Sets, Classes, and Russell's Paradox (Axiomatic Set Theory)

A formal explication of Russell's paradox and why it is a problem for axiomatic set theory.

This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.

Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!

Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)

[ahmedbellankas]

Thank you my dear philosopher, that was a great video and a good explanation.

[SigSelect]

Great explanations. Love the tone of the video as well, such a seriously rigorous topic goes well with a lighter tone.

[mohammedshoaib]

carneades, please create a playlist teaching the foundational mathematics for such videos. I am not able to understand your formulas

[MathCuriousity]

Hey kind being: may I pose a question: let’s say we have an equivalence relation aRb. Why can’t I represent this within set theory as set T comprising subset of Cartesian product of a and b, mapped to a set U which contains true or false? Thanks so much!!

[mothernature]

Here an example. If you have the number 4, 4 can be considered a number but also a set of possible ways to write 4. So 4 is 2×2. 4 is 4*1. 4 is 1+1+1+1. For is 3+1 and so on. And 4 is just 4.

[calebmonaco]

Can there be a class of all classes and sets and members?🤔

[gutzimmumdo]

This is my attempt to understand the paradox;

A = {S, B, ...}
The elements of set A are sets (S, B, ...)
A is also a set, so it could be inside the members of A, like this;
A = {A, S, B, ....}
The members of A share the following property (wich is the property they must satisfy to be elements of the set A):
"The sets that dont contain themselves"

So (S, B, ...) cannot be sets that contain themselves, like this:
S = {S, C, ...} S (the first element of the set S) is S itself, wich means that S contains itself, this cannot happend because of the previous property
("The sets that dont contain themselves") if we want S to be an element of A.
So, we know A is a set, and sinse A is a set wich elements are also Sets, then we could theoretically put A inside A, like this:
A ={A, S, B, ...}
But by the property of A ("The sets that dont contain themselves") this A ={A, S, B, ...} cannot happend, because A is inside A, therefore A is a set that contains itself wich violates the property ("The sets that dont contain themselves"), so, lets remove it and see what happends:
A ={A, S, B, ...}
A = {S, B, ...} removed A from Set A
but now, the set A is a set that doesnt contain itself sinse i removed A from A ={A, S, B, ...} leaving me this set A = {S, B, ...}
So now, the set A is a set that doesnt contain itself (wich satisfies the property of A, "The sets that dont contain themselves" so its now an element of A), therefore, i can put it back in A:
A ={A, S, B, ...} but wait, now A contains A... therefore, i must remove it again sinse A contains itself, it violates the property "The sets that dont contain themselves"
A = {S, B, ...} ...., now A can be put inside A again sinse is a set that doesnt contain itself... but then i have to remove it again...
Weirdchamp cringe

[martinhawrylkiewicz]

Great vid! Just wondering how would one prove that Russell class {x| x doesn't belong to x} is a proper class... My intuition tells me prove this by contradiction, assuming that Russell class is not a proper class, therefore it is a set....?

[mothernature]

What if you have a set of numbers that contains all possible sets of number (odd, even, real, unreal, etc) wouldnt the set of all possible numbers be a set initself?

[ToriKo_]

How did *you* learn about this whole set thing?

[Daetelus]

I don't feel like you actually explained how Russell's paradox poses a problem for set theory (axiomatic or otherwise). The set simply doesn't exist. What's the problem?

[Mtmonaghan]

Thank God for such a paradox. These positives were out to proof that numbers could be define with out appealing to the contingency of anything existent. This paradox is proof you can’t. There must be existence first, before the number 1 can be said to make sense. Something must be in a state of existence. So the primary question is, what is the meaning of Being. Read Heidegger, you find you will need many aspects of your existence, such as the relevance of poetry, art and many other ways of understanding, other than simple dead rigorous logic.

[ConceptHut]

Or the barber is outside of the set that is stated in which the barber shaves.

[jamestagge]

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.

[wprandall]

You might be surprised that the riddle Russel gave to Hege was actually illogical from a practical point of view. There was no reason for Hege to have a breakdown and die. Hege failed to see that if there is no answer to a riddle, then the riddle or formula has a problem. It does have a logical answer, but did Russell know this?


You see: everything in the universe has a reason for existing. Likewise, every problem has a solution. If not, then the problem is the problem, so to speak. There are problems with Russels riddle. First, a set of all sets is not a set at all. We are used to thinking of sets as circles with space inside, into which we place objects. They are actually just concepts with a heading. You should have at least two items to qualify a set. A set with no items is not actually a set, and has no meaning at all. Also, a set containing only sets is not a set at all either. It has no specific items. The items are already in sets, and don't need further defining. You must consider the sets themselves to be objects. But that to me seems to confuse things. It is contains sets, and then objects, there is an inconsistency.


As to the sets being "members of themselves", there is also a problem. Can a person be a "member of their self"? Actually - no. A "member" is a part of a group. I am not part of myself. I am ALL of myself. I can join a group, but not if I am the only member. I have in that case joined nothing. It is a redundancy. So Russel's description of "not members of themselves" is redundant and useless, as well as confusing. This is unless you say that a "part" can be 100%, and the group is 0%. Then the whole is a part, even though the terms are opposite.


Due to the above reasoning, Russell's last sentence is totally unreasonable. The set of all things is not there to begin with, and the sets that do exist ARE themselves already. So the answer is: the set of all sets (if it existed) does not have to become a member. It is already itself. Likewise, it is not a member of itself partially; just like all the other sets. So it is and is not a member of itself at the same time.


UPDATE: The conclusion to all this is that Yes - The set of sets (if you consider that it exists at all) can be a member of the sets who are not members of themselves because it is NOT a member of itself. It is ALL of itself. You can also say that it can't be a member of sets who are not members of themselves because it is 100% of itself and 0% of the group. Conceptually this could be accurate. (This might give some credibility to the idea of the empty set.) But 100% IS all, so it is both at the same time. However, as I said, the "set of sets" by itself doesn't really exist. If it does, the question becomes "can a set category be a member of the objects in the set?" No, it can't. But if the objects are sets, then it can be a member by that logic alone.

Of course the set names must be congruent in description, (I would think) and this brings us back to everything I said before. The answer to Bertrand Russel's unanswerable question that killed Hege is YES - NO - and therefore YES.

[wprandall]

I solved this puzzle. It is in basically three parts. My first position is that "a set of sets" doesn't exist. It has no objects. Sets aren't objects. But if given the benefit of the doubt, it can still be solved. I wrote the entire answer on another video comment section. At least I really believe I have solved it. Let me see if I can find it.
Has anyone else claimed to have found the answer? Is it known?

[jamestagge]

This makes no sense from the outset. It is ridiculous that anyone insists that it is valid, that mathematical formulae might proceed from it. If you are going to have a platform from which to test certain mathematical principles, formulate one that isn’t so sophomorically clumsy.
In any context that is defined by those concepts and principles employed, there is present a logic to its structure or architecture. This comes from the meanings of those used and the terms used to articulate them. Many have insisted that with regard to this it is simplistic as per the literal instructions of the sign, disregarding the implications of the logic implied. I am trying to say that the circumstances outlined by the context as defined and that which the sign instructs have to be seen in sympathy with each other. The sign’s instructions have no meaning otherwise. If you are going to talk about there being a barber in the town, you must respect that which he does and thus his function in the definition of the paradox. You cannot ignore the necessary relationships in a one dimensional interpretation of the instructions of the sign. If he is a barber, he shaves beards. If one of the sets of men does not cut their own hair, then they must come to him for that purpose, a logical understanding UNLESS you specify some alternative which none has, which the author did not, for my point is the only way his (the author’s) is permitted to be expressed. So, logically, if the men who shave themselves do that, they have no need of the barber which is what makes them unique, as per that set. Continuing on, the set of men who are shaved by someone else, which is by definition, the barber, or that alternative would have at least been alluded to by the author and was not. Correspondingly, the sign states that the barber shaves all those who do not shave themselves. That then is the end of it. So….we have two sets of men, i.e., those who shave themselves and those who are shaved. That the one shaves themselves and that the other is shaved, IS that which differentiates the two sets for in all other aspects, they are identical, e.g., they are all men, they all live in the town and they all must be shaved to abide by the law. In that this latter list of characteristics is the same for both when the former is not, by logic we know that the former is the critical factor which is the cause which separates them into two different sets. Also, by logic we know that if there were not two distinct sets of men, there would be no paradox. By the same necessary logic then, the critical factor which sets him (the barber) apart from the other two sets is that he shaves others. If you insist on denying this then you by that, deny that the other two sets exist as separate for that which separates them is the one characteristic which is not shared. So too, logically, with the barber. If you then make him a member of either set which you must do in order for the paradox to exist, i.e., that there might be this contradiction by which he cannot act to shave himself, you do so by a means which then nullifies the difference between the set of men who are shaved and the set who shave themselves. Either way you have no paradox.
The logic of the context in which the paradox is defined imposes itself via the concepts expressed and the meanings of the terms used in their expression. There is no escaping that. This is not a fascinating or mysterious or complex thought experiment. It’s really rather one dimensional. I simply don’t get the astonishment of all of you who try so hard to defend this construction. And this Fegel guy, or whatever his name was, couldn’t have been all that bright to have allowed this to have caused him such misery.

[ghirardellichocolate]

Like auditing taxes? Barber is not a member of the community, so he must file the taxes. Tax scams. A person employed by a corporation and also self employed decides to file the taxes. If he makes lots of money, why is he employed by another corporation - for safety reasons, in case he loses his job. While Government respects his will, someone can become suspicious of it, if they know you are also employed elsewhere. Imagine if the person is also on Welfare...

[johnsmith]

Spoken explanation is not your forte'.