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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?