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.
All Comments (21)

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.

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

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.

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!

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

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.

As a child I spent weeks writing "S, P, AO, Agent" and whatever else, under words for a language class (this was in a different country so abbreviations may not carry over)  its been 2 decades since, and today is the first time I have seen it used to explain something. It saved me 60, or maybe 90 seconds. Time well spent!

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!

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!

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, selfreferential system, if you dig deep enough.

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.

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.

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

Thank you for the brilliantly clear, insightful and extensive exposition of Russell's Paradox! Thank you too for not mentioning the dull, trite and deeply unhelpful 'Barber' analogy along the way either!

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….’

This reminds me of the "failure paradox" as well. In a nutshell; if one sets out with the goal to fail, then they can only succeed. Because if they fail then they succeeded at failing which invalidates the failure, but if they fail at failing then they succeeded at failing which is still a success.

Being told I don't have to remember certain things is surprisingly comforting

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

I've tried watching this twice now and I realise that I am a member of the set of people who don't care enough about Russell's Paradox to watch to the end.

As someone who has always sucked at math, I'm actually shocked that I pretty much understood everything you said.