# Functions operating in uncountable sets with cardinality $\gt\aleph_1$

A generic function $y=f(x)$ maps a number in the set of real number $X$ in another number in the set $Y$. It's well known that the irrational numbers are not countable. It's also known we can get a hierarchy of infinites starting from $\aleph_0$ with the simple operation: $2^{\aleph_0}=\aleph_1$, $2^{\aleph_1}=\aleph_2$ and so on. Is it possible to build a function mapping 'something' in a set $X_k$ with cardinality $\aleph_k$ with $k\gt1$ to another set $Y_k$ having the same cardinality? I can imagine something like a line, a plane or a solid having the cardinality of continuum, but intuitively I can't imagine something in the 'real' world having cardinality greather than $\aleph_1$.

• I feel as if this question has been asked and asked and asked again. – Asaf Karagila Jul 31 '14 at 14:19

Everything discussed below is within the $\mathsf{ZFC}$ axioms of set theory. This is just the most commonly used foundational axioms for mathematics. There are arguments to made that $\mathsf{ZFC}$ is a good formalization of most arguments of so called "intuitive" or natural mathematics. Hence if you hold such conviction, then if you manage to show that $\mathsf{ZFC}$ can not prove something, then you likely not be able to do this either using "intuitive" mathematics. Regardless of your philosophy concerning the relation of "intuitive" mathematics and $\mathsf{ZFC}$, the following response regards set theory as formalized in $\mathsf{ZFC}$.
First of all, within in $\mathsf{ZFC}$ is it unproveable that $\aleph_1 = 2^{\aleph_0}$, $\aleph_2 = 2^{\aleph_1}$, or more generally $\aleph_{k + 1} = 2^{\aleph_k}$. Letting $k$ vary of all ordinals, this statement is called the generalized continuum hypothesis ($\mathsf{GCH}$). This is independent of $\mathsf{ZFC}$, which means if there is a universe where $\mathsf{ZFC}$ holds, then there always exists two different universes such that $\mathsf{GCH}$ holds in one and fail in the other. In particular, your universe is one such that $\mathsf{GCH}$ fails, then there is some $\aleph_k$ which can not be obtain your hierarchy of iterating the power set operation starting from $\aleph_0$.
Now let us just suppose that $\mathsf{GCH}$ does holds. Then for instance $\aleph_2 = 2^{\aleph_1}$ and cardinality of $\mathbb{R}$ is $\aleph_1$. According to you, $\mathbb{R}$ is supposedly an "intuitive" set. The power set of the reals is then a set of cardinality $\aleph_2$.
(Now back to some philosophy) Now, you need to ask yourself: In the "real world" if something really exists (like $\mathbb{R}$ as you believe), should the collection of all its subcollections be something that "really exists"? Of course, the real line is not an actual real-world object, so I presume by "in the real world", you mean $\mathbb{R}$ is a valid object as a mental concept for whatever reason you have: such as it is an idealization or perfect representation of less perfect real world object or that it is a abstract notion which has by experience yielded practical use - science, engineering, etc.