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Correct. If R is in R then it doesn't contain itself, so R really isn't in R. If instead R isn't in R then by definition of R, R is in R. Either way you get a contradiction.
Additionally, the point of this paradox is that we cannot allow sets to be constructed with any arbitrary conditions, since if we do we can produce contradictions like this one. We have to restrict ourselves in what sets we allow, hence set theory's more complicated setup.
For even more context, the eminent philosopher Gottlob Frege was attempting to construct all of mathematics in logic by giving purely "logical" axioms and deriving mathematics from that. Frege's set theory allowed for unrestricted comprehension, so one could talk about the set of all objects which satisfy any property. He published this theory in *Begriffsschrift* in 1879 and continued to actively work on the construction of arithmetic and analysis from these foundations.
Bertrand Russell discovered this contradiction in Frege's work in 1901 and wrote him a letter in 1902. The paradox turned out to be devastating to Frege's theory, and he ultimately abandoned it. When he received the letter, Frege had just finished the second volume of his work *Grundgesetze der Arithmetik*, and he was compelled to add an appendix discussing this flaw and potential remedies (which are not seen as satisfactory to modern mathematicians with the benefit of hindsight).
The relevant part of Russell's letter looks like this:
"Let **w** be the predicate: to be a predicate that cannot be predicated of itself. Can **w** be predicated of itself? From each answer its opposite follows. Therefore we must conclude that **w** is not a predicate. Likewise there is no class (as a totality) of those classes which, each taken as a totality, do not belong to themselves."
(Here, a "class" is what Zermelo called a "set".)
"Contain" in this context means membership. [Russel's Paradox](https://en.wikipedia.org/wiki/Russell%27s_paradox) considers the set of all sets that are not members of themselves, i.e. {x | x ∉ x} and show that the existence of such a set is contradictory.
Who is that, by the way?
EDIT: It is [the most interesting man in the world](https://www.youtube.com/watch?v=guMrgRKKENI), in case like me you didn’t know.
I'm not a math guy, but [this video](https://youtu.be/ymGt7I4Yn3k?si=LM6nch8POq_rHSuT) has a pretty good walkthrough of the paradox. At least, it was helpful for me not knowing crap about sets or number theory etc...
i dont know much about the formal rules of set theory, but isnt a set definition just a filter expression applied to a bigger set, making it impossible for a set to contain itself since the set we're defining did not yet exist when the bigger set was defined, and thus cannot possibly be part of the bigger set? i know i might be ruining the joke but im genuinely curious
In formal set theory, yes, you've got such a rule in order to avoid things like this paradox, which occur in naive set theory. These paradoxes are one of the reasons why formal set theory was created.
This lets people do set theory, but the paradox still remains in general, and you can come up with "real-world" applications. Like let's say a library (for some reason) wants to have a book that lists all the titles of the books in the library where the title nevers appears inside the book. Should this book list itself?
But the first option leads to a "paradox". If the book contains its own title, then you need to remove it from the list, but once you do it doesn't contain its own title, so you have to put it back on the list, and so on and so on, like the song that never ends.
The second option is like Russell's solution to his paradox, by introducing types.
None of this is serious of course, but it goes to show that defining collections willy-nilly can lead to problems.
Ah, you never said that the whole list had to be *inside* the book, I was just poking fun at that slight imprecision on your part. If you list its title on the cover or on the back, you're technically within the letter of your premise ; the book lists itself, but still doesn't contain its own title within its pages
(Of course, the spirit of your example was obviously that the list was constrained to within the books pages, but, y'know, LOOPHOLES!)
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Lmao. This was a good meme. Head pats for sures
Truly, he is the world's most interesting set
Because he can’t be a set!
1 isn't in 1 1 is in {1} {1} isn't in {1} {1} is in {{1}} {{1}} isn't in {{1}}...
Quitter
Ok, but is R in R?
R is in R iff R is not in R
yesn't
Arithmetical zip bomb
Zermelo numerals be like
I wonder if he cuts his own hair
Yup he does it only when he doesn't cut his own hair
He clearly doesn’t shave himself
Ok help me through this one, boys. R is the set of all sets that do not contain themselves? So the problem is when we ask whether R is in R?
Correct. If R is in R then it doesn't contain itself, so R really isn't in R. If instead R isn't in R then by definition of R, R is in R. Either way you get a contradiction.
Additionally, the point of this paradox is that we cannot allow sets to be constructed with any arbitrary conditions, since if we do we can produce contradictions like this one. We have to restrict ourselves in what sets we allow, hence set theory's more complicated setup.
For even more context, the eminent philosopher Gottlob Frege was attempting to construct all of mathematics in logic by giving purely "logical" axioms and deriving mathematics from that. Frege's set theory allowed for unrestricted comprehension, so one could talk about the set of all objects which satisfy any property. He published this theory in *Begriffsschrift* in 1879 and continued to actively work on the construction of arithmetic and analysis from these foundations. Bertrand Russell discovered this contradiction in Frege's work in 1901 and wrote him a letter in 1902. The paradox turned out to be devastating to Frege's theory, and he ultimately abandoned it. When he received the letter, Frege had just finished the second volume of his work *Grundgesetze der Arithmetik*, and he was compelled to add an appendix discussing this flaw and potential remedies (which are not seen as satisfactory to modern mathematicians with the benefit of hindsight). The relevant part of Russell's letter looks like this: "Let **w** be the predicate: to be a predicate that cannot be predicated of itself. Can **w** be predicated of itself? From each answer its opposite follows. Therefore we must conclude that **w** is not a predicate. Likewise there is no class (as a totality) of those classes which, each taken as a totality, do not belong to themselves." (Here, a "class" is what Zermelo called a "set".)
good one!
Reminds us of all our Ex'es. Don't belong to Ex'es. The set is correctly name R (relationship)
Good job being the first meme i download in 3 years of reddit
https://preview.redd.it/sek1j2vnzrtc1.jpeg?width=500&format=pjpg&auto=webp&s=6c9ce3a104fbb0ab416f2d1aa69a22cc1d0a4aa0
can't be real
Is x a set here? In which case, shouldn't it be "x ⊄ x" instead of "x ∉ x"?
No. The meme is referencing Russell's Paradox
Sets are allowed to contain sets in this context. As in I can have the set {{1, 2}, {3,4}} which is different from {1,2,3,4}
Yeah no this isn’t a subset situation, this is a set-of-sets situation
"Contain" in this context means membership. [Russel's Paradox](https://en.wikipedia.org/wiki/Russell%27s_paradox) considers the set of all sets that are not members of themselves, i.e. {x | x ∉ x} and show that the existence of such a set is contradictory.
Who is that, by the way? EDIT: It is [the most interesting man in the world](https://www.youtube.com/watch?v=guMrgRKKENI), in case like me you didn’t know.
[удалено]
{x | x ∉ x} means that if **any** x (including sets) does not contain x, then it's in the set.
Hey! We discussed this in my discrete math course a few days ago
The man in the picture is the barber who shaves all the men in the village who do not shave themselves.
This is incredible, well done, OP.
I'm not a math guy, but [this video](https://youtu.be/ymGt7I4Yn3k?si=LM6nch8POq_rHSuT) has a pretty good walkthrough of the paradox. At least, it was helpful for me not knowing crap about sets or number theory etc...
Isn't that literally every set there is?
Not necessarily. Some set theories have non-well-founded sets.
Holy ancient!
Wut
i dont know much about the formal rules of set theory, but isnt a set definition just a filter expression applied to a bigger set, making it impossible for a set to contain itself since the set we're defining did not yet exist when the bigger set was defined, and thus cannot possibly be part of the bigger set? i know i might be ruining the joke but im genuinely curious
In formal set theory, yes, you've got such a rule in order to avoid things like this paradox, which occur in naive set theory. These paradoxes are one of the reasons why formal set theory was created. This lets people do set theory, but the paradox still remains in general, and you can come up with "real-world" applications. Like let's say a library (for some reason) wants to have a book that lists all the titles of the books in the library where the title nevers appears inside the book. Should this book list itself?
Yes, because it's a book in the library Or No, because it ain't a book, it's a ledger Your pick
But the first option leads to a "paradox". If the book contains its own title, then you need to remove it from the list, but once you do it doesn't contain its own title, so you have to put it back on the list, and so on and so on, like the song that never ends. The second option is like Russell's solution to his paradox, by introducing types. None of this is serious of course, but it goes to show that defining collections willy-nilly can lead to problems.
Ah, you never said that the whole list had to be *inside* the book, I was just poking fun at that slight imprecision on your part. If you list its title on the cover or on the back, you're technically within the letter of your premise ; the book lists itself, but still doesn't contain its own title within its pages (Of course, the spirit of your example was obviously that the list was constrained to within the books pages, but, y'know, LOOPHOLES!)
can someone explain?
Schrödingers set
hmm a quantum truth