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Suppose A1, A2, . . . have the same cardinality as R. (There are denumerably many of them.) What is the cardinality of A1 ×A2 ×···?

Not really sure how to approach this. Any help is greatly appreciated.

user770210
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  • What if there were only the two listed $A$s: $A_1$ and $A_2$? Then if there were only three? Then if there were only four? What do you get in the limit as the number of $A_i$ goes to (countable, since indexed by subset of integers) infinity? – Eric Towers Apr 09 '20 at 15:08
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    If you are allowed to use basic cardinal arithmetic properties, then the set in question has cardinality $c \cdot c \cdot c \cdot \ldots$ (a countably infinite product), which equals ${\aleph_0} \cdot c \leq c \cdot c = c.$ – Dave L. Renfro Apr 09 '20 at 15:17
  • i am not sure how to find what it would be as it approaches infinity. Would it still be R? – user770210 Apr 09 '20 at 15:34
  • So in this case, it is still equal to R? – user770210 Apr 09 '20 at 15:34
  • Possibly $0$, unless you're assuming the axiom of choice. – Asaf Karagila Apr 09 '20 at 18:39

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