Mathematics - Russell's Paradox

In this video, I show you the basics around Russell's Paradox and how to overcome it. Enjoy:)

[boltfan]

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.

[random_content_generator]

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

[awesomedancer]

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

[lishlash]

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.

[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.

[smarvysmarv]

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!!!!

[glennhitzert]

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....

[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.

[nathanielhellerstein]

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.

[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?

[paulkomladarku]

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"

[bits]

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

[peterjones]

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.

[Γιώργος-ετ]

Thanks, I finally understand the paradox

[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.

[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.

[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.

[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.

[McGavel]

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