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Is there a bijection between $\mathbb{Z}\times\mathbb{Z}\times\dots$ for countably infinitely many $\mathbb{Z}$'s and $\mathbb{R}$? That is, is $\mathbb{Z}\times\mathbb{Z}\times\dots$, repeated countably infinitely many times, uncountable?

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    What other sets do you know to be uncountable? What other sets do you know to be "bijectable" with $\mathbb{R}$?2012-12-06

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First note that as joriki commented there are many uncountable sets which are not of the same size as the real numbers. There might be uncountable sets which are strictly smaller than the real numbers.

To both your questions, however, the answer is yes. Note that:

$\mathbb{Z\times Z\times Z\times\dots = Z^N}\\\mathbb{R\times R\times R\times\dots = R^N}$

Now we have this: $|\mathbb R|=2^{|\mathbb N|}\leq|\mathbb Z|^{|\mathbb N|}\leq|\mathbb R|^{|\mathbb N|}=2^{|\mathbb{N\times N}|}=2^{|\mathbb N|}=|\mathbb R|$

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    Obviously we can. Any set below the cardinality of $\mathbb R$, as a matter of fact.2012-12-06