no. We are allowed to set y to 2 by some other means before we encounter the if statement. Similarly, the barber can shave himself at 6am, before applying the "paradox" rule at 12 noon (for himself). In other words, time is a factor as well. Since everything takes time, the paradox can be defeated.

[@jimkokko5]

You’re missing the point. Time is not an issue at all here. When the barber shaves, he’s doing so under which property? As a barber, or as a non-barber? That’s the interesting question. Furthermore, this is not about barbers or scheduling or anything of the sort, it’s about sets and what they can contain - that’s why time is not an issue.

[@davidjames1684]

@@jimkokko5 But in reality, time IS a factor. Many computers for example will evaluate instructions one at a time. We can assign each person (of 12 let's say) into a array position such as P[1], P[2] (person 1, person 2...). The computer program would check: If not Shaved(P[1]), then BarberWillShave(P[1]). If the barber is P[6], he could shave himself (or not), before the check for P[6] happens. Time is part of our universe and our everyday lives, so ignoring it in set theory is impractical. What would that set theory be modelling? Not reality. Rather some make believe universe where everything happens simultaneously and instantly.

[@jimkokko5]

@@davidjames1684 it’s interesting how you consider set theory to be a completely artificial human construct (which it is), and yet time is “real” :) I’m not so sure as you are about what is real and what is not; isn’t everything an imperfect model at the end of the day? Anyway, if you want the hard maths of Russel’s paradox you can check them out on Wikipedia or elsewhere on the Internet, you’re getting way too hung up on the barber story. The *actual* paradox cannot be resolved in the way you are suggesting. Example: how can your method handle the definition of a set that contains all sets? If it contains all sets it must also contain itself, which leads to infinite recursion (see how time is in no way a factor here?).

[@davidjames1684]

@@jimkokko5 To me, this is a "forced" paradox (Barber), when in reality, considering time, it is really not. This is not a set containing all other sets, that is a different problem. The barber paradox can be solved if you apply the "check" once per person and at some set time (which would happen in reality since a barber cannot shave everyone at once). We should be allowed to assume real world constraints such as one person walks into the barber shop and the barber simply observes if they have been shaven already or not and acts accordingly. Maybe the barber takes a break at 12 noon and possibly shaves himself, cuz he didn't get up early enough before his first scheduled customer (at 7am in my example scenario) to shave himself. When you introduce time (something we are all familiar with), the "paradox" seems to "unroll" itself as easily resolvable. Why would you ignore time to try to solve this problem when in reality, the barber cannot shave everyone, nor even evaluate everyone at the same time? The problem didn't state "ignoring time...". Do you just arbitrarily ignore real word things when trying to solve problems? If someone asked you which is heavier, 10 pound weight or a 20 pound weight, would you say they are the same, ignoring Earth's gravity? Where does it say in that problem to ignore Earth's gravity? Also, there is a flaw in the way this paradox is presented here. The barber shaves everyone who doesn't shave himself. What if Bill and Joe are non barbers, and Bill shaves Joe (and Bill also shaves himself)? How would the barber know from looking at Joe that Joe didn't shave himself (Bill did)? And if Joe didn't shave himself, what would the barber then shave on Joe? Bill is allowed to shave someone else without being a barber, just like if someone cuts a piece of wodd, they are not automatically a professional carpenter. A flawed puzzle is presented in this video.

[@jimkokko5]

@@davidjames1684 damn, so Russel’s paradox is a “flawed puzzle” after all huh? I guess set theory is doomed...

[@Γιώργος-ε6τ]

Thanks, I finally understand the paradox

[@McGavel1]

The barber fell in vat of acid one time and no longer grows hair :)

[@NickMirro]

Does the fact that this scenario can be drawn with a Venn diagram but not with a Euler diagram somehow reflective or indicative of the paradox?

[@DecelDefeo]

Russell's Paradox is a famous problem in set theory, discovered by the British philosopher and logician Bertrand Russell in 1901. It reveals a contradiction in the naive understanding of sets, particularly in the idea that any definable collection can form a set. The Paradox: The paradox arises when we consider the set of all sets that do not contain themselves as members. Let's call this set R. If R contains itself as a member, then by its definition (being the set of sets that do not contain themselves), it should not contain itself. Conversely, if R does not contain itself, then by its own definition, it must contain itself. This creates a contradiction: whether we assume that R contains itself or not, it leads to a logical inconsistency. Therefore, such a set cannot exist within a coherent system of set theory. Impact of Russell's Paradox: Russell’s Paradox demonstrated that the naive set theory, which allowed for sets to be freely defined without restrictions, was flawed. It led to the development of more rigorous formal systems of set theory, such as Zermelo-Fraenkel set theory (ZF), which includes rules to avoid self-contradictory sets like the one in Russell's paradox. One such rule is the axiom of separation, which restricts how sets can be formed. Real-world analogy: A common analogy is the "barber paradox": Imagine a barber who shaves all and only those people in town who do not shave themselves. The question is: does the barber shave himself? If he does, then by definition he shouldn't; if he doesn't, then by definition he should. This leads to a similar contradiction. Russell's Paradox exposed fundamental issues in logic and set theory, prompting significant developments in mathematics and philosophy.

[@elenatess2288]

I am confused. Even if the Barber shaves all the men who cannot shave themselves, why does this automatically mean he cannot shave himself?

[@ottevervoort3672]

+Elena Tess The Barber is only allowed to shave men who cannot shave themselves. He can shave himself so he isn't allowd to shave himself because he is only allowed to shave men who can't shave themselves.

[@TheStacey0147]

if the barber would have some else shave him THEN some random person, let's call them X, has to shave him. However, this can NOT BE the case because then the BARBER is now included in the set of "all men that do not shave themselves" AND the ONLY person who shaves those men has to be the barber. The barber can not be person X.

[@i8ET]

+Otte Vervoort You just blew my mind. Thank you.

[@v3le]

Elena Tess who "don't" shave themself

[@peterjones6507]

The Barber 'paradox' is not a paradox. It's just an impossible situation. It's as daft as the idea that sets are containers that contain themselves. .Daft ideas are often paradoxical. It's the reason why they're daft.

[@ktrtz7066]

If I understand this, a mitigation would be: a gun can shoot other guns but it can't shoot itself. Right?

[@jimkokko5]

Kwstas kantartzis Well, no. It's more like, a list of everything...let's try a simple example: Let's say that I live in an imaginary land, where there are only apples, oranges and rocks, nothing else, and let's say I want to make a list of all the things that exist in this imaginary world. My list: -apples -oranges -rocks -me Ah, but the list does not actually contain everything; There is now one more thing in this world....the list itself! So, let's try this again: My list: -apples -oranges -rocks -me -My list: -apples -oranges -rocks -me -My list: -apples -oranges ... and the list goes on and on and on. So, actually, I can't make a list that contains EVERY SINGLE THING, because when I create such a list another item actually pops into existence, resulting in me not being able to ever actually make a list.

[@NathanRichan]

+jimkokko5 Why is it not sufficient to just say 'This list', without actually reproducing the list?

[@jimkokko5]

Nathan Richan because 'This list' is not an item, for it contains infinitely many items. We want every item there is to be able to be listed.

[@NathanRichan]

***** So is the lesson here that whenever you want to list everything of one type, the thing you use to list them can't be of that type? For example, you wanted a list of all items, so the 'List' itself can't be an item, or we get a paradox. But you can list all the natural numbers, because that 'List' itself isn't a natural number.

[@jimkokko5]

well, talking about it that way is misleading...try to think of it as sets (and elements of sets). A set can be an element of another set, no problem there. The problem is that you're trying to define a "list of things" which does not only contain infinitely many things (which is fine, take the set of the real numbers for example), but also "creates" infinite "sublists". All this talk and analogies is kinda pointless, we are getting carried away from the actual, mathematical problem. If you want to have it explained to you via an analogy, think of it the box way (a box which contains everything, and thus must contain itself, which is clearly imposssible). The analogies are here to make it clearer, though, and not define it. If you want to get deeper down to why we can't have a set of all sets, I highly encourage you to read a little bit about a more mathematical way to approach this (I think it's in the 2nd paragraph): en.wikipedia.org/wiki/Russell%27s_paradox

[@tomalator]

does a set of all sets not containing itself contain itself?

[@nathanielhellerstein5871]

It does just as much as it does not.

[@kanjimanji]

So the paradox that does not exist has a name?

[@opentrunk]

"I see", said the blind man as he picked up his hammer and saw.

[@Dyslexic-Artist-Theory-on-Time]

If the the Universe is explained as a continuum based on an emergent process there will always be new sets coming into excistence. Because this is a continuous process it is incomplete!

[@christopherkimball2264]

The same argument can be made about god. If god created everything, But nothing created god then who created god? Everything is intersecting

[@jamestagge3429]

Just to reiterate for clarity. I really want to understand this…. 1. There must be two sets of men for the paradox to exist. > The set of men who shave themselves > The set of men who are shaved by the barber 2. The barber is a component of the paradox and it defines him specifically and unequivocally as the source of shaves for the set of men who are shaved. The paradox leaves no doubt that if you are a member of this set, you go to the barber. If not, the paradox does cannot exist. 3. Of the two sets of men, all characteristics of the men of each set are identical, but one. > they are all men > they all live in the town > they all must be shaved. The above too is unequivocal or the paradox cannot exist. 4. So, that which defines the sets as separate is the only characteristic which is not shared (thus two sets), i.e., that one is shaved by another and that the other shaves themselves. 5. If you deny that there must be a third set, the barber defined as such by the only characteristic he does not hold in common with the first two sets (shared characteristics - he too is a man, lives in town and must be shaved), that he shaves others and as such is not included in the definition of the first two sets (because he logically could not be), the paradox cannot exist because in the denial of this logic, the first two sets would be only one set (that which makes them separate would not be relevant to do so). 6. So, the barber logically cannot be a member of the other two sets so there is no paradox. So, you can go all around the houses by appealing to the instruction of the sign to try to establish that there is a paradox by which to be so astonished, but it does not change the fact that to define this paradox in the manner it was is an affront to the very logic which it attempts to display as paradoxical. The sign states that the barber shaves only those who do not shave themselves. He “shaves” those who do not shave themselves, a characteristic (he shaves others and no mention is made to qualify how he is shaved - only that he shaves others is that which defines him aside from that he is a man, lives in town and must be shaved) defined which is not shared by the other two sets and thus, the one that defines him as not of those two but of a third. You cannot have it both ways. If the unshared characteristic is not what defines membership to a set then there are no sets but on and the paradox cannot exist. Remember, the act which is key to the supposed paradox is shaving or being shaved. But here we have one that does the shaving when those who are shaved do not. Clearly, this is a third distinct set so the paradox does not exist. This is illogical in its definition and meaningless and frankly, not very impressive.

[@MikeRosoftJH]

This is completely missing the point. You are inventing conditions that are said nowhere in the thought experiment. It's not said anywhere that all men in town must get shaved; it's not said anywhere that nobody but the one barber shaves anybody but himself; and it's not said anywhere that if somebody shaves himself, he can't also get shaved by the barber. We treat "A shaves B" as a binary relation; either "A shaves B" is true, or it's not true. (If we want to make the definition of "A shaves B" unambiguous, we could define it to mean "A has shaved B at least once during the past month"; and indeed for any A and B this either is, or isn't the case. When A and B is the same person, we use the same definition: "A has shaved himself at least once during the past month".) The barber claims that he shaves all men in town who shave themselves, but doesn't shave those who do shave themselves. The claim can't be true. Let's ask him: "Do you shave yourself?" No matter how he answers, he contradicts his claim: if he shaves himself, he shaves somebody who shaves himself; and if he doesn't shave himself, then he doesn't shave somebody who doesn't shave himself. (Remember: the question is: "Does the barber shave himself"; not "Who shaves the barber?") Let's distill it: Suppose there are three people in a room: A, B, and C. Is it true that any of the three shaves those, and exactly those, of the three who don't shave themselves? Let's ask them: "Who do you shave?" A says: "I shave myself, and I shave C." B says: "I shave myself." C says: "I don't shave anybody." It can be seen that only C doesn't shave himself; and none of them shaves C and nobody else. We could change the three's responses (for example, A could say: "I shave myself, and I shave B", B could still say: "I shave myself", and C could say: "I shave A"); but the answer is still the same: none of the three shaves those and exactly those who don't shave themselves. The scenario serves as a visualization of a mathematical result: Naive set theory, which implicitly assumed that for any proposition P(x) there exists a set of all elements (sets) for which P(x) is true, is inconsistent. Let P be: "x is not an element of x"; the set of all sets for which this proposition is true cannot exists (because it is an element of itself if and only if it is not an element of itself).

[@jamestagge3429]

@@MikeRosoftJH I missed the point? Lol. I think you did not even read Russell’s paradox. It states quite clearly that all the men in town must be beardless by law, that there are two sets of men, those who do not shave themselves and those who do and that there is a barber who services the former, whose sign instructs that he shaves only those who do not shave themselves. If you deny this, i.e., the logical context of the two sets of men and that the barber is constrained by the instruction on his sign, you deny the very means of the paradox which we are discussing. Did you never read Russell on this? Apparently not. The point is, and I was quite explicit in my post, that the paradox fails as per Russell’s definition because the logic and means by which the first two sets of men are defined as such and not a single set of men is violated. The three membership criteria which each of the two sets shares with the other, they are all men, live in town and must be shaved would define them as a single set of men but for the single criterion of their respective relationships to shaving, i.e., that one shaves themselves and the other not. By that logic then, if it is to remain true throughout which it must, the barber must be defined as a third set of men, those who shave others. If you deny this then you deny the means by which the first two sets of men are two and not one set. The paradox would then fail. If you accept my proposition, then there is no paradox to consider and it again, fails. The point being that one cannot formulate a conceptual contradiction from which to launch mathematical propositions if they are to be trusted or considered on their entire merit. If one cannot even get this sort of proposition right, if it is not readily understood for the contradiction it is, how is the math to be trusted?

[@jamestagge3429]

@@MikeRosoftJH you seem to deny the means and manner of the paradox from the start which defies its function and purpose and Russlell's intent in proposing it to begin with.

[@jamestagge3429]

@@MikeRosoftJH no response????

[@MikeRosoftJH]

@@jamestagge3429 Sure, I haven't read anything by Russell; but it's not clear where you are aiming with this, because it's not Russell who came up with the barber paradox. You claim: "It states quite clearly that all the men in town must be beardless by law, that there are two sets of men, those who do not shave themselves and those who do and that there is a barber who services the former, whose sign instructs that he shaves only those who do not shave themselves." Where is this stated? None of this is needed for the paradox. The only thing that is needed is that "A shaves B" is a binary relation: for any A and B, even if A and B is the same person, "A shaves B" is either true, or false. And even under your assumption - that everybody gets shaved, and nobody except the barber shaves somebody else than himself (which, as I repeat, isn't needed for the paradox at all) - it follows that the barber must shave himself, and therefore shaves somebody who shaves himself. Then you claim: "By that logic then, if it is to remain true throughout which it must, the barber must be defined as a third set of men, those who shave others." By doing that you are changing the setup - it's no longer the case that the barber shaves those and exactly those men in town who don't shave themselves. (In other words, the sign doesn't apply to the barber himself, even though he is a man and lives in the town.) So it's not a solution to the problem as stated.

[@saftheartist6137]

Thank you

[@jadolphson]

As phrased in the video, it’s not a paradox. The barber “shaves every man who does not shave himself”, does not exclude that he shaves additional people. Had the statement been “shaves every man, and only those, who…”, then the barber needs to go out of town to get a shave.

[@PaoloCaglioLMT]

Go for a walk in the woods and burn one down, you'll soon quit thinking about paradoxes

[@kiyanamiri9786]

My listening isnt well could some one explain for me?😢

[@vishwasshankar3929]

What component will be left out of our system?

[@jimkokko5]

what do you mean?

[@vishwasshankar3929]

@@jimkokko5 At 2:00, you mentioned there is a component left out so it is incomplete. I did'nt understand that

[@jimkokko5]

@@vishwasshankar3929 If the set does not contain itself then it does not contain EVERYTHING. The component that is left out is the set itself.

[@vishwasshankar3929]

Oh! Ok. And so at the end did was the paradox resolved?

[@jimkokko5]

@@vishwasshankar3929 Yeah, basically the solution is that you cannot have a set that contains all the sets.

[@suruxstrawde8322]

Ohhh. So this would be used to debunk an all powerful being being all powerful. Though, if an all powerful being were to defeat it's own power. Wouldn't the simplest way just be to step down from power? Then, somehow if needed get back into said power?

[@JacobHall-rd2yv]

Like politics

[@DavidBadilloMusic]

If there are ANY men that shave themselves without needing the barber (as the question implies, because the barber only shave those who do not shave themselves, which means, there might be some that DO shave themselves), who's to say that the barber doesn't get shaved by one of those men that shave themselves? I mean, they know how to shave, as they shave themselves. Any of them could very easily shave the barber.

[@Mdebacle]

The premise is only one barber. Then no other man could shave the barber because that would make another barber.

[@johannesberger8641]

There's a slight mistake though: You say "The barber shaves all the men who do not shave themselves", but you should have said (and you surely meant this) that "The barber shaves all and only the men who do not shave themselves"

[@davidjames9935]

There is also another solution/resolution to this paradox. First of all, let's set this up more properly saying that it is a village of people, not an entire county. Whoever posed this question using a country is an idiot as it would be impossible for the barber to shave more than perhaps a few dozen people each day due to time constraints, fatigue.... So let's suppose there are 40 people in the village, half of which are men and thus there need to be 20 shaves per day. First of all, everyone has the right to shave themself according to the original question. So let's suppose 19 of the 20 men get shaved by the barber since they like his work and the price is good. Towards the end of the day, the barber is tired and sees he is still not shaven so then he decides to sign over his barber certificate to someone else (since all male villagers are capable of shaving). Now the paradox has been circumvented because the original barber (call him A) has "passed the torch" to some other male villager B who sees A is not shaven, applies the rule, and decides to shave A. Nowhere in the original question did it say that the barber remains constant for all eternity. What do I win?

[@jimkokko5]

+David James Man, are you fucking dumb. What the fuck does fatigue have to do with mathematical concepts? We can suppose that he shaves every fucking hydrogen atom in the universe, and it would still work. I suggest you go check out the proper mathematical definition, because this all refers to sets and not people. But to tackle your bullshit argument: 1. Where the fuck is it written that the barber can "sign over his certificate"? 2. This is not a circumvention, this is a bullshit irl analogy, tackling a problem whose solution is painfully obvious (and who no one fucking asked for, because it is so fucking obvious.) 3. You win a free Shut-The-Fuck-Up™ award, for your immense stupidity and inability to understand that this is a simple story that's used to explain a fucking *math* problem. 4. You have a very serious problem of understanding definitions in problems. When I say "I am going to specify the exact abilities of the barber", I'm essentially saying "the barber cannot do anything else". Sorry for the harsh language, but you're just asking to be made fun of. Try again when you've actually passed highschool maths 101. Cheers

[@davidjames9935]

***** Where it is written he cannot sign over his barber certificate to someone else? This is a lesson to me on poorly worded questions that can be easily attacked. There is also no rule that says a person cannot both shave themself and someone else so someone could shave the barber and since he didn't shave himself he also shaves himself (2 shaves in one day that is). The question is who shaves the barber and my answer is anyone in that country/village can including the barber himself including "overlap" where 2 people shave the barber in the same day. I live in the real world not some fantasy mathematical world. By the way I am a college grad and have taken several college level math classes. One big problem I have with "mathheads" is they tend not to state problems clearly with clear restrictions and clear definitions. This is a near perfect example. I am poking holes in this problem that has been around for a long long time.

[@jimkokko5]

Then I am fucking sorry because you can't reason for shit and fail to understand a mathematical definition of a simple fucking function. Oh, and by the way, you saying you are something on the internet doesn't actually prove that to be true. Just saying. You may live in any world you like, but this is a mathematical problem which is translated into a simple story, so that many people can be able to understand it without needing to know advanced set theory. Can you get what I'm saying, or is it too much to ask? First of all, the use of the word "mathheads" indicates you are simply too fucking stupid to realize that people who study maths aren't kids who just read math books and brag about it. They actually fucking *study* maths in a university. I have stated time and again why my definitions are mathematically and thus logically pure. You on the other hand are trying to point out flaws that don't exist. I do try to show you *why* they don't exist, but it seems that your fantasy of you being some God-sent math problem correcting superhero is just too strong for you to realize how fucking dumb you sound.

[@davidjames9935]

***** Maybe to you but I don't see anyone else chiming in here agreeing with you that I am dumb. I can't be as dumb as you say cuz I circumvented the paradox without breaking any of the original rules. Just keep changing barbers in a "roundrobin" fashion and the person that shaves the previous barber (who is no longer a barber at that point) can be the new barber (who is the only barber in the village/country at that point). Because the barber changes daily, the answer to the question needs to specify which barber (on what day) they are talking about. Also, where in the problem did it say any attempts to "solve" this paradox have to be done mathematically? If a non mathematical solution works then it should be considered valid.

[@davidjames9935]

***** - Another comment is that I am not very fond of mathematics because it seems like there are fundamental flaws in it and many things are "forced" to try to work. I am not math savvy enough to give multiple good examples but I have seen mathemeticians "poke holes" in modern mathematics and I pretty much understood their points. One simple example is how they define x factorial to be all the factors from x down to 1 (inclusive) multiplied together yet they "force" 0! (by convention) to be 1 but there are no factors from 0 "downto" 1.

[@MGHOoL5]

I can't help but see Russell's Paradox as absurd for the simple fact that it opposes identity principle. How can you say "x ∉ x"? or the barber shaves "non-shavers" and he is a "non-shaver" so does he shave himself and contradict being a non-shaver, or go to the barber whom is himself and thusly contradict himself since he will be shaving himself? This is also like saying "This sentence is wrong" which again cannot be true nor false because it is internally contradictory. The simple matter is that what we are saying really is just unrealistic and shows how our minds can create abstractions that do not correspond to reality. We start off on a wrong premise that "x ∉ x" is something that you can say, that "a town that necessarily shaves has a man who necessarily cannot shave". Am I missing something?

[@jimkokko5]

What you're missing is the paradox as expressed in terms of set theory (because words tend to be confusing, even though the barber analogy is funnier): A set cannot contain all sets, because it would then have to contain itself.

[@MGHOoL5]

@@jimkokko5 Aha. I guess it could be said that one should differentiate between the absolute and the relative. One cannot call set R an absolute containing infinite sets relative to it whilst at the same time having the set R here be present again in the same logic as relative. What we are saying really is that absolute = relative which although could be true, it is only so in different scenarios or it will be a contradiction (i.e. law of non-contradictory in where a thing cannot be its opposite at the same instance). The example of: "This sentence is wrong" is just like that. We call the sentence absolute and the content relative, whilst at the same time allow the content to be an absolute and the content as relative and instead of making them a tautology which would allow that e.g. "this sentence is true", we create a contradiction i.e. true = false, which is duo to the relative being an absolute and a relative and vice-versa.

[@michaeldepaulis917]

Wow, Godwin's Law was fulfilled in only 15 seconds.

[@SathReacts]

I'm 8 years from your future and am glad SOMEONE made this comment. Cheers.

[@lawrence1318]

There is indeed an answer, and it's quite simple. There is no proof that anyone shaves himself until after he has shaved himself, at which point it is too late to not shave himself. So the barber shaves everyone but himself, and because there is no proof that he shaves himself until he has shaved himself, he also shaves himself. So there is no paradox, but rather, the supposed paradox is produced by deficient category definitions which fail to distinguish between action and state.

[@lachlanloveridge2053]

I found a solution this paradox. He said “the barber shaves all the MEN that do not shave themselves” but he never said that the barber was a man.The barber could be a women (some women grow facial hair), or a teenage boy, or a clone, cyborg or alien

[@BrandonJPeck]

Explanations that contain mistakes and use typed in corrections make things all the more confusing

[@jimkokko5]

+Brandon Peck this is youtube. It's free, This video is 3 years old. Get fucking over it

[@MashAllPotatoes]

+jimkokko5 wewlad

[@cosmictapestry3437]

Ahhhaha you just got rolled on son. Admittedly, as a simpleton this actually gave me some insight into this interesting paradox. I don't think this video was for you, in all of your infinite wisdom. Thanks @jimkokko5

[@jetzeschaafsma1211]

I saw someone do a 20 minute one this. Is there more money if you get people to watch for 20 minutes?

[@jimkokko5]

Maybe? I do know that at one point KZbin favored longer videos but I’m not sure it’s the case anymore. That said, if you wanna go into extreme detail you could talk about this for hours to be honest.

[@novaastronomia8720]

Simple and wise thing is to ignore this fact of Russell or it will crush the wall of mathematics and sets.

[@jhaaglund4518]

People getting too caught up in the analogy (not helped by the OP failing to say that the barber shaves those who do not shave themselves and nobody else). Let's call the set of all sets which do not contain themselves R, if R does not contain itself then it is a set which does not contain itself, which implies that it should, in fact, contain itself. This is the actual paradox, the barber is just a fucking analogy.

[@Valdakyr]

Where is the paradox? Let's say from A: 'a man does not shave himself' follows B: 'the man is shaved by the barber'. A and B are not aequivalent. So if the barber shaves himself the statement is true, because it doesn't say that the barber doesn't shave men who shave themselves. If he doesn't shave himself, however, the statement is simply false.

[@jimkokko5]

Well, yeah, it actually does say that the barber shaves ONLY those who do not shave themselves. But in the end we come to the conclusion that there's an intersection between the set of the people who shave themselves, and the set of the people who are shaved by the barber.

[@Valdakyr]

***** yep, that's right. Thank you.

[@jimkokko5]

You're quite welcome:)

[@benheideveld1152]

What Valdakyr correctly observes is that claiming that A. "Everybody who isn't shaving himself is shaved by the barber" is fine, B. "The barber is only shaving people who do not shave themselves" is also fine. A and B can both be true at the same time! That just excludes the possibility that the Barber shaves himself. I don't have a problem with that, let him join ZZ-Top

[@benheideveld1152]

+Ben Heideveld Adding "in a country where every man is shaved on a daily basis" makes the setup impossible, but not a paradox.

[@Beemerboy324]

So is the plural of "paradox" "paradoces"?

[@MarilulaneMusic]

secondly, can someone pls explain why it's impossible to prove (1.35) whether or not there can be a set that contains all the sets? to me that's prima facie - no, clearly that statement cannot be true. why is that impossible to prove? again, it seems his mistake is due to the fact that he has been selective in his scope of reference - 'is there a set that contains all the sets?', i can equally ask 'is there one planet that can fit through a keyhole?' no.

[@veselin4504]

Very easy solution! You said that people who can't sheve THEMSELVES go to barbers.Barbers can shave other people who can't shave THEMSELVES. So a barber who can't shave HIMSELF goes to an another barber.=> Barber shaves to barbers? Am I wrong?

[@jimkokko5]

Yes. There's only one barber.

[@xXLeAkErXx]

0:55 that's wrong though because somebody else can shave him and still shave themselves

[@jimkokko5]

No, because the only one who can shave other people is the barber.

[@RazorM97]

is that a set or a multiplicity of sets, different things give different stories ig :/

[@sortehuse]

The barber shaves all people in the town that don't shave themself - that is the statement. Noone says that the barbar isn't allowed to shave someone that shave themself, so the answer to who shaves the barber is: Himself

[@EsamAmm]

That’s why set theories are under a context or governed by laws thus it’s classified! …if you leave the set theories without any rules then it’ll create a paradox!

[@EsamAmm]

In the barber case…it should create a set for barbers who are shaved by other barbers thus the paradox is eliminated!

[@dunisanisambo9946]

Every set is a subset of itself.

[@jimkokko5]

Dunisani Sambo yes, but a set cannot contain all sets, because then it should contain itself.

[@dunisanisambo9946]

jimkokko5 it will keep on containing itself.

[@jimkokko5]

Dunisani Sambo that is precisely the problem that this paradox illustrates

[@dunisanisambo9946]

jimkokko5 Yes, the paradox is there because they want the set to be precisely defined and it’s not, but we know it contains itself.

[@bigkidz3612]

The parafox doesn't exist when the paradox exists

[@riverjoe128]

Good stuff

[@jimkokko5]

A weird one:P And that's because I'm greek, which means that english is not my native language.

[@KrisKitchen]

The barber got sick of people asking the question, made a machine and had it shave himself.

[@BairdBanko]

So the barber is both the person and the machine. Perhaps the machine is the person. (This is a misleading conclusion since a single property of the machine being able to shave the barber may not define all the characteristics of the barber. But it may so I am leaving it there.)

[@Spoif]

The barber shaves everyone that does not shave themselves... with the exception of himself. 8)

[@glutinousmaximus]

You don't need to bother with set theory. Math and allied logic _Must_ follow rules consistent with those (human) rules agreed upon by everybody when manipulating any quantity. It's rather abstract, but see Kurt Godel's take on this. Anyway, this 'paradox' simply _does not_ follow the rules.

[@aman2426]

This isn't a very good example of the paradox. Out in simple words, the paradox is just saying, "If you want to create a set which contains all other sets possible, i.e. a universal set, say 'A', then that set A must also include itself to be truly universal, it cannot leave itself out, once it includes itself, then this new set, say 'B' must also include itself, and it becomes a new set 'C', and so on...

[@Vladislavchooo]

The barber shaves people who don't shave themselves, so why can't another barber shave our character ? he doesn't shave himself he only shaves other people who don't shave Themselves, so what is the problem in this case the people who shave themselves don't really matter but their lack/absence does

[@jimkokko5]

+Vladislavchooo Let's suppose that we do have another barber (although by this point we're beyond the initial rules of the paradox). Let's say he shaves the first one, and everything's fine- until it's his turn to shave. What now? It's the same problem. Again, as many people in the comments, you got stuck in the story. The point is to try and see the actual mathematical problem rather than trying to find a real-world bodge that would "fix" the story.

[@Vladislavchooo]

what ? the guy he shaved will shave him and so what ?

[@Vladislavchooo]

i'm not trying to see the real world scenario I go by the presented means, atleast he said the barbers shave people who don't shave themselves and that counts the barbers as well since they don't shave themselves, so what is the problem if another barber shaves them and they shave the other barber ? write it down and see if any of the properties change (the ones for the paradox to occure)

[@jimkokko5]

+Vladislavchooo So, first of all, there's only one barber. That's a strict rule. Second of all, you're saying "the barbers shave people who don't shave themselves and that counts the barbers as well since they don't shave themselves"- no, the barber does indeed shave himself, and there's the actual paradox. Remember the rules? If you can shave yourself, fine, if not, you're gonna get a shave from the barber. Let's suppose that the barber can shave himself, which he does. What category does he fall in then? He is being shaved by the barber (it matters little that the barber is actually himself) AND he shaves himself at the same time, something that should be impossible, and hence, is a paradox. Now, if we play by your rules and thus have two barbers, there's a slight alteration to the rules: You say that each barber should shave the other one; but you forget that that implies that neither of the two can shave themselves, which wasn't originally a rule. If you allow them to shave themselves, well...there's the paradox again. The actual mathematical thing is really quite simple if you have some basic knowledge of set theory: Is there a set that can contain all the sets there are? If it must contain *everything*, then it must contain itself too, right? How can that be?

[@Vladislavchooo]

if there is only one barber then no one can shave him since shavings by a barber are allowed on people who don't shave themselves. so it's contradiction thus no one can shave him (unless someone from the guys comes and shaves him and I don't see how that would defy the rules as well)

[@titanarmy4116]

What happens when Pinocchio says, "My nose will grow"....???

[@abdalln8554]

The blue fairy comes down and smacks him for being a smartass.

[@glenarth]

the barber or the whole town do not consider himself(the barber) to be the barber at the time he shaves himself. problem solve!

[@sunnybarmes5512]

Maybe the barber doesn't shave... he simply grows his beard indefinitely?

[@javyahdavla5087]

What if the barber just lets his hair grow unrestricted?

[@MarxMuru]

Been trying to understand this and make correlation with barbers paradox- Just cant understand and comprehend it :(:(

[@jimkokko5]

Well, tell me what it is that you don't understand!

[@MarxMuru]

Well, why is the set of all sets a paradox in first place. A set can contain other sets is perfectly normal ?

[@jimkokko5]

Let me ask you this: Is it possible to have a box that contains EVERYTHING? The first answer that comes to mind is "well, yes, if the box is big enough.", but think about the fact that EVERYTHING includes the box itself! That means that you have to put the box in itself. As I've said in the video, this is impossible, because in order to prove it (in mathematics) you need to take as a fact the very thing you are trying to prove. Point being: you can't put EVERYTHING inside a box.

[@MarxMuru]

I guess i just dont get it..

[@MarxMuru]

too low IQ! :)

[@dnickaroo3574]

If the Barber of Seville happened to be a woman, then there is no Paradox. We assume that ALL the men can be assigned to either one of 2 distinct, non-intersecting sets. However, the Barber of Seville cannot be assigned to a set without a contradiction arising. The Barber is used is used in the definition of the sets; and this leads to the Paradox. If you consider ALL the men in Seville, EXCEPT for the BARBER, then no Paradox arises. Definitions which are self-referential will often lead to similar problems.

[@jimkokko5]

D Nickaroo The actual mathematical question is wether or not a set can contain all the sets there are. Don't get carried away by the story :)

[@dnickaroo3574]

Yes, that is what Russell's Paradox is really about: the assumption that there is a set of all sets leads to a contradiction.

[@rjravaz8283]

It might not be a paradox. Who said the barber was a man? It could be a women therefore the women would not need to save problem solved! (solved by RLV ) lol

[@csfi3979]

Why not say the existence of a "barber that shaves men who don't shave themselves" is impossible.. and there will be no paradox ... like the existence of "an unstoppable force and an unmovable object at the same time " .... or " a married bachelor" .. they are impossible to exist by definition.

[@jimkokko5]

+abdeljalil nachi Firstly, a "barber that shaves men who don't shave themselves" is a quite real and not at all impossible situation, contrary to "married bachelor". When someone goes to the barber for a shave, it is the barber who shaves him and not the man himself. Secondly, you're picking on details that have nothing to do with the actual paradox. You need to remember that the story serves one sole purpose: to "materialise" and give a different point of view on a mathematical (and thus theoretical) problem.

[@csfi3979]

[ a "barber that shaves all men who don't shave themselves" is a quite real and not at all impossible situation] ... i disagree, I think it's impossible to shave all men that don't shave themselves .. because the barber himself is a man.

[@jimkokko5]

abdeljalil nachi Now you're either approving my point or the paradox.

[@NeoHoshi]

+jimkokko5 Both.... and none.... It's Schrodinger! lol

[@jimkokko5]

Amineo YOU BROUGHT QUANTUM PHYSICS INTO THIS, YOU HAVE DOOMED US ALL

[@marclawson6144]

So no definition is absolutely 100% accurate. Wow. Is that an idea hot from the presses of Duh Magazine? So what if we can’t label things perfectly?

[@junkowl7539]

It says that he shaves the men that don't shave themselves. It didn't say that he ONLY shaves men that don't shave themselves

[@jimkokko5]

Oh, not again ;_; I've discussed this before with another guy in the comments...I didn't say the word ONLY, because the barber story serves as a generic representation of the paradox (for you to understand it better), rather than the definition of it. Towards the end of the video, I use the "boxes example" to define the paradox.

[@jimkokko5]

No. You just said that it doesn't contain itself.

[@chriskarampa]

Να φανταστω διαβασες το logicomix?

[@Itsant33]

I hate to not necessarily debunk the paradox but rather question it. Saying the barber can't shave himself makes no sense. It's just an analogy that doesn't work.

[@michael2974]

No, you're right. Barbers are not required to follow the rules of set theory.

[@Observe3]

But what about the existence of an unknown black swan, why is it that we can conclude that because we cannot currently prove things to be true then they must be wrong, i disagree with this method of falsification!

[@jimkokko5]

Could you be a little bit more specific? I don't see why this method isn't correct.

[@Observe3]

***** My point is that yes for a simple problem such as the barber problem you have presented, the existence of a the paradox means that it cannot be proven, however you cannot apply this method of falsification to all things, we need not conclude that something cannot be proven only because we do not currently have the tools to prove it, that would be like saying ...we know god doesnt exist, because we cant prove that he does exist

[@jimkokko5]

You're right; I'm not making myself clear in the end of the video. I meant to say: "Claiming that the paradox exists is false, because any way you try to prove it is true requires you to use as proof the same thing you are trying to prove. Therefore, the paradox does not exist."

[@Zephaniah0207]

The barber must pluck his beard hair by hair. This is truly a travesty of nature.

[@FatalNoogie]

This comment is a subset of all the words I said that are NOT in this comment. Therefore, if this comment is true, then it has no words. If it has no words, then it does not need any proof to be asserted as true, since it has no content. Therefore, I assert it as true. Therefore it has no words and you didn’t really just read this. ... Why are you still staring at this white patch of screen anyway?

[@user-kx7do4fh2j]

The barbar example is hilarious.

[@rockon4211]

It's not really a paradox due to the fact of that everyone can cut there own hair meaning that the barber can cut his hair

[@kudos4201]

The barber story is purely analogical and is utilised to define the logical parameters of the paradox as it is defined mathematically

[@johnl2448]

why add music to something you want to hear?!!!!

[@midiarennie6373]

the answer is in the question. the barber only shaves the men qho do not shave themselves. therefore the barber shaves himself because he is one of the men who shave themselves!

[@marusdod3685]

simple answer. the barber is bald

[@jimkokko5]

you're gonna lose it when you find out bald people can have beards

[@marusdod3685]

@@jimkokko5 a bald woman

[@jimkokko5]

@@marusdod3685 women can also have beards. Also a beard doesn’t have to be huge to be a beard. Also you’re maybe the millionth person to make that joke in this comment section, I guarantee you have everybody dead rn

[@nathanielhellerstein5871]

How about _two_ barbers? They shave each other, and between them shave all those, and only those, who do not shave themselves.

[@GnomicMaster]

Poor dear Bertie, all disembodied mind!!

[@johnraymma]

The barber isn't from that country

[@camuD4RTF4EZFEF4CRV34RF]

traduction ??

[@gastongr568]

all the men of the village ate barbers

[@Jeronus1]

How do we know the barber isn’t Amish? If he is he wouldn’t want to shave or be shaved.

[@shubhrajit2117]

So mathematicians discarded the universal set for this silly reason?!

[@jimkokko5]

They didn't "discard" the universal set, they found better definitions for set theory fundamentals.

[@averagejohnson3985]

Heisenberg for mathematicians

[@bash3471]

Why this is so dam hard

[@megawattapps]

The answer is he doesn't shave...

[@buddykidd5441]

I have a solution, the other men shave the barber, because the other men don't shave THEMSELVES it said nothing about shaving others, and he wouldn't be a barber because he doesn't charge the barber