Are cardinal numbers well defined? Or could we have something like $\aleph_{1/2}$?

Question

From what I understood, cardinal numbers are defined as:

$\aleph_0$ = the cardinality of $\mathbb{N}$
$\aleph_{n+1}$ = is the least cardinal number greater than $\aleph_n$

The continuum hypothesis states that the cardinality of $\mathbb{R}$ is $\aleph_1$. I more or less understand this cannot be proven (does not follow from ZFC).

However, can it be proven dat there in fact is a 'least cardinal number' greater than $\aleph_0$?

Is it possible that there are sets with cardinality $\aleph_{1/2}$, $\aleph_{1/3}$, et cetera, so to speak?

Or expressed differently: is it possible that for every set A with cardinality C_A > $\aleph_0$, there exists another set B with cardinality C_B where C_A > C_B > $\aleph_0$?

Answer 1

Cardinal numbers are special ordinal numbers, and the class of ordinals is well-ordered. So it does make sense to talk about the first ordinal that is not in a bijective correspondence with $\omega$, this is a unique and well-defined ordinal number and is defined to be $\aleph_1$. By definition all smaller ordinals are countable, so there is no intermediate cardinality between $\aleph_0$ and $\aleph_1$. The same goes for $\aleph_\alpha$ and $\aleph_{\alpha+1}$, mutatis mutandis.

Answer 2

Yes, cardinals are well-defined. Under the axiom of choice, every sets are well-ordered. Especially, every set is equipotent (that is, has same size with) some ordinal number, so we can define cardinals as ordinals. The detail of the definition might be technical for beginners to set theory, but the idea of the definition is same as you have known.

More precisely, $\aleph_0$ is defined as the least infinite ordinal, called $\omega$, and is same as the set of all natural numbers set-theoretically. $\aleph_1$ is defined as the least uncountable ordinal (in other words, least ordinal whose cardinailty is not less than $\omega$) $\omega_1$, and $\aleph_2$ is defined as the least ordinal whose cardinality is not less than $\omega_1$ and so on. $\aleph_1$ is the least cardinal greater than $\aleph_0$, and there is no cardinal between them.

I haven't heard about such as $\aleph_{1/2}$, and I strongly suppose such objects are not treated in set theory. However as far as I know, the surreal number contains 'fractional ordinal' such as $\omega_{1/2}$. If you want, it might be okay to say that $\aleph_{1/2}$. Of course, they are not real cardinals and no set has such 'cardinality'.