Russels Paradox

Super fact 111 : Russel’s Paradox is a logical contradiction discovered in 1901 that showed that the mathematical discipline of “Set Theory” was fundamentally flawed. Mathematicians had naively assumed that any definable property can be used to form a collection (or set) of items, but that is not true. An example of the Paradox is “A male barber shaves all men who do not shave themselves and only men who do not shave themselves. Does he shave himself?” Both “yes” and “no” are impossible answers. That is an example of an impossible set. Set theory needed an exclusion of such impossible sets.

Bearded client visiting barber shop. Barber scissors and straight razor, barber shop, suit. Vintage barber shop, shaving. Portrait bearded man. Mustache men. Brutal guy, scissors, straight razor. | Russels Paradox
Bearded client visiting barber shop. Shutterstock asset id: asset id: 1821348236 by Body Stock.

Russell’s paradox is a famous logical contradiction discovered by the philosopher and mathematician Bertrand Russell in 1901. To solve the contradiction, you need to remove the assumption that any property can form a set. In other words, not every set is possible. Basically, self-reference cannot be allowed.

To take the example above “A male barber who shaves all men who do not shave themselves and only men who do not shave themselves.” Is something that cannot exist. If the barber shaves himself then he is shaving someone who shaves himself, which was not allowed. If the barber does not shave himself, then he is not shaving all the men who do not shave themselves. Either way, it does not work. Such a barber cannot exist. In general, you cannot define a set anyway you like.

I consider this a super fact because it shows that contractions can be hidden even in mathematical disciplines, and it is important because you certainly don’t want contradictions hidden in a mathematical or scientific discipline. Contradictions lead to more contradictions and lots of problems.

It is a black and white photo of the mathematician Bertrand Russel.
Bertrand Russell portrait. Honourable Bertrand Russell.jpg: Photographer not identifiedderivative work: Conquistador, Public domain, via Wikimedia Commons

A Crazy Barber Story Involving Our Children

This happened soon after the September 11 attacks in 2001. In addition to planes crashing into buildings, there were attempts at biological warfare by spreading anthrax through the postal service. This is something we paid special attention to at my work because we were making postal sorting machines. It is also the reason I do not like people who write addresses in cursive.

Anyway, my wife called me at work, and she was very upset because our daughter’s hair was falling out. She touched her hair and it just fell off. She did not know what could be causing her hair to suddenly fall out, but she thought that it might have been biological warfare. I told her to call our doctor who had the good sense of suggesting that perhaps the kids had been playing barbershop. As it turned out they had. Our son confessed to cutting off our daughter’s hair. He had realized that this was bad, so he tried to put her hair back as well as he could. Afterwards, she was walking around with loose hair on top and that’s when my wife found her.

Was our son the barber who cut everyone’s hair and only those who did not cut their own hair? No because that barber can’t exist. Some sets can’t exist and you need to include that in the definition of what a set is, or in this case what kind of groups of barbers you can have.
Our son is cutting his sisters hair. The picture is generated with the help of ChatGPT.
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Author: thomasstigwikman

My name is Thomas Wikman. I am a software/robotics engineer with a background in physics. I am currently retired. I took early retirement. I am a dog lover, and especially a Leonberger lover, a home brewer, craft beer enthusiast, I’m learning French, and I am an avid reader. I live in Dallas, Texas, but I am originally from Sweden. I am married to Claudia, and we have three children. I have two blogs. The first feature the crazy adventures of our Leonberger Le Bronco von der Löwenhöhle as well as information on Leonbergers. The second blog, superfactful, feature information and facts I think are very interesting. With this blog I would like to create a list of facts that are accepted as true among the experts of the field and yet disputed amongst the public or highly surprising. These facts are special and in lieu of a better word I call them super-facts.

26 thoughts on “Russels Paradox”

  1. That’s a cute story about your kids, Thomas. The anthrax in the mail thing was quite worrying at the time. I recently listened to a podcast about it; as I recall, the primary suspect was a scientist who possibly had some mental health problems.

    I have to admit complete ignorance about set theory, so this post did remedy that to some extent!

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    1. Thank you so much Audrey. You are right about anthrax scare. I remember that in the end we found out it was likely a scientist with mental health problems. However, at the time we did not know that and at work we had to take all kinds if special precautions and come up with ideas for how to protect people from the possibility of spreading germs and viruses via mail.

      I just thought it was funny how we needed to pay attention to the fact that allowing a collection/set to be part of the collection/set itself could lead to a logical disaster and no one had seen it for several decades. Words like “all” and “only” can be dangerous.

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  3. I’ve heard the term “set theory” but knew nothing about it until now. Your barber analogy explained it well! 👌 That’s the second sad hair story I ‘ve read today. What is it about kids and haircuts? 😆

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  6. I’d not heard of this and it was interesting to think about. I like odd things like this. haha Your story of the kids and the hair cutting. Glad it was only them playing around and not something bad causing the hair to fall out. That was a scary time of wondering if mail had anthrax in it. 🙂

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    1. Thank you so much Barbara. It was big news among mathematicians back then and Gotlieb Frege, the leading mathematicians in the topic at the time, accepted it but got depressed. He had been doing this type of math wrong his whole life. You are right about the antrax Barbara. We who were making the mail sorting machines had to take it very seriously. It was panic.

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    1. It is a clever paradox and it overturned the Set Theory at the time and made the leading Set Theory mathematician, Gotlieb Frege, depressed. He accepted it, but it was with a big “damn it”. You are right, my wife panicked, and that was not fun, but it was short lived. I sort of panicked too. Why would a little girl’s hair suddenly fall off? However, I thought there must be another explanation. I just couldn’t thing of one when she called me at work. Oh boy, putting the hair back after cutting it off, that was sneaky.

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    1. Thank you so much Denise. Yes it is clever paradox that changed mathematics. What I did not mention is that at the time set theory was seen as the most fundamental of mathematics topics so having it overturned by a simple a paradox was big. It also effected logic and theology, self reference, and words like all and only are more dangerous than what was thought at the time. Kids playing barber is definitely something that can happen. Putting the hair back and people panicking about biological warfare adds something to it.

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