# why is the set of all binary sequences not countable?

What is wrong with this reasoning: The union (finite or infinite) of countable sets is countable. The set of all binary sequences is the infinite union of the sets $S_n$ of all the binary sequences of length n, which are finite, hence countable.

• You haven't accounted for all the infinite binary sequences, only the finite ones of any length. – Alex J Best Mar 22 '13 at 17:47
• Also, it is not true that an infinite union of countable sets is always countable. A union of countably many countable sets if countable, but the union of uncountably many sets each of which is countable may not be. For example $\bigcup_{x\in [0,1)} (\mathbb Z+x)$ is $\mathbb R$ which is not countable. – hmakholm left over Monica Mar 22 '13 at 18:04
• @AlexJBest thanks for the response; I was fearing it will exactly this response, as I cannot make the difference between the two.. – Radu Mar 22 '13 at 18:48
• @Radu probably one cannot understand the difference between countable and uncountable without first understanding the difference between finite and infinite – Trevor Wilson Mar 22 '13 at 18:56
• @TrevorWilson indeed, I am under the (obviously wrong, as it has already been pointed out) impression that my finite sequences can grow without bound, hence cover all infinite sequences.. – Radu Mar 22 '13 at 19:09

• @Radu: I’m not sure what you mean by where they come from. They are simply functions from $\Bbb N$ to the set $\{0,1\}$; the existence of that set of functions follows from the usual axioms of set theory, as does the proof that there are uncountably many such functions. – Brian M. Scott Mar 30 '13 at 22:27