Barber & Russell Paradoxes (History of Undecidability Part 2) - Computerphile
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The Barber Paradox: Professor Brailsford continues the history of undecidability.
History of Undecidability Part1: • Undecidability Tangent...
Turing & The Halting Problem: • Turing & The Halting P...
/ computerphile
/ computer_phile
This video was filmed and edited by Sean Riley.
Computer Science at the University of Nottingham: bit.ly/nottscomputer
Computerphile is a sister project to Brady Haran's Numberphile. See the full list of Brady's video projects at: bit.ly/bradychannels
You know it ain't tobacco in the pipe when a few guys are discussing the twoness of 2.
Dude .... like what if ... like .... 2 was just like .... the set of all subsets of ummm ... like ... the set .... of all subsets ... of like a strong ... limit ....... cardinal?
Man ....... you're stoned.
+Kd8OUR The strongest drug of all:
Crystal Math
@@metallsnubben 👏👏👏👏👏👏
it gets hard with two. One is easy. It's all alone and ever more will be so.
So you're questioning the tobacconess of their tobacco?
I can imagine Godel trolling everyone while saying :
'Problem?"
Wikipedia has a better description of Russel's paradox:
According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox.
that hit it home for me, although I have read it many times, this time it hit, thanks. although, he does a great job starting from prop 5, in the previous vid. be ez
How can a set not be a member of itself? All I can think of is an empty set, cause that's like the definition of emptiness.
So is it like a set(R) that would contain all of these (empty) sets is empty cause this something is filled with nothing thus it is nothing, but if it's nothing it can't contain anything even if what it contains is nothing, thus it's a paradox?
@@janisstrods4404 think about a set of all the squares in a plane. The set itself is not a square in a plane hence it is aset that is not member of itself.
A lot of set can be thought of in a similar fashion
Credits - wikipedia
@@janisstrods4404 the set of all things that begin "the set of..." - the set of all things you can think about - etc...
Thanks mate, now I get it
Consider a set of all the sets that have never been considered...
Wait now they're all gone.
The set of all sets that have never been considered, is not , as in of itself, a set that has never been considered. Therefore considering it does not modify its purpose as a container of unconsidered sets. Meow.
Then consider the set of all sets that have never been contained in any set which has been considered.
It's not easy being entertaining while talking about mathematical logic, but this video and the one before has managed to do that beautifully. More videos on these kinds of topics please :)
I could listen to this bloke all day, he explains so well, and so effortlessly.
I'd really love to hear more about Goedel's incompleteness theorem. I know this is only tangentially relevant to computer science but if you guys and Professor Brailsford have time I'd really love to see a full length video on that. I learned about it in the first year of my philosophy degree but I feel like I'd really love to hear professor Brailsford talk about it as he explains things so well (far better than any professor I've ever been taught by). Obviously if you don't have time that's fine but I was kinda sad that this video ended where it did and I'd love to hear a bit more about it
addictedtoflames read the book GEB
Goedel's incompleteness theorem is only tangentially relevant to CS? WTF??? Don't tell Alan Turing!
@@marcthatcher if you're one of those people who says CS isn't "using programming language" he's barely wrong.
Great video. These result are known for me for a long time but it was really interesting to hear how different mathematician reacted to them. Thank you very much for keeping this channel going.
I loved this series of videos
I'd have loved to have a professor like this guy teaching me this stuff! Great video!
" To me and you we'd be like wut? " literally laughed out loud. I love how mathematicians never forget how humbling math can be and just in one second and relentlessly take the position of a completely clueless person, that is their power.
Hofstadter's G.E.B. is one of the best books I've ever read. It's got a wonderful proof of Godel's imconpleteness theorem
A proof by construction, no less.
Brady, I would love to hear more about Gödels Incomepleteness Theorem - perhaps on numberphile?
Gödel is a critic hero: he is the destroyer of "objective" mathematics. A true philosopher.
Marcelo Siqueira Lima lol, nobody's destroyed objective mathematics mate
Gödel DESTROYS Frege, Russell, Whitehead and Hilbert with LOGIC and FACTS
@@MarceloSiqueiraLima_CdC well mate you don't know about Godel and doesn't understand incompleteness theorems
Or perhaps the barber lives outside of town.
+JalapenoStudios HEARD!
Write an addendum that allows the barber to shave himself.
I could listen to that voice for hours
This kind of stuff is of my favorite!
I learned about this in my first set theory class, what a wonderful exercise!
Ohhh, little bit sponsor! Little bits are awesome and would make for a great Christmas present to any parents out there :D
Thank you professor and Brady. Good things come to those who wait (I am a mathematician and love these videos) Do we get a part III :) thanks guys
interesting but complicated
I really enjoyed listening to the story from you
thank you
"The village barber will shave all those who do not shave themselves." You didn't say he will ONLY shave those who do not shave themselves.
It doesn't matter. If he shaves himself, he's still shaving himself, which means that he can shave himself so therefore he doesn't have to see the barber, but he is the barber so therefore he either must see the barber (himself) or grow a beard. It's not that he can only shave other people because the logic says so, it's just that he's not supposed to see a barber unless he can't shave himself that the problem is undecidable and therefore paradoxical...
That was the first thing to leap out at me, too.
I love reading Russell. Maybe not profoiund, but highly provocative with his logical analysis in the vein of Lewis Carroll.
beautiful video. thank you very much, love these kinds of vids. more please ^^
This reminds me of dividing by zero, or taking the square root of a negative number...
If you divide not by zero, but by numbers approaching zero, you get closer and closer to infinity, whatever that is.
If you represent the sqrt of -1 as i .. you get a complex number which represents an extension of dimension in a coordinate system.
What if this paradox is not a problem at all, what if it's a solution?
What if.. the undecidability is actually what causes existence to exist in the first place?
You are a good teacher
This is awesome :)
The statement describing Godel's theorem is actually wrong. Godel showed that if a first-order logic together with a set of axioms can derive a kind of simple arithmetic, then such an axiomatic system can express a statement which is not provable by the system. -- The video mistakenly claims that the system only has to be able to have "one thing derive from another" which is true of all axiomatic systems. But not all axiomatic systems are amenable to Godel's construction and in fact infinitely many such systems are decidable. "Decidable" means that every true statement expressible in that system has a proof in that system.
should say "express a true statement". sorry.
Ditto. Vanilla propositional logics, of which there are infinitely many, all have sound and complete proof systems. Propositional logic isn't theoretically interesting but it's important and one almost everyone is familiar with.
@@anon8109 It does not sound quite right either, unless you assign a specific meaning to "truth". Better, Gödel showed that such a theory cannot have one unique model. Either the system is inconsistent or incomplete. But this only applies to sufficiently simple systems. E.g. if you take every true statement of standard arithmetic as an axiom, your system will indeed be consistent and complete. The bummer is that we cannot know those axioms.
I have one correction to what you said about Goedel's incompleteness theorem. It's not that *any* system of axioms will necessarily be incomplete, because there are a lot of systems you can jot down on a table napkin which you can derive things from but that are complete, i.e. any logical sentence expressed in that system can be proven true or false within the system. Rather, it's that any system expressed in a particular mathematical language (set of symbols) which comprises a set of axioms that can be expressed in a finite number of those symbols *and* is sufficiently complex enough to encode natural number arithmetic will have valid logical sentences for which you can't derive a proof in that system.
Captivating
Okay, more serious issue - the question "Does barber shave himself?" isn't undecidable. For problem to be undecidable we actually need an answer to exist. For every decision problem there is an "algorithm" which gives us the answer: "If the answer is yes, return yes. Otherwise, return no". But which of these could possibly answer the question - the solution _does not exist_. This is why it's called a paradox.
Indeed. "Decidability" implies there is an actual valid decision to be made, which there isn't in the case of a paradox where there is a direct contradiction involved. The barber paradox is ultimately no different from saying "1 + 1 = 2 and 1 + 1 =/= 2". No such barber can exist in the first place so the question is meaningless.
My solution: We're speaking of transitive verbs, shave - praise - follow - paint - play badminton with - include in a set (whatever you want it doesn't matter) that have a logical tense. This kind of tense isn't necessarily overtly marked by any grammatical suffix like "-ed" in English, or by a word like "then" or "now" or "tomorrow". But it is indicated by the relative position of the verb. An artist paints all and only those who don't paint themselves. Each of the two occurrences of "paint" have a different logical tense since they occur at different times in this sequence of clauses, a difference that can be made explicit: An artist will paint tomorrow all and only those who didn't paint themselves yesterday. So if the artist didn't paint herself, then she will. And if she did, then she won't. Simple. Logical. The fallacy in the paradox consists in failure to differentiate with respect to logical tense.
I think Gödel's incompleteness theorems require as a hypothesis that the system be able to construct Robinson arithmetic (Peano arithmetic minus axiom schema of induction), not just derive anything at all. The incompleteness theorems do not apply to any proper fragment of Robinson arithmetic, such as one without multiplication.
You should talk about Zermelo-Fraenkel set theory.
Correct me if I'm wrong, but the Barber's paradox and the parallel lines are two distinct problems. In the first, who shaves the barber is not undecidable. Both options lead to a paradox, hence both are wrong. This is because the set of axioms (or as you call them, propositions), is inconsistent.
Whereas, in the parallel lines problem, the axiomatic is too weak to decide whether the statement is true or false, and the need for another axioms emerges.
The main difference is that in the first case, addition of axioms with never solve the problem, and in the last case it does.
Talking about the other form of the barber's axiom: The set problems. This was solved by creating a completely new theory, called set theory, which solves this problem, very vaguely said, by increasing the "dimension" of what is understood by a set, whenever you talk about a set that contains all sets. Hence it does not need to contain itself, since it is not called a set, but something else.
After reading a bit more about this subject, I can confidently say that the use of paradox and undecidable as synonyms is not at all correct, since, by Gödel's incompleteness theorems, they are indeed opposite: If a system is consistent, i.e. it does not allow paradoxes to be deduced from the use of logic and the axioms (e.g. the barber paradox is not consistent), it most be incomplete, i.e. it allows undecidability.
"OH NO... OHHHH NOOOOE" hahahah that was great :)
"This sentence is false."
The crux of the matter always comes down to self-referential properties, when something starts referring to itself in the middle of its propposition, proof, axiom or whatever you run into trouble.
Little Bits! I saw those at Maker Fair last year. They were pretty neat. I still think they should team up with Minecraft and do a redstone themed version.
More on Gödel, please!
Nice!
A small comment on the barber paradox. "The barber will shave all those who do not shave themselves". I.e: if you do not shave yourself, you must be shaved by the barber. But the law says nothing about the converse. This should allow the barber to shave himself then, since the law doesn't have anything to say about people who shave themselves?
Of course, the paradox arises again as soon as you demand "the barber will shave ONLY those who do not shave themselves".
Not that it pertains to this, but I'd thought I'd share it: I enjoy the omniscience paradox, which goes something like this; Can (insert omnipotent being here) create a stone that they cannot lift? (explanation below)
If the being can't, they're not truly omnipotent. If they can, they can't lift the stone and are still not truly omnipotent, or never were to begin with.
There are several scenarios which "solve", "argue" or otherwise avoid the paradox, which I'll let the people of the internet find for themselves, I just thought it was interesting, and useful should you happen to be atheist or the like :P
One way to formalize this argument is roughly like this:
A is given,
A implies anything.
Does A imply some C such that A does not imply C?
undecidable!
You could argue that the answer is no, because A clearly implies anything, so there can't exist such C. However, if you say that, then is "anything" really anything?, Not really, it is "anything that exists", which is strictly less than at least "anything imaginable".
You could argue that yes, A imples such a C, but then you have proven that A does not imply everything.
More to the point, neither of these arguments are contained in the theory described by the two sentences, so you could say that there's nothing wrong with "A is given; A implies anything", in a vacuum. However, it would be very hard to add any extra valid reasoning tools to that theory, since if by doing so you make that particular question decidable, then you've either weakened what the word "anything" means, or you've created a paradox.
I've found that when people are confronted with the formally contradictory behavior of omnipotence they tend to, without thinking they're doing so, weaken potency or they pull some nonsense about their deity not being bounded by logic full stop. If someone wanted to posit that there is a being with maximal potency, i.e. that being had as much power as a being could have, ok fine, but good luck actually figuring out what that is.
If someone wanted to say that the axioms of logic that apply to a deity shouldn't include the law of non-contradiciton AND they proposed a good axiomatic system for reasoning about them then I'd be interested to see it. But no one defending a deity really thinks about it that deeply and just say its absolutely inscrutable.
brocoli
That's also both a poetic and a logically consistent way to put it. We can ask for "anything" as by definition, we can only ask for things "that can possibly exist and are logically consistent". :)
I love the conversation you guys started :D and that you guys kept it to a debate and it didn't turn into a heated argument that starts a fight, there's already too much of that on the internet. In the end it truly comes down to semantics and definitions, i.e.: Can (deity) create a triangle without non-Euclidean geometry whereby the angles do not equal 180, etc, etc. It goes on and on. Personally, I'll just leave it to facts, logic, experiments, testing, and retesting, etc, to find out about the universe and the small corner of it that we live in. Thank you all :)
"Tempus Fugit" ~Time Flies
Wyatt G Thanks for noticing.
Remember, even scientists, mathematicians and others ask questions about the universe and deeper meanings within it. We do not need to give up on logic or facts to answer these questions.
Sometimes the universe surprises us with answers we cannot explain (such as the undetectability problem), or that even seem impossible at first. We should always look to improve our own understanding.
From this we can grow and lean. :)
My question would be whether the 'set of all sets not a member of themselves' is actually a "thing"; we can talk about a red blue, or a square circle but that doesn't mean that they're things, even imaginary things. And if it IS a thing does that mean things like the largest prime number are also things?
Great vid, it would be awesome if the next one were about non computable problems/functions, such as Wang tiles, predicting halt of game of life, kolmogorov complexity, busy beaver, and so on.Also, a further discussion on the impact of non computable functions/problems in many fields of knowledge.As well as, how they are connected to godel's theorem.
The question of who shaves the barber is not undecidable. The solution to the barber paradox is that the statement that the barber shaves only those who do not shave themselves is self-contradictory. The proof is that from this assumption you can derive that the barber shaves himself if and only if he does not shave himself.
The barber paradox is solved if there are two barbers. "The barber shaves only those who do not shave themselves." Each barber does not shave themselves, but may turn to each other to be shaved.
Have you tried moving item between sets to resolve source item issues? And can't you just make a model in which you have a perfect sphere sitting next to a ruler and you draw infinitely many lines the width apart won't they all technically all go on forever looping back in on themselves without any deviation as long as theres the distance is constant which when drawn with a ruler it would be. So where's the problem with the lines?
Please make the "LittleBits" logo clickable in the video, thanks :)
wait, who came up with the idea of sets first? gottlob frege or georg cantor?
+Aryan Arora Neither. The notion is fairly intuitive. I'd say that no one really came up with sets. The first successful attempt at axiomatizing it was by Zermelo.
Cantor
If you want to know more about undecidability, barber and russell paradoxes, there is a very famous, an legible book that covers this topic: Gödel, Escher, Bach
en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach
unfortunatelly the coupon doesn't work. I tried with both capital and small letters. It doesn't decrease the price by $20. (though it must be an error in the website because when I tried "numberphile" as coupon it wrote it doesn't know that coupon)
Just being picky here - "set" of all 2-element sets isn't actually a set - it's a proper class.
The original law says "You must be clean-shaven. If you cannot shave yourself, the barber will shave you."
Amended law: "If you are able, you must shave yourself. If you cannot, the barber will shave you."
The barber is able to shave himself, therefore he must.
I really don't like the way the Barber Paradox is worded. It's a way to illustrate Russell's Paradox in layman's terms, but it depends on the barber making up this nonsensical rule of shaving exactly the people who don't shave themselves.
I prefer the version with the indices that don't include themselves. After all, it's perfectly possible that an index of books might include itself, so you could make an index of exactly those indices that _don't_.
Now the question becomes whether that index should include itself or not which I always found quite easy to understand.
Even easier to understand if we say "I have a book with every book ever written compiled in it", then title that book "Every Book Ever Written Volume 1". We would have to make Volume 2 to contain a compilation with volume 1, as that is also a book! So we cannot contain "all books ever written" in on "book", we would keep needing new volumes.
There is an easy way out. We put "every book ever written" in a "library". Now we have no paradox. ;)
TechyBen No, that is not the same thing as Russell's or the Barber paradox. What you stated is not a paradox at all, and there is actually a whole branch of mathematics which likes to throw out the Axiom of Foundation, which states that there exists no set which is a member of itself.
Ah ok. Though are both not contradictions? Though being contradictions of different types?
TechyBen Neither are contradictions. A contradiction is when you have a statement which implies something already assumed to be true is false, but the barber paradox and Russell's paradox are just examples where certain logic statements can't be proven true nor false given a particular set of assumptions. Most mathematics you're probably familiar with is done within the parameters of the ZFC axiomatic system, which just gives as an axiom that sets cannot contain themselves as members. There's no reason you *have* to assume that though, and there are logically consistent systems which allow sets to contain themselves.
Philip No, that's not true. Both those paradoxes (which are actually one and the same) leads to contradictions. If you accept the axioms :
1) All men in town that doesn't shave themselves are shaved by the barber.
2) All men in town that aren't shaved by the barber shave themselves.
3) All men in town must be shaved clean.
4) The barber is a man of the town.
Demonstration that this theory is inconsistent :
3) and 4) => 5) The barber is shaved clean.
5) and principle of excluded third => 6) The barber shaves himself or is shaved by someone else.
But both this option leads to a contradiction.
Those laws cannot be enforced.
Calculus assumes that a function is infinitely reducible. That is were we to take any small portion of a function there would be contained in this portion and infinite number of smaller portions and so on and on and on forever. It assumes irreducibility. The question does the set containing all sets contain itself can be answered with infinite exapanability ie we can expand the set to ever increasing sets all the way to infinity. Calculus tends towards infinitely small (infinitely close to 0 without being 0) and this question tends towards infinity (infinitely close to containing everything without containing everything) . Consider also as you tend towards infinitesimally small you go down and down and down through sets of infinity carnality until you reach some bottom why could you not go the opposite direction and build up and up and up to reach the top from infinitesimal to infinite.
okay an obvious question then. What if there is another barber? Can barbers shave eachother?
Only if they're drunk.
Yes, having two barbers is a valid resolution of the paradoxe.
Yet another one would be to say that the barber actually does not live in or originate from the village (he's like a traveling barber), and hence he is not bound by the laws of the village, so he isn't prevented from shaving himself.
That will create new problems, like "To which barber do you go?". Sure you can now give a solution to that, but they may throw more problems on the table.
Garbaz Plot twist: The barber is a scientist that created a super secret syrum that will remove all facial hair permenantly.
Sandwich247 That will give you many many more problems, so not a nice solution, but a "possible" :D
New Axiom: All unproveable true statements are true.
For sets, to define a set you must have a rule or set of rules that can be used to determine whether or not a candidate for membership is or isn't a member. The proposed set of all sets that aren't members of themselves has no such rule so there is no such set.
I've saved mathematics, right?
Proposition: Within the laws of the village, the barber must be shaved. If he really is the only barber, then he will have to shave himself. The logical combination of, shaving himself and being the barber at the same time should transform the double "Yes" or "1" to the initial question to a logical superposition akin to quantumphysics
he is really intersting.
I don't think I enjoyed hearing someone talk that much since ever.
It's like the barber has dual nature. He is a citizen of the village and at the same time he is a barber. Kind of like dual nature of light.
can you do the dining philosophers problem
Ah yes. _"My system works perfectly well if you exclude all the things that break it."_ ..Technically correct.
Do a video on lambda calculus!
Please
Why was Hilbert angry with Gödel? Because he had identified with the mind that wanted to dominate reality. Mind being a part of reality cannot dominate its greater whole. Look around, or within, and check it out for yourself. Pause and be peaceful.
About male and female...
XX and XY
The variations are rare and that's it...
2:39 TRIGGERED
THEN WHO WAS PHONE?
Just means we are missing a factor, or one of our factors are variables.
Barber and villagers travel or live in different places.
I know it's all just semantics, but I would "solve" the barber paradox by separating roles and people; person who just happens to be a barber shaves himself; it doesn't mean he's acting as "the barber" at that point. I doubt this "solution" scales up to the set of all sets though.. =)
Clearly class Barber extends Person
For what's it worth, this actually kinda is the modern solution. You can construct sets of people, but by default you don't know if you can make sets based on roles. Sometimes you can, sometimes you can't. There is a class of all sets, but there is no set of all sets. Sets can't contain classes, but classes can contain elements.
Why not say that the barber is not a member of the set of shaveable people? --> Oh, he mentions that right after.
or A = the set of all sets that do not contain themselves, or are of this type
So just be careful when you define sets that rely on some rule which its own membership can influence
Russell's theory of types may be inelegant and artificial, but axiomatic set theories such as ZFC and NF are just as ad-hoc.
The job of barber rotates among the populace.
Dude, you are playing with the axioms à la Wittgenstein. Gödel was saying given what we have (the rules of the game, if you will), there are simply things that cannot be determined.
Why do I feel the incompleteness theorem as similar to the proof of infinite primes🤔
:D 8:25 - It's not a bug, it's a feature :D
If we can't prove a proposition, how do we know it is true?
3:48 ... what?
Agreed.
Where is History of Undecidability, Part 1?
How did I solve the barber paradox spelled like this:
in the village:
1)every male have to be shaved
2)barber shave those who does not shave himself
3)there is only one barber and he grows a beard
Who shave the barber?
(as you can see some of the ways "out of the box" have been already closed)
I solve it this way: every morning every man wakes up and decides to shave or to go to the barber. The barber have no option of somebody else shaving him, so he only left with the option to shave himself, because it's forbidden not to be shaved.
In my experience any paradox looses it's mystery and undecidability if dealt with practical ways. In some cases you discover, that given rules just create infinite loop, so in those cases there's no practical approach, you learn, that input data contradicts itself.
E.g. "if almighty god can create unmovable stone". The answer to that is: god can not interact with the system he create, he can only be almighty within that system, but he is not almighty within the system he belongs to.
the barber goes to a different village
WintersunForever
Nah, there is the solution within the borders of *one* village in my answer.
the barber has alopecia
The essence of twoness? Sounds like cologne for mathematicians
ok; i understood the other representations of undecidability quite well i believe but I FAIL at understanding why the barber could even be imagined to not shave it's self. how could the shaver not maintain it's own beard? could anyone help me by chance? i watched it three times to make sure i was not listening wrong.... why would the barber not shave him/her self when the need arised in that system?
wait...parallel lines meet?
It depends. For planes, no. But for lines on the surface of a spherical manifold they will. A "great circle" is a circle drawn on a sphere that have the same radius (roughly speaking). Additionally, if you drew one on a large enough sphere it would look like a straight line. If you have two straight lines (i.e. great circles) that are parallel according to a transversal line test at one point they would actually both intersect each other at two points and loop back on themselves.
parallel lines only have to be parallel at one point?
BariumCobaltNitrog3n For a typical plane you'd only have to guarantee that the lines have the same trajectory and each had at least one point not contained in the other line to also guarantee that the lines were parallel (as opposed to colinear or intersecting).
However, the generalization for other geometries fails.
It is the essence... of TUNAS!
GLUB GLUB
massive
*EVERYBODY, PRESS "2"!!!*
I claim that there is only 3 solution top this problem.
1: The barber don´t grow beard (woman, boy, cancer patient)
2: The law don't apply to the barber (amedment, don´t live in the vilage and so on)
3: Someone else shaves the barber (a other barber or someone in a other village)
Is this true, is it provable? Can someone answer?
If the Barber is required not to have a beard, beards are grown or not grown. Grown beards shaved are no longer beards. The barber doesn't have a beard if he doesn't grown one, or it is shaved.
The question we are trying to answer is who shaves the barber, in the case that he doesn't grow a beard or isn't required not to have a beard, no one shaves him (1 and 2), if he does grow a beard and is required to have it shaved, someone else must shave him. This covers all cases, so there are only two answers, no one or someone else.
***** Well, yes, if you try to answer the original question, that is true, but was thinking about the intermediate step, now may ways its possible to cheat the system with categories.
Now one or someone else is only the answer to how does it, its not a answer to how its done
The case of the barber may allow too many variables for a definitive answer, that's the problem with metaphores. With sets there's no doubt possible.
If we assume that there is a set x wich contains all sets that doesn't contain themselves then there is no way around a contradiction, because x either contains himself or not. If it does the we have x containing a set that does contain itself, contradicts x's definition, but if it doesn't contains itself then there is a set, x, that doesn't contain itself whil not being contained by x. Also contradicts x's definition.
Hence, we can't assume the existence of such a set x, we can't enforce that law. Same thing about the town, the laws cannot be enforced.
Finally! We got to Kurdle Gurdle! GEB-EGB! MU! And all that rot...
Maybe I'm missing something, but it seems like the barber interpretation is terribly misleading. Choosing to express the "paradox" in that way makes it very easy to brush off: it's simply an error of mistaking OR for XOR. Each man shaves himself OR is shaven by the barber. The paradox only arises when we think it means each man shaves himself XOR is shaven by the barber.
The barber checks the state of himself. He has not shaved himself. So he shaves himself. While he is shaving himself, he isn't checking his state. There is no "half shaven". A person is either clean-shaven or not. When he is finished, he checks the state of himself again. He is shaven (by himself). He does not need shaved, so he doesn't shave himself. Everything is just fine.
little bits are great, but a bit too pricy for my liking... I wish they had things like little bits but just made out of transistors and resistors and LEDs to make all logic gates... I would love such a kit...
Logician:) "Required to grow a beard" ...lol...very clever
What if there are 2 village barbers?
one of the other people in the village
Frege was really upset.
I think one day we will have to admit there is no one algorithm or way of thinking in any system that solves everything. It's akin to saying that a hammer is the only tool you'll ever need for any manual labor job. The truth is, no, the hammer has things it can do, and things that it cannot. So to solve the things it cannot, you use a different tool.
Who shaved the barber? Well some other citizen (not being the barber) shaves the barber. It won’t be professionally done but it is an easy way out of the situation but I haven’t found a solution to Russels paradox yet🤔💭😅
For the second way out of the Barber paradox, how does the barber become a barber? Everyone is required to be clean shaven and no one is born with a beard. So how does one become a bearded barber to begin with?
The idea of being shaved by a woman makes me shudder. In all innocence I submitted to the process at college for the benefit of a charity fundraiser. Each subject got a new disposable razor applied to their face and that's where it began to go wrong.
The idea had appeal at the time but the problem was that the razors shaved the bristles so closely that they removed a fine layer of skin as well. Not enough to quite bleed but enough to leave you seeping slowly and with red, stinging, crispy stripes all over your face later on. Nice one girls, cheers for that.
To be fair, they were inexperienced. Anyone, male or female, who's been licensed to cut hair has to also learn to shave facial hair. I've had my beard/mustache trimmed by a female haircutter once before with excellent results. In fact, I'm due for another tomorrow.
The barber is a prisoner for breaking the law. He shaves from inside his cell.
I didn't quite get the bit about things being "true but not proven". How can you know that something is true without proving it?
+TapetumLucidum Maybe one can assume it to be true, and it works fine, but one may never be able to prove it. Like the parallel lines axiom; if you assume it to be true, you get euclidean geometry. If you assume it to be false, you get non-euclidean geometries. It can be true for some geometries, but not proven true for all possible geometries. That's what I understood of it.
He was saying you can't cite a truth as evidence for the same truth. For example,
Guy: "Bush did 9/11."
Friend: "How do you know?"
Guy: "Because Bush did 9/11!"
*****
Well, that's not quite what he was saying, but he did prove that you HAVE to do that to something to get "started" on your mathematical system. I.E eventually, there will be something in your system which is only the case, because you knowingly or unknowingly set it to be that way in the start. So basically, you, as a mathematician, set Bush to do 9/11, and then went on from there ;)
+metallsnubben Well, that's true, but can you prove it?
+TapetumLucidum - Someone could witness a crime for example such as a beating and have it video recorded but when there is a trial for that criminal, the recording is confiscated and erased. That is an example of something that is known true but now cannot be proven.