[Guille]

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?

[michellemfry]

Can you try to substitute the words universe for set and universes for sets. Perhaps you will see a new slant on an old problem. Does the universe contain the universe or does it contain all universes as given by the universe of all universes?

[Guille]

Originally Posted by michellemfry Can you try to substitute the words universe for set and universes for sets. Perhaps you will see a new slant on an old problem. Does the universe contain the universe or does it contain all universes as given by the universe of all universes?

Yes, it's a good analogy. But do you mean universes as all the different examples if this same universe, or to you refer to really different universes all?

[michellemfry]

I'm not certain, I'm still in the Paradox. I'm not trying to be funny, but it kind of feels amusing.

[michellemfry]

To paraphrase: Imagine a universe which doesn't contain a universe. Now imagine a universe which is the collection of all universes that are like our original universe(i.e.that they don't contain universes). Does the universe of all universes, that do not contain universes, contain a universe?

My answer was yes. A universe which doesn't contain a universe is what existed before the Big Bang.

[harmonygirl]

this is discussed in Godel Escher Bach...

[Guille]

Originally Posted by harmonygirl this is discussed in Godel Escher Bach...

Great, I'm more convinced to buy the book every day. Does the author conclude anything which is interesting about this?

[harmonygirl]

right now I am at the place where he is noting the varios paradoxes of Bach's fugues and Escher's art...no conclusions yet...

[zeroca]

Originally Posted by <<>> 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.

Hello,
It was a consideration of my leisure time. I don’t pretend it to be right, but enjoy it:
The main idea of this post is: if we had a set, which contains itself and at the same time doesn’t contain itself, we would solve the paradox at once!
In spite of the fact that I studied in physical-mathematical school 31 years ago, I found that my knowledge in higher mathematics has vanished completely. But anyway I consulted the site - http://en.wikipedia.org/wiki/Russell%27s_paradox for several minutes to copy some extracts - definitions:
Black bold – Just the term, definition of which is given.
Red bold –the term, definition of which is given below.

1. A set can be thought of as any collection of distinct things considered as a whole.
A. Collection is any group of items that has one or more properties in common.
B. Holism is the idea that the properties of a system cannot be determined or explained by the sum of its components alone.
4. A system is an assemblage of related elements comprising a whole, such that each element may be seen to be a part of that whole in some sense.
5. The term, element, means "a constituent part".
6. Empty set is the unique set which contains no elements.

I think we must seek the essence of paradox within definition of set itself:
1. A set can be thought of as any collection of distinct things considered as a whole.
2. Empty set is the unique set which contains no elements.

The main question: do we have the right to make two definitions, which are mutually exclusive? (See below)
----Empty set is defined as a set, despite the fact that it contains no elements and that contradicts the definition of set, which generally must be collection of distinct things considered as a whole (i.e. any set must be collection of elements, i.e. we admitted, allowed the arising of paradox at the starting stage of definition).

----If we have the right to make two definitions, which are mutually exclusive, then let’s define a new set - binary set, which is the set, which contains itself and at the same time doesn’t contain itself. Don’t ask, please, which set could it be?!
O.k., let’s take again empty set, i.e. a set, which contains no elements. Does it contain empty set? Yes, it does, as empty set doesn’t contain elements, so empty set contains infinity number of empty sets, because they can be summed up to one empty set, which contains no element i.e. if any set contains infinite number of elements and at the same time doesn’t contain any element (i.e. empty set in our case), is just binary set, which is the set, which contains itself and at the same time doesn’t contain itself.

[subversion]

Originally Posted by zeroca Hello, It was a consideration of my leisure time. I don’t pretend it to be right, but enjoy it: The main idea of this post is: if we had a set, which contains itself and at the same time doesn’t contain itself, we would solve the paradox at once! In spite of the fact that I studied in physical-mathematical school 31 years ago, I found that my knowledge in higher mathematics has vanished completely. But anyway I consulted the site - http://en.wikipedia.org/wiki/Russell%27s_paradox for several minutes to copy some extracts - definitions: Black bold – Just the term, definition of which is given. Red bold –the term, definition of which is given below. 1. A set can be thought of as any collection of distinct things considered as a whole. A. Collection is any group of items that has one or more properties in common. B. Holism is the idea that the properties of a system cannot be determined or explained by the sum of its components alone. 4. A system is an assemblage of related elements comprising a whole, such that each element may be seen to be a part of that whole in some sense. 5. The term, element, means "a constituent part". 6. Empty set is the unique set which contains no elements. I think we must seek the essence of paradox within definition of set itself: 1. A set can be thought of as any collection of distinct things considered as a whole. 2. Empty set is the unique set which contains no elements. The main question: do we have the right to make two definitions, which are mutually exclusive? (See below) ----Empty set is defined as a set, despite the fact that it contains no elements and that contradicts the definition of set, which generally must be collection of distinct things considered as a whole (i.e. any set must be collection of elements, i.e. we admitted, allowed the arising of paradox at the starting stage of definition). ----If we have the right to make two definitions, which are mutually exclusive, then let’s define a new set - binary set, which is the set, which contains itself and at the same time doesn’t contain itself. Don’t ask, please, which set could it be?! O.k., let’s take again empty set, i.e. a set, which contains no elements. Does it contain empty set? Yes, it does, as empty set doesn’t contain elements, so empty set contains infinity number of empty sets, because they can be summed up to one empty set, which contains no element i.e. if any set contains infinite number of elements and at the same time doesn’t contain any element (i.e. empty set in our case), is just binary set, which is the set, which contains itself and at the same time doesn’t contain itself.

wow! Good job zeroca. That's the exact same solution I came up with in my post Russell's Paradox no more. Essentially I said that Russell's paradox is a question about the empty set and it is irrelevant to ask whether or not the empty set contains itself, which is the same thing that you just said but in different terms. That must mean that our solution is correct! I give us both a pat on the back. I don't know why my thread was moved to the your toe theory section. Can we put it under mathematics please?