Russell's Paradox

Back in 1902, mathematicians defined a set as follows:

A collection of distinct objects, where an object can be anything, including a set.

So, here are some sets under that definition:

{ 4, 5, 6 } { a, b, c, d } { 8, a, 🌍, “The concept of FOMO”, 🤔 }

But also:

A = { 0, 1, 2, A }

Where A is a set that has the element A, which is…itself.

For decades, mathematicians thought this was fine, until Bertrand Russell pointed out a paradox.

Bertrand Russell gave the label “extraordinary sets” to sets like A that refer to themselves. He called non-self-referential sets (like { 4, 5, 6}) “ordinary sets”. It follows that every set must be either:

  1. Extraordinary
  2. Ordinary

However! If you say that there is a set, S, that is a collection of all ordinary sets, is S ordinary or or is it extraordinary?

The answer is that it is neither, which is why this is a paradox. Here’s a verbal explanation that looks at the two cases.

  1. S is an ordinary set. Well, this can’t be right because if it “contains all ordinary sets”, and S is an ordinary set, then S contains S. If S contains S, S is a an extraordinary set.
  2. S is an extraordinary set. An extraordinary set contains itself. But S is the set of all ordinary sets. If it contains itself, then it contains an extraordinary set. So, it cannot be both an extraordinary set and the set of all ordinary sets.

So, yeah, it is neither.

I had a bit of trouble getting this to sink in. I made this face 🤔 for quite a while. I think it would have helped me to think of it this way:

There are two conditions that need to be met in order for it to NOT be a paradox.

  1. S has to contain all ordinary sets, as defined.
  2. You have to designate S to be either extraordinary or ordinary, and that designation has to hold up. (If you say it’s ordinary, it has to be ordinary.)

There are two things you can change to see if you can meet both of the above conditions.

  1. S can either contain itself or not contain itself.
  2. S can be designated as either extraordinary or ordinary.

Work it out for yourself

Below is a tool that:

  1. Lets you change the two things you can change. Click on “Ordinary” or “Extraordinary” to change the designation of S. Drag the “Reference to set S” inside and outside of the box that represents set S.
  2. Automaticaly evaluates the effects of your changes on the two conditions.

Set S

[ { 0, 1, 2 }, { a, b, c }, and all of the other ordinary sets ]
Designate S as:

Condition checkers

  • S contains all ordinary sets per definition of S:
  • NO
Reference to set S
(You can drag this.)

Go ahead and try out all of the possible combinations. I hope you’ll see that there’s no options that result in both the definition of S as the set of all ordinary sets being respected and S being either ordinary or extraordinary.

(Due to me not coming to this concept with fresh eyes, I can’t really know for sure whether or not this will help you understand Russell’s Paradox. Please email me and let me know if it does not or does!)

If you’re wondering what they did about this paradox, eight years after he introduced the paradox, Russell proposed redefining sets so that mathematicians could continue with their work. Sets were redefined so that they could not include themselves. Collections that included themselves are now called classes.

I learned about Russell’s Paradox from the excellent graph theory book Dots and Lines by Richard J. Trudeau, which is ~$5 used!