Does anyone know what the cardinality would be for an infinite Cartesian product of sets that all have the same cardinality as $\Bbb{R}$? Thanks. I really do not understand the problem, so an explanation would be greatly appreciated.
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3Depends how infinite – spaceisdarkgreen Apr 09 '20 at 01:11
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2As you’ve stated it, the problem is not well-posed: as spaceisdarkgreen said, if ${X_i:i\in I}$ is your family of sets that have the same cardinality as $\Bbb R$, and you’re forming $\prod_{i\in I}X_i$, the cardinality of this product depends on the cardinality of $I$. – Brian M. Scott Apr 09 '20 at 01:22
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What do you mean by how infinite? – user770210 Apr 09 '20 at 15:03
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This is the full question: Suppose A1, A2, . . . have the same cardinality as R. (There are denumerably many of them.) What is the cardinality of A1 ×A2 ×···? – user770210 Apr 09 '20 at 15:05
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@user770210 "denumerably many of them" is the answer to "how infinite?". (Brian Scott's comment is saying the same thing as I was saying, only more clearly.). You just said "infinite product"... it could have had more than countably many factors. – spaceisdarkgreen Apr 10 '20 at 05:06
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I really think this has to be a duplicate but I'll admit my quick search didn't yield anything. As a hint, what you want to compute is $|\mathbb R| ^{\aleph_0} = (2^{\aleph_0})^{\aleph_0}.$ I imagine you have learned some rules of cardinal arithmetic that will help you simplify this expression. – spaceisdarkgreen Apr 10 '20 at 05:19