Russell's Paradox - Tumblr Posts

kept thinking there was gonna be a joke about the glasses being diagonal, and then there wasn't one, and then that itself amused me

Michael Sheen: “full of edgy danger” 😇

@rainbowpopeworld ah yes the glasses being a joke about unpredictability makes sense, thanks! For the purposes of the paradox, I shall set that excellent explanation aside and assume that the only joke about the glasses was that there was no joke about the glasses. Here we go!

Russell's Paradox, glasses joke version:

1) I expected there to be a joke about the glasses

2) I saw no joke about the glasses

3) I considered the very lack of a joke to itself be a joke

4) Thus I do see a joke about the glasses

5) So now (2) is false. Which means that we can't say (3), which means (4) is false. But that means (2) is true again! Oh no!!

In short, there is a joke about the glasses if and only if there isn't a joke about the glasses. A paradox.*

Russell's Paradox, general version:

Define a "set" as a (possibly empty) collection of stuff. Also, sets are stuff.**

Define A as the set of all sets that do not contain themselves.

Does A contain A?

1) If A is in A, then A contains itself, so A is not in A.

2) If A is not in A, then A does not contain itself, so A is in A.

So A is in A if and only if A is not in A.

We broke math.

We fixed math by saying "sets are not allowed to contain themselves."

Which to me feels unsatisfying, but maybe there's more to it that I have yet to learn.

(Also, I've been trying to figure out how to define A such that the general version becomes the glasses joke version. Haven't found an entirely convincing one yet, so it's possible that the glasses joke paradox isn't exactly Russell's Paradox but has a similar vibe.)

*This is resolvable by changing the wording to "I expected him to refer to his glasses, but then he didn't, and thus he made a joke without actually referring to his glasses." So then (2) becomes "I saw no reference to glasses" and (4) is "I saw a joke about glasses" and there's no contradiction. But that's less fun.

**Indeed all stuff is sets, and can all be built from the empty set {}. Or at least that's one way of looking at math.

Michael Sheen: “full of edgy danger” 😇