olegt
Posts: 1405 Joined: Dec. 2006
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Here is my example in which I tried to explain to Joe the difference between an empty set and a set containing an empty set.
Quote | Here is the difference, Joe.
Let's go back to our favorite example featuring Denton's books, Evolution (E) and Nature's Destiny (N). Books are objects.
We can ask all kinds of questions about books such as this: "Which books did Denton write?" The answer to that question, {E,N}, is also an object, but it is definitely not a book. Rather, it is a set of books.
We can also ask "Which of these books did Joe G read?" The answer to that is also a set of books. It could be {E,N}, but most likely it is {E}. And when we ask "Which of these books did Oleg read?" the answer will again be a set of books, this time an empty set, {}.
So far so good. Now we can go one more level up and ask the Big Question: "What are the possible answers to the question Which Denton's books did you read?" The Big Answer is of course a set of answers. And since each answer is a set of books, the Big Answer is a set of sets of books. In this case, it is {{}, {E}, {N}, {E,N}}. Again, the Big Answer is not a book, and neither it is a set of books. It is a set of answers, or a set of sets of books.
Suppose now we go to a country where only one of Denton's books has been published. When we ask its citizens the question the answers (sets of books) we get are either {} or {E}. The Big Answer (a set of sets of books) is, accordingly, {{}, {E}}.
Finally we travel to a far corner of the Earth, where no one even heard of Denton. The answer to the question is invariably the set of books {}. The Big Answer to the Big Question is the set of sets of books {{}}.
Hope this example is pedagogical.
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That didn't help.
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