Old 09-24-2001, 05:14 AM
scm1001 scm1001 is offline
Join Date: Jun 2001
Location: cambridge
Posts: 2,340
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?
Old 09-24-2001, 05:32 AM
Viscera Viscera is offline
Charter Member
Join Date: Apr 2000
Location: Norman, OK
Posts: 441
Could be wrong (probably am ), but what about someone unshaved, shaved the barver, then the barber shaved that person?

Old 09-24-2001, 06:33 AM
DMC DMC is offline
Charter Member
Join Date: Oct 2000
Location: Atlanta, GA
Posts: 3,680
Welcome to the Straight Dope.

You might find this relevant.
Old 09-24-2001, 06:59 AM
Schnitte Schnitte is offline
Join Date: Feb 2001
Location: Frankfurt, Germany
Posts: 3,467
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.
Old 09-24-2001, 07:12 AM
scm1001 scm1001 is offline
Join Date: Jun 2001
Location: cambridge
Posts: 2,340
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
Old 09-24-2001, 10:15 AM
Telemark Telemark is offline
Charter Member
Join Date: Apr 2000
Location: Again, Titletown
Posts: 20,434
No one shaves the barber, she doesn't need a shave.
Old 09-24-2001, 10:49 AM
Cabbage Cabbage is offline
Join Date: Sep 1999
Location: Radford, VA
Posts: 2,387
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:

1. 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.

2. 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.
...ebius sig. This is a moebius sig. This is a mo...
(sig line courtesy of WallyM7)
Old 09-24-2001, 10:55 AM
Mangetout Mangetout is offline
Charter Member
Join Date: May 2001
Location: England
Posts: 57,102
The barber is a woman and cannot grow a beard.


Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is Off
HTML code is Off

Forum Jump

All times are GMT -5. The time now is 09:47 AM.

Powered by vBulletin® Version 3.8.7
Copyright ©2000 - 2018, vBulletin Solutions, Inc.

Send questions for Cecil Adams to: cecil@straightdope.com

Send comments about this website to: webmaster@straightdope.com

Terms of Use / Privacy Policy

Advertise on the Straight Dope!
(Your direct line to thousands of the smartest, hippest people on the planet, plus a few total dipsticks.)

Publishers - interested in subscribing to the Straight Dope?
Write to: sdsubscriptions@chicagoreader.com.

Copyright 2018 STM Reader, LLC.

Copyright © 2017