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julypraise
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Let X be a set which is countably infinite. Then is there any example, on earth, of the power set of the power set of X?
julypraise said:Let X be a set which is countably infinite. Then is there any example, on earth, of the power set of the power set of X?
julypraise said:Because X is countabliy infinite, you can take any set that is countably infinite such as Q or Z^+ or Z, etc. And as most people know, P(X) in this case is equivalent to R. But my question here is what on Earth P(R) for example? Is it an, like, example of large cardinal?
julypraise said:Let X be a set which is countably infinite. Then is there any example, on earth, of the power set of the power set of X?
julypraise said:Let X be a set which is countably infinite. Then is there any example, on earth, of the power set of the power set of X?
The power set of the power set of an infinite set is the set of all subsets of the power set of the original infinite set. In other words, it is the set of all possible combinations of subsets of subsets of the original infinite set.
The main difference is that the power set of the power set contains subsets of subsets, while the power set only contains subsets of the original set. This means that the power set of the power set has a larger cardinality and contains more elements.
Yes, the power set of the power set of an infinite set is always infinite. This is because the original infinite set has an infinite number of elements, and the power set of an infinite set already has a larger cardinality than the original set. Therefore, the power set of the power set will also have an infinite number of elements.
No, the power set of the power set of an infinite set cannot be empty. This is because the power set of the original infinite set is never empty, and the power set of the power set contains all possible combinations of subsets of subsets of the original set. Therefore, it will always have at least one element.
The power set of the power set of an infinite set is related to Cantor's diagonalization argument in that it illustrates the concept of uncountable sets. Cantor's argument states that the cardinality of the power set of a set is always greater than the cardinality of the original set. This is also true for the power set of the power set, further demonstrating the infinite nature of these sets.