Russell’s Paradox

Appendix A tells the paradox of a barber, it might help someone with Russell’s Paradox.

Now, for Russell’s Paradox. Assume that we have learned enough to realize that a set that has itself as an element causes bad things that, mathematically disgust us.

A={A,B}

Nasty! That set A is just downright nasty!

Along come two other sets:

  • C={D,E}
  • F={G,H}

Hey, these sets aren’t nasty, they are clean! n=nasty, \neg n=clean

Let S be the set of all sets that are clean:

C is in S and F is in S

S={C,F}

Hey, S doesn’t include itself so it is clean. Well, if it is clean we need to put it in any listing of clean sets…

S={C,F,S}

But in that moment, S goes from clean to nasty.

  • S can’t be clean and nasty at the same time.
  • S can’t (be in) and (not be in) S at the same time

Appendix A

A barber shaves all those who don’t shave themselves. Then someone asks, well who shaves the barber?

Some people didn’t shave today, for different reasons, some of which we say “yeah, we understand”. The kind and sympathetic barber spends a couple of hours shaving those people, while wearing a mask.

Then someone, perhaps a person of malice, says “Hey, the barber didn’t shave himself this morning, right?”

We look at him. “Yeah, he isn’t shaven so he didn’t shave himself.”

“Well, the rule says, he shaves all people who don’t shave themselves, so he needs to put his name on that list of people he will shave.”

“But… but… the moment he starts shaving himself he is disqualified from being on the list. He can’t (not shave himself) and (shave himself) at the same time.

Another reference words it like this: “the barber is defined in a way that he shaves himself and does not shave himself.”

Possibly we might have worded it better to write it longer like this: “he shaves all people who don’t shave themselves and he only shaves people who don’t shave themselves.”

Right now I’m arguing that we don’t need that extra wording because when we say “he shave people who don’t shave themselves” we put him to work at a single task and exclude him from doing anything else. Your thoughts?

Actually, this idea of “if we only list one thing or one action, that’s the only thing or action that is true in our scenario” might be something that we want to turn the spotlight on it. There have been times in writing that I felt a need, after saying “A”, to explain that it was A and only A. We’ll put this on the back burner.