User blog comment:Edwin Shade/Turing Machines and a Problem by Vel! Posed Half a Decade Ago/@comment-1605058-20171010173654/@comment-1605058-20171011151653

There is a thing called the aleph hierarchy, which is exactly what you ask about at the end. For all ordinals \(\alpha\) we define cardinal numbers \(\aleph_\alpha\) as follows: \(\aleph_0\) is the cardinal of countable sets, which is the least cardinal number, and for any other \(\alpha\), \(\aleph_\alpha\) is the least cardinal greater than \(\aleph_\beta\) for all \(\beta<\alpha\). Thus, for example, \(\aleph_\omega\) does make sense.

It is standard in set theory to identify a cardinal with the smallest ordinal of that size, so for example \(\omega=\aleph_0,\omega_1=\aleph_1\) and this way we can speak of \(\aleph_{\aleph_0},\aleph_{\aleph_{\aleph_0}},\dots\), and the limit of those cardinals exists. (by the way if you just say \(\aleph_{\aleph_{\aleph_{\dots}}}\) with \(\omega\) times \(\aleph\), then there is no botton for \(\aleph_0\) to be at. I am being pedantic but I find this extremely annoying)

Moving to your first question, if we assume continuum hypothesis, then we can't really say anything about \(\aleph_2\), except for it being (by definition) the least cardinal above continuum. There is an analogue of continuum hypothesis, called the generalized continuum hypothesis, which says that every \(\aleph_{\alpha+1}\) is the cardinality of the family of all subsets of a set with cardinality \(\aleph_\alpha\) (for \(\alpha=0\), this says that the family of all subsets of a countable sets is equal to \(\aleph_1\); on the other hand, this family has, provably, size continuum). Under this assumption, \(\aleph_2\) corresponds to the family of all subsets of real numbers. Unfortunately, this way or another, this doesn't correspond directly to any natural number system.

Last but not least, as I've said above, complex numbers and real numbers have the same cardinality - continuum, or \(\beth_1\). Instead you really want to have a meaningful notation for your machines, I would recommend just using \(\mathbb R,\mathbb C\) there.