Is there a set whose power set is countably infinite?

Question

Does there exist a set whose power set is countably infinite?

I know for sure that if a set has a finite number of elements, then its power set must have a finite number of elements, and if a set has an infinite number of elements, then its power set must have an infinite number of elements (possibly uncountably many elements). Then, there must not exist something like that (which I stated at first). Am I right? Please someone clarify it.

Answer 1

No.

Let's call a cardinal $\kappa$ a strong limit cardinal, if whenever $A$ is a set of cardinality strictly less than $\kappa$, also $\mathcal P(A)$ has cardinality $<\kappa$.

It is easy to see that $\aleph_0$ is a strong limit cardinal, exactly because everything smaller is finite, and the power set of a finite set is finite.

Now we can prove a general theorem:

Suppose that $\kappa$ is a strong limit cardinal. There is no set $A$ such that $|\mathcal P(A)|=\kappa$.

Proof. Recall Cantor's theorem, for all sets $A$, $|A|<|\mathcal P(A)|$. If $|\mathcal P(A)|=\kappa$, then $|A|<\kappa$. But now by virtue of being a strong limit cardinal, $|\mathcal P(A)|<\kappa$ as well. $\square$

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In particular, it means there is no set whose power set is countably infinite.

Answer 2

The size of the power set of a set is larger than the size of the set itself. This statement is called Cantor's theorem.

If $n$ is finite, then the size of its power set is $2^n$ which is finite. So, the desired set has to be infinite. But then an infinite set has to have a set of the size of natural numbers (countable) inside it.

By Cantor's theorem again, the size of the power set of $\mathbb{N}$ is therefore greater than the size of $\mathbb{N}$ itself. This means that the size of $\mathcal{P}(\mathbb{N})$ has to be strictly larger than countable, i.e. uncountable.

Since the set you are looking for has a set of the size of natural numbers inside it, its cardinality must be strictly larger than countable. This shows that such a set whose power set is countable cannot exist.

Answer 3

No, and this follows from these three observations:

• A finite set has a finite power set.
• $|\mathcal P(A)|$ is is always larger than $|A|$ (Cantor's theorem).
• $\aleph_0$ is the smallest non-finite cardinal number.