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if product of two countable sets is countable, then we can proceed by induction, countable product of countable sets is countable. But countable infinity product of two element set is uncountable. Why?

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    $\begingroup$ Induction proves that any finite product of countable sets is countable. Countable products are uncountable, as you observe. $\endgroup$ – Ethan Bolker Feb 6 at 2:27
  • $\begingroup$ @EthanBolker but the induction principle implies that holds for all positive integers. can I say that "infinity" is not included in positive integers? $\endgroup$ – user642007 Feb 6 at 2:44
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    $\begingroup$ "Infinity" is not an integer. There are infinitely many integers, but no particular integer is "infinitely large". $\endgroup$ – Ethan Bolker Feb 6 at 2:47
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    $\begingroup$ Infinity is not an integer, no. Even if we did add an integer $\infty$ that is larger than all other integers, how would induction help us? We couldn't use the $\infty - 1$th case to prove the $\infty$th case, because we don't know the $\infty - 1$th case is true, because we don't know the $\infty - 2$th case is true, etc. There's no path back to the base case! $\endgroup$ – Theo Bendit Feb 6 at 2:49
  • $\begingroup$ @EthanBolker thanks $\endgroup$ – user642007 Feb 6 at 2:51
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Let the numbers in [0,1) be expressed in base two.
Each of those is now a sequence of 0's and 1's.
Use that to create an injection from [0,1) into {0,1}$^N$.
Thus as [0,1) is uncountable, so is {0,1}$^N$.

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