Russell's Paradox - a simple explanation of a profound problem

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This is a video lecture explaining Russell's Paradox. At the very heart of logic and mathematics, there is a paradox that has yet to be resolved. It was discovered by the mathematician and philosopher, Bertrand Russell, in 1901. In this talk, Professor Jeffrey Kaplan teaches you the basics of set theory (a foundational branch of mathematics dating back to the 1870s) in 20 minutes. Then he explains Russell’s Paradox, which is quite a thrilling thing if you are learning it for the first time. Finally, Kaplan argues that the paradox goes even deeper than Russell himself realized.

Also, I should mention Georg Cantor, Gotlob Frege, Logicism, and Zermelo–Fraenkel set theory in this description for keyword search reasons.

[nyc-exile]

My teacher told me that "all rules have exceptions" and I told her that that meant that there are rules that don't have exceptions. Because if "all rules have exceptions" is a rule then it must have an exception that contradicts it.

[louismartin]

I started reading Russel’s “the limits of the human mind” and I found out mine lasted one paragraph.

[scottmitchell]

This paradox is one of many paradoxs in a set of known paradoxes.
That make up the set of all paradoxes.

[jimstack]

At my age (77), I am not going to wade through 18, 643 comments to check if someone else has made the same comment as I am making here! I apologise in advance, however, if that is, in fact, the case.

When I first came across Russell's Paradox, more than 50 years ago, I explained it to myself as follows: if A is a set, then A is not the same thing as {A}, the set containing A. A set, in short, cannot be a member of itself, and the Paradox arises because the erroneous assumption is being made that a set can be a member of itself - your Rule 11.

On the few occasions in the last 50 years when I have thought about this again, I have come to the same conclusion.

I concur with the other comments about the quality of your presentation. Well done!

[KittchenSink]

For a 57 year old man who cannot even recite his times tables (my head just doesn't do maths), I'm stunned I actually followed that, I really did!!
That speaks volumes about this guys ability to convey information. I applaud you Sir, especially for the ability to hold my attention for the entire video. I quite enjoyed that!! I've no idea what use it is to me personally, but it was fascinating!

[joshwah]

I asked my girlfriend if we could have sets and she told me no because I didn't contain myself.

[kingfisher]

I have no interest in mathematics and no advanced training in mathematics, but i can follow the concepts --and more to the point - I love listening to characters who love what they do, and Jeff, you are a fascinating character. And that is a compliment.

[priscillawrites]

When I was in 7th grade, we were taught set theory in math class (yes, an advanced level geek class). The set theory we were taught included ‘a set cannot contain itself.’ Yale University wrote our curriculum.

Shrödinger’s veterinarian walked into the waiting room and said to Shrödinger ‘I have good news and bad news….’

[alexanderthegray]

I really didn’t expect LeBron James to be so crucial to the fundamentals of set theory. What a legend.

[jonathanbenton]

Unlike many of your commenters, I don't have anything pithy to say about your presentation. I had never heard of Russell's Paradox or anyone else's Paradox. All I can do is tell you how much I appreciate how you described it. I did have to go back and review a couple of sections near the end, but I got it!

You are passionate about sharing your knowledge with everyone who cares to learn. Even, and perhaps especially, people incarcerated in prisons. You are a gifted teacher, so thank you for sharing your knowledge with ALL of us.

[gm]

Russell was so excited with his discovery, he just couldn't contain himself.

[MartinCohen-yevo]

About 20 years ago I wrote a book about this (and other) paradoxes called 101 Philosophy Problems. It's really not complicated. See the tale of the Barber - given sole responsibility to shave everyone in the village EXCEPT those who normally shave themselves - but who will shave the Barber? However, Jeffrey is right that SOLUTIONS to it create new problems about how we both talk and think about the world. People - philosophers! - even say things like "such a barber cannot exist… Put another, way, the cures are worse than the disease. The problem for Frege and also Russell (as he mentions) is that it shows the limits of maths and logic. The more intriguing problem is that it shows the limits of how we think.

[HxTurtle]

I speak German and understand the letter Russell wrote to his colleague. the level of confidence he put into his writing that his recipient will just understand him amazes me.

[UnbornLifeMatters]

Never thought I could have such an enjoyable time watching a 30 min video on advanced mathematical theory. I chuckled and even laughed multiple times. Well done sir

[jeannes.]

I recently discovered your channel, and it has helped me a lot to understand philosophical thoughts. I am a philosophy major studying in Southeast Asia. We learn things in our own language, which is Thai. I have to say, this can cause a lot of misunderstandings, so I try to approach them in English instead. I’m not a native English speaker and sometimes I have to translate word by word, but I think it’s much easier to understand than in my own language. Thank you so much, Professor

[Faladrin]

No paradox. The description doesn't matter per rule 5. {x: x is the set of sets that do not contain themselves} -well, the whole "x is the set of sets that do not contain themselves" is just a description. It might tell you a human what the set is, but it is just a description. What is actually in the set though is what makes up the set (rule 2). Human's being unable to make a set that accurately maps to that description is our problem and is a language problem, it's not a problem with sets or math.

And if you're confused, this is exactly the same as what he means when he says '4' is not the number four. It is a symbol that we use to represent the number four. The set notation is itself a symbol for a thing (perhaps real or perhaps only in our minds), but it is not the thing itself. The "set of all sets" is a specific set, but the words "set of all sets" is just a descriptor which per rule 5 doesn't matter. In this case the descriptor does match the actual set it is attached to, but it doesn't matter if it does or not.

So, the "set of all sets that does not contain themselves" can be a real set, but it can't be true to it's description, but that's fine by rule 5. If you really want to define that set and have a descriptor that works though, you can do it.

y={x: x is the set of all sets that do not contain themselves, y}
or if you'd rather
z={x:x is the set of sets that do not contain themselves except for z}

So you get to choose. y and z here are two different sets because their membership is different, and they both work and are true to their descriptor. The descriptors do not matter.

[KaiserAegis]

As soon as you got to explaining to the paradox, I knew exactly what the issue was because it's conceptually identical to several other paradoxes I've studied, including the Liar's Paradox and the Grandfather Paradox. I've noticed that this sort of problem tends to arise in almost any kind of abstract, self-referential system, if you dig deep enough.

[dqpwczx]

Being told I don't have to remember certain things is surprisingly comforting. What a wonderful video. I truly enjoyed your presentation of Russell’s paradox..

[chrisjackson]

The way this is explained by Jeffrey is massively engaging. Sometimes my thoughts were a little ahead of him which was really satisfying when he confirmed my thoughts. Other times I was having to play catch up, which again was gratifying when things clicked into place after his explanation.

[maeog]

As someone who has never been good at math and gets anxious at basic addition and multiplication, thank you. You explained everything in a way that was quick, easy to understand and actually giving me a time frame on how long it will take you to explain something and giving the sort of cliff notes was really awesome. Literally every time you said, “don’t worry, you won’t need to remember that” I felt relief. And I actually learned something without feeling fucking dumb as bricks lol came for the philosophy, stayed for your awesome way of educating!