[zeroca]

Originally Posted by harmonygirl okay, I may be missing something, why would the realization of a zero set "solve" Russells paradox? It doesn't address the set of all sets contain itself question...

Originally Posted by <<>> For Zeroca, the idea of binary set which he develops and wabout which I have big doubts leads to the set of all empty sets contains the set of all sets that don't contain itself and therefore is a set of empty set's, sub interprets this is also being an empty set

Originally Posted by subversion 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?

I think I was misinterpreted:
----I didn’t say that empty set is the solution of Russel’s paradox. I say that empty set isn’t the solution of Russel’s paradox. The set, which doesn’t contain itself isn’t empty set: for instance the set, called “unity”, the name of which is described with five letters and is the set of names described with four letters (i.e. the set-“unity” doesn’t contain itself) isn’t empty set, but is the set of elements: “mama”, “papa”, “home” etc… And the set of all such sets can’t be empty set, as it isn’t collection of empty sets, but is collection of sets, each of which isn’t empty.
--But I pointed out the ground for solution of paradox, when mentioned, that the possible answer must be of a binary nature like an empty set, and it wasn’t me who adopted a conception of a binary set. I was one of them, who noticed a binary nature of empty set: empty set doesn’t contain itself, as it doesn’t contain elements, but at the same time it contains itself, as it (as consisting element) is empty.
---I also noticed that at the stage of definition of sets mutually exclusive suppositions were made:
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.
I think that conception of empty set needs to be revised. I’ll try to explain on the example of universe: if universe was defined as something that consists of one, or infinite number of elementary particles, then the universe ceases to exist at once when there isn’t even any single elementary particle. We can not say – empty universe to characterize such state, i.e. complete absence of elementary particles, i.e. complete absence of universe can’t be called empty universe, so
I wanted to show that wrong definitions lead to wrong conclusions.
Anyway, under a present condition of sets’ definition, we can seek solution of paradox, i.e. under present conditions it is solvable.
Russel’s paradox, as I understood, consists in following:
If the set-A of all sets-B (each of which is “self-not-contained”) doesn’t contain itself, then we must include A among other B sets; if we include A among others, then it becomes the set, which contains itself, so we must exclude it from the list of B sets.
I’ll try to solve paradox with illustrative examples:
At starting stage let’s take a set of all possible sets, all possible variants of sets and call it universal set. As you see, universal set contains in itself all existing sets (infinite variants of sets).
----Let’s take a set A, which is collection of all names, each of which consists of four letters, and let’s call this set “Home”. Home is example of the set, which contains itself as element, because its name also is composed of four letters.
----Let’s take a set B, which is the collection of all names, composed of four letters, and let’s call it “Unity”. “Unity” is the example of the set, which doesn’t contain itself as element, as mandatory for including within itself is that its name must be composed of four letters.
----Let’s take a set B1, which is the collection of all names, composed of five letters, and let’s call it “Unito”. “Unito” is the example of the set, which contains itself, and at same time “Unito” contains the set B, called “Unity”, as the name of latter is composed of five letters (and which is self-not-contained).
----Let’s take a set A2, which is the collection of names, composed of five letters, and call it “Mama”. “Mama” is the example of sets, which doesn’t contain itself, but it contains the set “Unito”, as it’s the collection of all five-letter names (and which is self-contained).
I.e. any set within universal set is represented in two forms: when it is consisting part of self-comtained and part of self-not-contained sets. The “trick” of Russel’s paradox consists in following: when you take the set of all sets, each of which doesn’t contain itself, you take at the same time all sets, each of which contains itself:
In our example:
When you are collecting all self-not-contained sets and take a set “Unity” (which is the example of the set, which doesn’t contain itself), you should remember that “Home” is consisting element of “Unity” (and “Home” is the example of the set, which contains itself and you can’t “touch” it, in spite of the fact that you need it, as it is consisting element of “unity”), you can’t as well “touch” the set “Mama”, as you take at once all its consistent five-letter-named sets, each of which are self-contained i.e.
You can not take separately a set of all sets, each of which doesn’t contain itself; you at once infringe the right of all sets (or parts of them), each of which contains itself, as they both are organic consisting parts of each-other.
I.e. any set (be it self-contained, or self-not-contained) is represented in two different forms of sets within universal set: within some sets- as part of self-contained, within another sets- as part of self-not-contained (for instance, the set “Mama” is the part of “home”, which is self-contained, but the same “Mama” ” is the part of “Unity”, which is self-not-contained)
I.e. Set of all set’s (each of which doesn’t contain itself)=Set of all sets (each of which contains itself)=Universal set. As universal set has the binary nature, so it can serve as a solution of Russel’s paradox.
I.e. the solution of Russel’s paradox can be:
You can’t collect all self-not-contained sets separately within one set, as you at once take all self-contained sets together with them, as they are consisting parts of each-other.

[subversion]

Originally Posted by harmonygirl Sub, I don't agree that a set would contain a zeroth element which is itself. When you define a set in this way, it defeats the question about whether a set can contain itself. I don't think that the set of all apples contains as a zeroth element the notion of the set of all apples. In my definition of the set of all apples, it ONLY contains all apples.

and all apples is itself, therefore it contains itself. Oh, and BTW defeating the question about whether a set can contain itself is the whole point. That's what you call the absolution of the paradox.

[subversion]

Originally Posted by dustin_archibald My reasoning was stated in the post which shows that the definition of the example set doesn't allow for the ideas you are describing. The example set of B={A} has elements with properties exactly the same as A. If what you are proposing was true if B = {A} then B = {{A},A} then {A}={{A},A} However {A} <> {{A},A}. In other words the set of A does not contain the set of A since the set of A's properties are not exactly the same as those of A. The definition of the set of A is that it contains all elements whose properties exactly match those of A. A, however, is not a set, it is an item or entity. Thus the properties of A do not match the properties of the set of A Something is not made up of itself. It is made of a number of other substances to create it. For example, a cookie is not made up of cookies. It is made up of the ingredients that, when combined, give it definition as a cookie.

You are right, that a cookie is not made up of cookies. But a cookie is made up of a cookie. It has to be. And also I discount your proof above because you talk about the set A and then you say that A is not a set. SO which is it? A set or not? Your logic is wrong because it assumes that something is not itself.

[subversion]

Originally Posted by zeroca I think I was misinterpreted: ----I didn’t say that empty set is the solution of Russel’s paradox. I say that empty set isn’t the solution of Russel’s paradox. The set, which doesn’t contain itself isn’t empty set: for instance the set, called “unity”, the name of which is described with five letters and is the set of names described with four letters (i.e. the set-“unity” doesn’t contain itself) isn’t empty set, but is the set of elements: “mama”, “papa”, “home” etc… And the set of all such sets can’t be empty set, as it isn’t collection of empty sets, but is collection of sets, each of which isn’t empty. --But I pointed out the ground for solution of paradox, when mentioned, that the possible answer must be of a binary nature like an empty set, and it wasn’t me who adopted a conception of a binary set. I was one of them, who noticed a binary nature of empty set: empty set doesn’t contain itself, as it doesn’t contain elements, but at the same time it contains itself, as it (as consisting element) is empty. ---I also noticed that at the stage of definition of sets mutually exclusive suppositions were made: 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. I think that conception of empty set needs to be revised. I’ll try to explain on the example of universe: if universe was defined as something that consists of one, or infinite number of elementary particles, then the universe ceases to exist at once when there isn’t even any single elementary particle. We can not say – empty universe to characterize such state, i.e. complete absence of elementary particles, i.e. complete absence of universe can’t be called empty universe, so I wanted to show that wrong definitions lead to wrong conclusions. Anyway, under a present condition of sets’ definition, we can seek solution of paradox, i.e. under present conditions it is solvable. Russel’s paradox, as I understood, consists in following: If the set-A of all sets-B (each of which is “self-not-contained”) doesn’t contain itself, then we must include A among other B sets; if we include A among others, then it becomes the set, which contains itself, so we must exclude it from the list of B sets. I’ll try to solve paradox with illustrative examples: At starting stage let’s take a set of all possible sets, all possible variants of sets and call it universal set. As you see, universal set contains in itself all existing sets (infinite variants of sets). ----Let’s take a set A, which is collection of all names, each of which consists of four letters, and let’s call this set “Home”. Home is example of the set, which contains itself as element, because its name also is composed of four letters. The set A1, called “Papa”, which is collection of all names, each of which consists of four letters is the same sort of example: it also contains itself. ----Let’s take a set B, which is the collection of all names, composed of four letters, and let’s call it “Unity”. “Unity” is the example of the set, which doesn’t contain itself as element, as mandatory for including within itself is that its name must be composed of four letters. ----Let’s take a set B1, which is the collection of all names, composed of five letters, and let’s call it “Unito”. “Unito” is the example of the set, which contains itself, and at same time “Unito” contains the set B, called “Unity”, as the name of latter is composed of five letters (and which is self-not-contained). ----Let’s take a set A2, which is the collection of names, composed of five letters, and call it “Mama”. “Mama” is the example of sets, which doesn’t contain itself, but it contains the set “Unito”, as it’s the collection of all five-letter names (and which is self-contained). I.e. any set within universal set is represented in two forms: when it is consisting part of self-comtained and part of self-not-contained sets. The “trick” of Russel’s paradox consists in following: when you take the set of all sets, each of which doesn’t contain itself, you take at the same time all sets (or part, or even single set), each of which contains itself: In our example: When you are collecting all self-not-contained sets and take a set “Unity” (which is the example of the set, which doesn’t contain itself), you should remember that “Home” is consisting element of “Unity” (and “Home” is the example of the set, which contains itself and you can’t “touch” it, in spite of the fact that you need it, as it is consisting element of “unity”), you can’t as well “touch” the set “Mama”, as you take at once all its consistent five-letter-named sets, each of which are self-contained i.e. You can not take separately a set of all sets, each of which doesn’t contain itself; you at once infringe the right of all sets (or parts of them), each of which contains itself, as they both are organic consisting parts of each-other. I.e. any set (be it self-contained, or self-not-contained) is represented in two different forms of sets within universal set: within some sets- as part of self-contained, within another sets- as part of self-not-contained (for instance, the set “Mama” is the part of “home”, which is self-contained, but the same “Mama” ” is the part of “Unity”, which is self-not-contained) I.e. Set of all set’s (each of which doesn’t contain itself)=Set of all sets (each of which contains itself)=Universal set. As universal set has the binary nature, so it can serve as a solution of Russel’s paradox. I.e. the solution of Russel’s paradox can be: You can’t collect all self-not-contained sets separately within one set, as you at once take all self-contained sets together with them, as they are consisting parts of each-other.

but according to the theory of definition the name you choose to describe your set is completely arbitrary. So if you called the set of all things that are four letter words "unity" I could call it "plap" which would mean it does contain itself. It doesn't matter what you call it. The important point is that the idea of the set is always a self contained idea no matter what. So the set of all sets which do not contain themselves must be empty.

Here's another example, let's take the set of all things which are cool. If all the things in the set are cool then the set itself should be cool too. Let's take the set of all things which are red. There's no reason to make the set of all things which are red a green set. That would be considered an error of definition.

So give me a better example of a set which doesn't contain itself because if you rename the "unity" set with a four letter name it does contain itself without changing the contents or nature, of the set. No matter what you name it, the idea of a set is always a self-contained idea no matter what. All this means is that something must always be considered the same thing as it is.

[subversion]

Zeroca, I completely agree that you can not take the set of all sets which do not contain themselves. That is because it is the empty set no? It cannot be the full set because the full set contains itself. Therefore I think that the set of all sets which contain themselves is the complete set and the set of all sets which do not contain themselves is the empty set. I think this is a very useful way to define it. Don't you?

[zeroca]

Originally Posted by subversion Zeroca, I completely agree that you can not take the set of all sets which do not contain themselves. That is because it is the empty set no? It cannot be the full set because the full set contains itself. Therefore I think that the set of all sets which contain themselves is the complete set and the set of all sets which do not contain themselves is the empty set. I think this is a very useful way to define it. Don't you?

I don’t think so: we can put it this way: if you can’t collect all elements of any set (i.e. if you can’t collect the whole quantity of them, but only part of them), it surely mustn’t be an empty set. You can call empty set only a set without any single element.
I’ll try to be more precise with my explanations:
If you collect green things into the bag, so accordingly you have a set of green things, then regarding to the matter of “self-containness” you have two possible alternatives: either this bag enters among the elements or not:
If the bag is green so you have self-contained set, if the bag isn’t green, the set is self-not-contained. I’d like to emphasize that you aren’t obliged to analyze all qualities of bag: is it big or small; is it red, black, or yellow; is it new or old. It’s enough for you to check if it is green, if not – then all other infinite qualities of bags enter within quality “not-green”.
Let’s take refrigerator, which’s turned on. It contains cool things, but it itself isn’t necessary cool, especially if refrigerator is somewhere in Africa, i.e. if we collect the set of some cool things, this refrigerator doesn’t enter in requirement, so the set refrigerator in this particular case isn’t example of self-contained sets, but if we place it on the pole, it at once takes up position within self-contained sets, as it is also cool, together with its content.
Let’s take all refrigerators in Africa. Their number is definite, as there’s not infinite number of refrigerators in Africa, so the set of all turned on refrigerators in Africa is the example of set, which is all self-not-contained sets in Africa in regard to refrigerators (i.e. within some definite region with some definite qualities of set).
The trick of Russel’s paradox is to thrust infinity within consideration. I.e. we must collect not definite number of self-not-contained sets, but all of them! I.e. Russell complicated the task for us, but I don’t think so, as it’s enough to seek one example of self-not-contained set, which for some reason (for any reason) doesn’t enter these requirements, that the task is almost solved, so cited by me above example of set “Mama” in previous post is good example of it:
The set “Mama” is the part of “home”, which is self-contained, but the same “Mama” ” is the part of “Unity”, which is self-not-contained.
And it doesn’t’ matter how you call any set. With infinity you are free to choose any variant, because infinity can contain all credible ones: You can call unity as unito, as unitu and despite their being the same set with different names, anyway they are considered as different sets.
I showed above (in previous post), that
Any set (be it self-contained, or self-not-contained) is represented in two different forms of sets within universal set: within some sets- as part of self-contained, within another sets- as part of self-not contained.
That’s enough to assert that we can not collect separately all self-not-contained sets into one single set, as the same, infinite quantity of the same sets at once remain as elements of single "container", defined as set of all self-contained sets and this quantity of sets is the same as that of universal set, which as you see, is of binary nature, I.e. it contains each consisting set in two different forms: as self-contained and as self-not-contained ones.

[subversion]

in the example you used of a bag full of green things I would say that the set is defined as the bag full of green things, in which case the bag is part of the contents. So even though the set is the bag that's only full of green things, the set includes the bag no matter what color it is. So for sake of descriptive labelling, you might want to make the bag green and that way you know what's in it. But regardless of what color the bag is it is always part of the set of the bag full of green things.

[zeroca]

Originally Posted by subversion in the example you used of a bag full of green things I would say that the set is defined as the bag full of green things, in which case the bag is part of the contents. So even though the set is the bag that's only full of green things, the set includes the bag no matter what color it is. So for sake of descriptive labelling, you might want to make the bag green and that way you know what's in it. But regardless of what color the bag is it is always part of the set of the bag full of green things.

Sub, I haven’t dealt with higher mathematics for 30 years now. I recently consulted the site http://en.wikipedia.org/wiki/Russell%27s_paradox, skipped through some definitions of sets and then tried to offer my opinion of my leisure time about it. My explanations aren’t that of professional mathematician’s. If all this is as you say, it means, I interpreted everything incorrect way.

[dustin_archibald]

Originally Posted by subversion But a cookie is made up of a cookie. It has to be.

Your assertion is that this cookie is made up of itself, correct? Let's follow this example further. Let's say that the cookie (C) being referenced has, at least, these ingredients: flower (f), sugar (s), eggs (e), butter (b).

My assertion is:
C=f+s+e+b

Your assertion (as I understand it) is:
C=C+f+s+e+b

If the formula is correct then f,s,e,b must each equal 0 in order for it to be true. So that would mean the cookie has no ingredients except for itself. One could rectify this by adding -C, but that just leads to my assertion anyway.

I'm not saying that a cookie is not itself. I am saying that a combination of components leads to the definition of the cookie. I see your argument as saying "a cookie is a cookie because it's a cookie" which is circular logic.

Originally Posted by subversion And also I discount your proof above because you talk about the set A and then you say that A is not a set. SO which is it? A set or not?

Your understanding of the terminology I use is wrong. I talk about the set of A which is indicated by {A}. The set A would be defined as A={some elements}. So:

A - [An element with properties]
{A} - [The set of all elements with properties exactly the same as A. Or simply the set of A]
A={some elements} - [The set A. A is an identifier used to reference all the elements in the set.]

[dustin_archibald]

Originally Posted by subversion in the example you used of a bag full of green things I would say that the set is defined as the bag full of green things, in which case the bag is part of the contents. So even though the set is the bag that's only full of green things, the set includes the bag no matter what color it is. So for sake of descriptive labelling, you might want to make the bag green and that way you know what's in it. But regardless of what color the bag is it is always part of the set of the bag full of green things.

I think it's important, as pointed out earlier, to get correct definitions. From my understaning the sets we are referring to here are defined as a grouping of elements based on the common properties (or a common property) of those elements.

Is a major issue still "Do sets contain themselves?" or has this discussion progressed from there?