This problem is about proving that the power set $\mathcal{P}(X)$ of any set $X$, satisfies the inequality $|\mathcal{P}(X)| > |X|$. The challenge particularly comes in when $X$ is infinite, and the proof requires the use of a diagonalization argument.

(a) Contradiction via Diagonalization

To prove this, let's assume that there exists an onto function $g: X \to \mathcal{P}(X)$ from the set $X$ to its power set $\mathcal{P}(X)$.

Now, consider the set $D = \{ a \in X \mid a \notin g(a) \}$. This set $D$ is in the codomain of $g$, meaning there must be some element $d \in X$ such that $g(d) = D$, since $g$ is assumed to be onto.

Now we ask: Is $d \in D$?

• If $d \in D$, by the definition of $D$, it follows that $d \notin g(d)$. But since $g(d) = D$, this means $d \notin D$, a contradiction.
• If $d \notin D$, then by the definition of $D$, $d \in g(d)$, but $g(d) = D$, meaning $d \in D$, which is again a contradiction.

Thus, the assumption that $g$ is onto must be false. Therefore, there can be no onto function from $X$ to $\mathcal{P}(X)$, implying that $|\mathcal{P}(X)| > |X|$.

(b) Are there uncountable sets with greater cardinality than $\mathbb{R}$?

Yes, there are uncountable sets with cardinality greater than $\mathbb{R}$. From the previous result, we know that for any set $X$, the cardinality of its power set $\mathcal{P}(X)$ is strictly greater than the cardinality of $X$. In particular, if we take $X = \mathbb{R}$, then $|\mathcal{P}(\mathbb{R})|$ is greater than $|\mathbb{R}|$, meaning that the power set of the real numbers has a strictly larger cardinality than the real numbers themselves.

Since $\mathbb{R}$ is uncountable, this implies that there are uncountable sets with cardinality larger than $\mathbb{R}$.

Let me know if you'd like a more detailed explanation of any part!

Follow-up Questions:

1. What is the cardinality of the power set of a finite set?
2. Can you explain why diagonalization is necessary for infinite sets but not for finite ones?
3. How does the Cantor-Schroeder-Bernstein theorem relate to this proof?
4. What is the relationship between the cardinality of the continuum and other infinite cardinalities like $\aleph_0$?
5. How does the Axiom of Choice relate to uncountable sets and cardinalities?

Tip:

When working with proofs involving infinite sets, keep in mind the distinction between injective, surjective, and bijective functions, as these are key tools in comparing set sizes.