Saturday, February 25, 2006

Russell's Paradox

For those who don't know the extent of my geekdom get ready.

I was introduced to Russell's Paradox as a kid in the form of Who Shaves The Barber?. It tells of a town where a barber shaves everybody in the town who does not shave himself. Most people in the town neatly fall into a category of either shaving themselves or not. But things get tricky when you try to categorize the barber. If it is considered that he does not shave himself then he would have to be shaved by the barber, himself, which contradicts.....

In set theory the paradox is stated as the set of all sets that do not contain themselves.

For my analysis we will use the definition of the set to make a system: a test set, and a validator. Each will have a yes or no value for contains self. If the test set has the 'contains self' value marked as 'yes', then the test set is a member of its proper set. If the validator is marked as 'yes' for 'contains self', then the test set must contain itself to be valid. This can be expressed in a table like this:





Contains Self

YesNo
ValidatorX
Test SetX

So the validator says the test set has to contain itself and it is marked as such. If only that were the end of it. Let's read the definition of the set again: the set of all sets that do not contain themselves. But the test set does contain itself, so it should not be part of the set. That is to say now our validator is not marked properly. So we can then mark the validator property of 'contains self' to 'no'. And we get this:








Contains Self

YesNo
Validator
X
Test SetX

But it can easily be seen that the test set is now invalid as it does not agree with the validator. But that can be easily fixed by making the test set not contain itself.








Contains Self

YesNo
Validator
X
Test Set
X

So now the set is valid, but is the validator? Again the definition: the set of all sets that do not contain themselves. Yeah, it has to be changed. Since the test set does not contain itself the validator should be marked as yes. And we get this:








Contains Self

YesNo
ValidatorX
Test Set
X

But test set has become invalid again so it has to be changed to no to make it valid like this:








Contains Self

YesNo
ValidatorX
Test SetX


And if you were paying attention you'll notice this is where we started.

To summarize the rule for the set is that it must match the validator. But the validator must not match the set. So the set chases the validator and the validator runs away giving us the four states we saw above, though never to settle on any one of them. It is a virtual perpetual motion machine. When I think about it this way it is strange to think of it as a paradox. Consider the set of all sets that do contain themselves. Is this not considered a paradox? If we did a similar experiment on this definition we would find that it could stabilize in either of two states (Either the first or third state above). If everything that cannot achieve a stable static state is a paradox where does that put, well, everything in the natural world that continuously needs energy and minerals and puts out waste. Is life a paradox? Perhaps, but I like to think not.

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Links: Russell's paradox

2 comments:

  1. hm, i like this entry.

    i get what you are saying for the most part, unforunately an insane lack of sleep is preventing me from actually absorbing it. i'll have to come back and re-read this and actually grasp it when this crazy weekend is over.
    my weekend has been overrun with remembering who has paid cover and catching everyone trying to sneak past me while i'm preoccupied with the honest ones who are paying cover and waiting for their change/stamp. not an easy task. i have so much more respect for people who work the door at bars. although, i've always had a respect for them. it's just on a different level now.
    wow. i'm even rambling in your comments. my apologies. i need to go nap now.

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  2. Some ideas are better sought with slight impairments. I suspect this might be one of them.

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