second question for the day
Just reading a book on impossibilities and it mentioned Russell’s paradox on the barber in town who shaved everyone who didn’t shave themselves. Problem - who shaved the barber?
I can’t see the problem. Obviously such a town or Barber can not exist. Its a bit like saying there is a town where 1=2. But 1 can’t equal two. Paradox!
Can anyone enlighten me to what was worrying dear Bertrand?
Could be wrong (probably am ;)), but what about someone unshaved, shaved the barver, then the barber shaved that person?
Vis
If the barber does not shave himself, he has to shave himself because the assumption is the barber shaves everyone in town who doesn’t shave himself - and only those guys. Thus, assuming he does not shave himself results in the consequence that he has to shave himself --> contradiction
If he does shave himself, he violated the rule that he shaves only those people who do not shave themselves - he just shaved someone who did indeed shave himself. Thus, we have another contradiction, and the paradox is the fact that whatever you do or assume, there will be always a contradiction that cannot be solved.
DMC
thanks for the link, it answered my problem, though I imagine I will spend the next few months trying to work out what the answer means
No one shaves the barber, she doesn’t need a shave.
Before Bertrand, set theory came basically from an intuitive approach–if you can describe some collection of objects, what the hell, we’ll call it a set. Bertrand’s paradox showed there’s a flaw in that, somewhere.
Consider the set S of all sets that don’t contain themselves. Does S contain itself? If it does, then, by definition of S, it doesn’t. If it doesn’t, then, again by definition of S, it does.
Clearly something’s wrong here, namely, the idea that anything can be a set. However, it’s not immediately clear on how to remedy this. Just what types of things should we not allow to be sets? It’s not at all trivial to determine that.
So the idea was that we should axiomatize set theory (and all of mathematics, since set theory is the foundation of all of mathematics). What we would really like to do is set up a collection of axioms that describe just what are (and, by default, are not) sets. Ideally, we’ll like to do this in a way such that:
The collection of axioms are consistent. In other words, we don’t get a paradox like Russell’s, which started this whole mess to begin with.
We don’t “lose” any mathematics. We want our collection of axioms to be strong enough to prove all the theorems of mathematics.
So we’re kind of looking for a balance, we want enough axioms so we actually have something worth investigating, but we don’t want so many axioms that they end up conflicting with each other and generating contradictions.
Various axiomatized set theories have come along, but the most common is ZFC set theory (named after Zermelo, Fraenkel, and the Axiom of Choice. In a roundabout way, it avoids Russell’s Paradox by simply not allowing sets to contain themselves.
So were 1. and 2. achieved? Well, one of the things important in mathematics is that we want to be able to study infinite sets (like the integers or the real numbers, for examples). ZFC allows for there to be infinite sets, but once you introduce infinite sets, things get a lot screwy.
In the 30’s, Kurt Goedel proved that any axiomatic system that includes the natural numbers will have two problems:
a. It can’t prove that itself is consistent (which shot 1. down). The page DMC linked to is inaccurate here. Goedel didn’t prove that number theory (and so on) is inconsistent, he proved that it cannot prove its own consistency. Of course, that still leaves it wide open on whether it actually i consistent or inconsistent.
b. There will be undecidable statements. You got a “true/false” question about the natural numbers you’ve been wondering about? You figure the answer can be found using mathematics? Not necessarily; it could an undecidable statement, meaning using logic and the axioms of ZFC, it’s impossible to prove your statement to be true or false.
The barber is a woman and cannot grow a beard.