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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 $\aleph_1$.

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

Given this point view, one approach to your question would then be, if some object is a so called valid mental concept, should the collection of all the subcollection of this object be as well a so called valid mental concept. Do you think it is possible to believe the existence of an object causes no paradox in reasoning but the existence of the power set of this object is a break in reason?

Of course, all these are philosophical question. Depending on your beliefs, the iterated power sets of the real line may be a sufficiently "real world" enough for you.

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