Theory of Everything - View Single Post

Guille · **Russell's Paradox -** 09-09-2005, 04:13 PM

This is one of those paradoxes that when you come along them, they make you think and think and think about them, wondering how and what lead to that point, and how to solve it. eno's paradoxes act equally.

Let me expose the paradox:

Imagen a set which doesn't contain itself. This means, that it isn't included in the description requiered for being part of itself. Now imagen a set which is the collection of all sets that are like our original set (i.e. that they don't contain themselves). Then, the question is: Does the set of all sets that do not contain themselves, contain itself? Let's say that it does contain itself. Then, it must be a set that contain itself, but, as it is the set of all sets that do not contain themselves, it must not contain itself. So, we come to the conclusion that it doesn't conain itself. But, if it doesn't contain itself, it is one of the sets that does not cotain itself, and so, it enters in the requierments to ente rin itself, and so, it now contains itself. This is the paradox.

It is much easier to understand with an example. Let's consider that set A is the set of all the sets that can be described with less than 10 words. I have described set A with 16 words, which is more than 10 words, and so, it doesn't enters the requiermnets of itself (being described with more than 10 words) and so it doesn't contains itself. Now, let's consider set R the set of all sets that don't contain themselves. Such as the set of all dogs (which isn't itself a dog) or the set previously described (set A). Then, doesn R contain itself? If it contains itself, it must not contain itself, and if it doesn't contain itself, it must contain itself.

Logically written, it can be reduced to:

R is the set of all set's that don't contain themselves:

R={x|x€/x}

Then

R€R<--->R€/R

R = Russell set
x = set that doesn't contain it self
€ = contains
€/ = doesn't contain
<---> = iff
| = which cover
{} = set

There have been many attempts to solve this paradox, and modern logic has fallen apart from it by inventing the concept of "type" in logic, and several other things that can be avoided if anothers olution is given.

Now,

What is your opinion about this paradox?
Do you think it's a paradox or not?
Do you have a solution for it?

	(#1 (permalink))
Guille The Thinker Status: Offline Posts: 3,278 Thanks Given: 14 Thanked 9x in 9 Posts Join Date: Mar 2005 Rep Power: 47	Russell's Paradox - 09-09-2005, 04:13 PM This is one of those paradoxes that when you come along them, they make you think and think and think about them, wondering how and what lead to that point, and how to solve it. eno's paradoxes act equally. Let me expose the paradox: Imagen a set which doesn't contain itself. This means, that it isn't included in the description requiered for being part of itself. Now imagen a set which is the collection of all sets that are like our original set (i.e. that they don't contain themselves). Then, the question is: Does the set of all sets that do not contain themselves, contain itself? Let's say that it does contain itself. Then, it must be a set that contain itself, but, as it is the set of all sets that do not contain themselves, it must not contain itself. So, we come to the conclusion that it doesn't conain itself. But, if it doesn't contain itself, it is one of the sets that does not cotain itself, and so, it enters in the requierments to ente rin itself, and so, it now contains itself. This is the paradox. It is much easier to understand with an example. Let's consider that set A is the set of all the sets that can be described with less than 10 words. I have described set A with 16 words, which is more than 10 words, and so, it doesn't enters the requiermnets of itself (being described with more than 10 words) and so it doesn't contains itself. Now, let's consider set R the set of all sets that don't contain themselves. Such as the set of all dogs (which isn't itself a dog) or the set previously described (set A). Then, doesn R contain itself? If it contains itself, it must not contain itself, and if it doesn't contain itself, it must contain itself. Logically written, it can be reduced to: R is the set of all set's that don't contain themselves: R={x\|x€/x} Then R€R<--->R€/R R = Russell set x = set that doesn't contain it self € = contains €/ = doesn't contain <---> = iff \| = which cover {} = set There have been many attempts to solve this paradox, and modern logic has fallen apart from it by inventing the concept of "type" in logic, and several other things that can be avoided if anothers olution is given. Now, What is your opinion about this paradox? Do you think it's a paradox or not? Do you have a solution for it?