Tuesday, January 13, 2015

Information underload

What do I mean by this, information underload? (I'm being influenced by Chaitin's essay from which I'm deriving this post). I mean that at times we are given information that seems to be constrained, but more is there to be seen. In other words, we must not quite break the rules as see around them.

Chaitin is discussing the foundations of math and gets to a famous paradox:
Anyway, the best known of these paradoxes is called the Russell paradox nowadays. You consider the set of all sets that are not members of themselves. And then you ask, ``Is this set a member of itself or not?'' If it is a member of itself, then it shouldn't be, and vice versa! It's like the barber in a small, remote town who shaves all the men in the town who don't shave themselves. That seems pretty reasonable, until you ask ``Does the barber shave himself?'' He shaves himself if and only if he doesn't shave himself, so he can't apply that rule to himself!
The barber shaves all men who don't shave themselves, or he wouldn't, in part, have a business. But that's just one facet, or constraint here. He also shaves himself, and by doing so, according to the setup, he must be shaving someone who doesn't shave himself...

But that's limited. We can also intuit that the barber, being the sole subject and object of this, is in a class by himself. That is, if we 'break' the rules. Is that cheating? Are there no exceptions?

The hinge, I think, is "all men" in this brief tale. Is it forgotten: "all men who" -- the qualification is there to be seen; it is another way. There needn't be contradiction.