Cardinality of the set of at most countable subsets of the real line?

Question

I'm exploring an unrelated question about power series with complex coefficients. While exploring this question, I wondered: What is the cardinality of the set of all such power series? Or with different language: What is the cardinality of at most countable subsets of $\mathbb{C}$ (or $\mathbb{R}$, if you prefer)?

I asked my advisor and he surprisingly wasn't sure, though he suspects that the set of subsets in question has a larger cardinality than $\mathbb{R}$.

Thanks a lot!

Edit: Certainly if we only consider finite subsets, then this set of subsets has cardinality equal to $\mathbb{R}$.

Edit2: Realized my wording was wrong. I'm actually looking for the cardinality of the set of sequences with entries in $\mathbb{C}$, not the cardinality of the set of at most countable subsets of $\mathbb{C}$. However, both questions are answered below, and both turn out to be $|\mathbb{R}|$.

Answer 1

(Note that $|\Bbb C|=|\Bbb R|$.)

If we assume the axiom of choice, then indeed the answer is that the set of countable subsets of $\Bbb R$ has the same cardinality as $\Bbb R$ itself.

The proof is simple. Using the axiom of choice, choose an enumeration of each such subset (i.e. a bijection between the countable set and $\Bbb N$). Next note the following cardinal arithmetic:

$$|\Bbb{R^N}|=(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0\cdot\aleph_0}=2^{\aleph_0}=|\Bbb R|.$$

Therefore the set of sequences of real numbers has the same cardinality as the real numbers. Now send each countable set to its enumeration, and we're done.

Without the axiom of choice, however, it is consistent that there are strictly more countable subsets. For example in Solovay's model where every set is Lebesgue measurable this is true.

Answer 2

The set of all countable subsets of $\mathbb R$ has size $\mathbb R$. This is because there is an injection from this set into the set $\mathbb R^{\mathbb N}$ of countable sequences. Now, $\mathbb R$ has the same size as $\{0,1\}^{\mathbb N}$ (for instance, see here), and we see that $|\mathbb R^{\mathbb N}|=|(\{0,1\}^{\mathbb N})^{\mathbb N}|=|\{0,1\}^{\mathbb N\times \mathbb N}|=|\{0,1\}^{\mathbb N}|$, since $\mathbb N$ and $\mathbb N^2$ have the same size.

Perhaps surprisingly, the answer depends on the axiom of choice. Tarski proved (without choice) that, for any set $X$, the collection of well-orderable subsets of $X$ has size strictly larger than $X$ (For a proof, see for instance here, or the references linked to here). It is consistent without choice that the only well-orderable subsets of $\mathbb R$ are countable (that is, that $\omega_1\not\le|\mathbb R|$). Whenever this situation holds, we have that the collection of countable subsets of $\mathbb R$ has size strictly larger than $\mathbb R$.