We all know that there are the infinite sets $Q$ and $R$. I have been taught, and also read that there is one to one correspondence between the elements of both sets, and thus both are regarded as equivalent.
But not long ago, I read something new for me from Tom Crilly's '50 Mathematical Ideas You Really Need to Know' page 104. Here is what it says word by word:
Although both sets are infinite, the set $R$ has a higher order of infinity than $Q$.Mathematicians denote $card(Q)$ by) $\aleph _0$, the Hebrew ‘aleph nought’ and $card(R) = c$. So this means $\aleph _0 < c.$
Doesn't this contradict the earlier statement I stated ?
First of all, for cardinality of $\mathbb{R}$ and $\mathbb{Q}$, we have $|\mathbb{R}| > |\mathbb{Q}|$.
For your main question, I suggest you to check Cantor's Theorem, which states that for any set $X$, cardinality of power set $P(X)$ of $X$ is larger than cardinality of $X$, which is $$|P(X)| > |X|$$ This means, if you give me $\mathbb{R}$, $|P(\mathbb{R})|$ is a distinct infinity from $|\mathbb{R}|$ and $|\mathbb{Q}|$. And you can extend this result even more with $P(P(\mathbb{R}))$ for instance.