[michellemfry]

I like to substitute words and see if any logic falls in my lap. Do seeds contain themselves? Yes, there is the information for a seed inside a seed. Does this interest you at all in the discussion? I am not a great philosopher, so be gentle with your response, please.

[harmonygirl]

Michelle,
I don't think that seeds contain themselves, they contain information that lead to their formation. I think Dustin's example was a sound one: a cookie contains ingredients (flour, sugar, eggs and butter-and hopefully chocolate!) but the ingredients of a cookie do not contain a cookie. A set of things (whatever those things are) may or may not contain itself (a bag of bags, etc) but does the set of all sets contain itself?

[michellemfry]

Who would have guessed that a discussion that includes seeds, cookies, and bags would be actually about Russel's paradox? I'm just along for the ride, harmonygirl.

[harmonygirl]

I know, it's interesting what we choose as examples. I am sure there is a pyschology paper in there somewhere...

[Guille]

Come back to the theme.

Originally Posted by harmonygirl

Michelle,
I don't think that seeds contain themselves, they contain information that lead to their formation. I think Dustin's example was a sound one: a cookie contains ingredients (flour, sugar, eggs and butter-and hopefully chocolate!) but the ingredients of a cookie do not contain a cookie. A set of things (whatever those things are) may or may not contain itself (a bag of bags, etc) but does the set of all sets contain itself?

None of these four forms are russell's paradox:

'a cookie contains ingredients' is a set that contains it's members. Well then yes, because all sets contain the members that contain themselves.

'but the ingredietns of a cookie do not contain a cookie' this is a deep metaphysical discussion. Many philosophers believe that they do, that everything contains it's own cause, or, in this case, it's container. And anyway, weren't those ingredients determined to end up forming part of the cookie? If I had seen them, they woul dbe determined, I couldn't have stopped it.

'a set of things may or may not contain itself ' well this is harder to tell. The thing is that int he example of the cookie it actually doesn't contain itself, it contains it's members. There is a wrong interpretation by you all: all sets contain themslves in that they contain all their members and these are the set, but this is not containing itself, it's just being itself. And every set is itself, it's members. The containing about which russell' paradox is about is that a set's cnditions for members includes the set itself. A cookie doesn't contain itself in here, for a cookie doesn't condition a cookie, but the ingredients. You may set 'well yes, but it could be simply conditioning the cookie' well this is true, but irrelevant. The problem in the whole of this discussison is what we consider member and what we consider set. Because normally I look at cookies more like entities, like members of a set of cookies, than sets of ingredients. And here is the main problem of set-ting.

'but does the set of all sets contain itself?' This again isn't russell. Russell's paradox is: The set of all sets that do not contain themselves contains itself if it does not contain itself, and does not contain itself if it contains itself. The set of all sets contains itself obviouslyl as it is a set. of all sets means all, including the empty sets, and the set of sets and even russell's set.

[dustin_archibald]

I don't know if someone already said this, in this thread; there is a lot to read and understand.

One way to avoid Russell's Paradox is to prove that it isn't a paradox at all.

One way we can do this (prove it's not a paradox) is by proving that it's based on a false assumption; that sets can contain themselves, more specifically, sets have the ability to contain themselves. How do we do this? We need to prove that a set cannot contain itself; that it cannot exist if it contains itself.

A second way to do this (prove it's not a paradox) is by proving that it's based on a false assumption; that a set cannot contain itself; a set does not have the ability to contain itself. How do we do this? We need to prove that a set must contain itself; that a set cannot exist without containing itself.

If you can prove that either (one or the other) assumption is incorrect, the paradox will fall apart since it is based on a false assumption

Answering the question "Does the set of all sets contain itself?" is important since, in my opinion, it is the only set with the possibility of containing itself.

[Guille]

Originally Posted by dustin_archibald

I don't know if someone already said this, in this thread; there is a lot to read and understand.

One way to avoid Russell's Paradox is to prove that it isn't a paradox at all.

One way we can do this (prove it's not a paradox) is by proving that it's based on a false assumption; that sets can contain themselves, more specifically, sets have the ability to contain themselves. How do we do this? We need to prove that a set cannot contain itself; that it cannot exist if it contains itself.

A second way to do this (prove it's not a paradox) is by proving that it's based on a false assumption; that a set cannot contain itself; a set does not have the ability to contain itself. How do we do this? We need to prove that a set must contain itself; that a set cannot exist without containing itself.

If you can prove that either (one or the other) assumption is incorrect, the paradox will fall apart since it is based on a false assumption

Answering the question "Does the set of all sets contain itself?" is important since, in my opinion, it is the only set with the possibility of containing itself.

I had thought of disprovng a starting assumption. In some threads I've stated clearly that the logicians have foughtthe paradox by saying that sets dont' contain sets, and that if there is a collection of any kind of sets, it is to be called a group. A collection of groups is a class...etz, there's a point where it all stops, but that's nto important: the important thing is that the claim to be false the assuption that sets can contain any set at all.

Now, both of the assumptions you propose that could be proved false are good points for they are not as strong as the other assuptions (there are a few other: that a set can contain sets, that a set can contain sets that don't contain themselves, that a set can contain sets that do contain themselves....). But any attack to them is simply impossible. They are right by definition. If you read my starting post, I give two examples one of a set that contains itself and one that doesn't.

I give you these proof: the set of all Dustins is not a Dustin itself, therefore it doesn't contain itself. The set of of all sets that have infinite to 0 members contains itself, as it has a number of members (even if it's infinite). By the first I proof that sets have the ability to not contain themselves, and by the second I proof that sets have the ability to contain themselves.

[dustin_archibald]

Does the Set of All Sets contain itself? The question is moot since the Set of All Sets does not exist nor can it exist. In addition, no set can contain itself. Here's why:

I.
A set is an association of elements matching certain properties. This means that all elements of the set must have certain properties in common and those properties are the criteria for forming the set. It is a method for associating elements in meaningful context. You do not add elements until it is complete. If you do that it is a different set until the last member is added, at which point it becomes the defined set. Some examples are:

1. The set of all green crayons
2. The set of all humans of age 55
3. The set of the first 10 animals you can list.
4. The set of all sets containing fur bearing animals*.

*Note that this set itself does not contain fur bearing animals. It only contains the sets that have fur bearing animals as their elements.

II.
A set is incomplete until all the conditions for it's definition are met. An incomplete set is useless; without substance; undefined. You cannot make any comparisons with it nor can you extrapolate any meaningful data. A set truly is the "sum of it's parts". If you do not group all the required elements according to its definition it is not a set. So in the examples from I none can be considered the defined set until its conditions are met:

1. Cannot be the set of all green crayons until it contains all green crayons
2. Cannot be the set of humans age 55 until it contains all humans of age 55
3. Cannot be the set of the first 10 animals you can list until it contains the first 10 animals you can list.
4. Cannot be the set of all sets containing fur bearing animals until it contains all the sets of fur bearing animals.

III.
An element cannot be an element of a certain set unless it has the neccesary properties in order to be an element of the set. In addition, an element must exist in order to be considered a candidate for being an element of a set. What about the set that only contains elements that do not exist? It is the null set. There is no method for grouping something that does not exist.

Let's look at the Set of All Sets:

1. It must contain only sets.
2. A set must exist in order to be an element of this set.

The Set of All Sets is the "sum of its parts". We cannot define a set as the Set of All Sets until it contains all sets. If we associate all the sets in existence as a group of sets it will not be complete (it will still not be the Set of All Sets) because there is one set that does not exist yet: the Set of All Sets. Furthermore, the Set of All Sets will not be complete until the Set of All Sets exists. However, the Set of All Sets will not exist until the Set of All Sets is complete. As you can see the concept of the Set of All Sets is paradoxical, rather than recurrisive as is normally thought.

Can a set contain itself? No, because a grouping of elements is not considered to be the defined set until all elements meeting the definition of the set are grouped. An element cannot be a part of a set until it exists.

What does this mean for Russell's Paradox? Russell's Paradox assumes that it is possible for a set to contain itself. Since a set cannot contain itself Russell's Paradox is not substantiated.

[subversion]

Originally Posted by harmonygirl

Michelle,
I don't think that seeds contain themselves, they contain information that lead to their formation. I think Dustin's example was a sound one: a cookie contains ingredients (flour, sugar, eggs and butter-and hopefully chocolate!) but the ingredients of a cookie do not contain a cookie. A set of things (whatever those things are) may or may not contain itself (a bag of bags, etc) but does the set of all sets contain itself?

Yes, while a cookie contains ingredients it does not mean that ingredients always contain cookies. However, a cookie always contains itself, a cookie, no matter what. Anything that exists must physically be itself which is to say it contains itself and nothing else. Therefore a set, in order to be a real thing, always contains itself by default as it's own zeroth member. Therefore all sets must contain at least themselves and any set which does not contain itself must be empty by default. Therefore the empty set is the set of all sets which do not contain themselves. This is the solution to Russell's paradox point blank. On the other hand, the set of all sets which do contain themselves is the set of all non-empty sets which is the opposite of the empty set, the full set. The full set has not been invented yet but I still know about it because I am a genious.

[subversion]

Originally Posted by <<>>

'a set of things may or may not contain itself ' well this is harder to tell. The thing is that int he example of the cookie it actually doesn't contain itself, it contains it's members. .

You're wrong, the cookie always contains itself in order to be a cookie. Why do you say that the cookie does not contain it's own self? Do you mean to say that the cookie does not belong to itself? You might not want to tell the cookie that

The simple answer to the paradox is that all things logically contain themselves and if something doesn't it is nothing, i.e. the empty set. Doesn't that make perfect sense?