## Is this set uncountable

##S = \bigcup _{i=1}^{∞}\left\{{0,1}\right\}^i ##

methinks yes because:

##S = \bigcup _{i=1}^{∞}\left\{{0,1}\right\}^i \equiv \left\{{0,1}\right\}^\mathbb{N}##

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 Quote by Bachelier ##\bigcup _{i=1}^{∞}\left\{{0,1}\right\}^i \equiv \left\{{0,1}\right\}^\mathbb{N}##
This equality is false. Furthermore, the set on the right is uncountable. The set on the left is countable.

 Quote by micromass This equality is false. Furthermore, the set on the right is uncountable. The set on the left is countable.
So how do we look at ##\left\{{0,1}\right\}^∞##?

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## Is this set uncountable

 Quote by Bachelier So how do we look at ##\left\{{0,1}\right\}^∞##?
What do you mean with $\infty$? The notation you are using now is not standard at all.

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 I think that $S = \bigcup _{i=1}^{∞}\left\{{0,1}\right\}^i$ is countable all right. The mapping with $\mathbb{N}$ is quite obvious.