If we subtract all the non-computable infinite sequences from the set, does cardinality change?

Question

I may not have expressed the question correctly.

My question is:

Suppose, the set of $A$ contains countable infinite length sequences/functions, which is consist of elements $\left\{0,1\right\}$. Then, in this set we have countable infinite computable and countable infinite non-computable infinite sequences. So, the set of cardinality equals to $\aleph_0$. Then, if we subtract all the non-computable infinite sequences from the the set $A$, does cardinality change? Are the number of elements "reduce?"

Thank you.

Answer 1

No, it doesn't change - you have $\aleph_0$-many computable sequences left. And this has nothing to do with computability theory: this is just the fact that $\aleph_0+\aleph_0=\aleph_0$.