Cantor compares the infinity types of the set of naturals N, with
that of their powerset 2^N : the collection of all its subsets,
using combinational concepts like set, subset, intersection and union.
This is the weak point of his presentation, because a much clearer
point of view is the sequential genration aspect of the sets involved.
Namely: N is sequentially generated by +1, the succesor *function* of
Peano, in short: N = (+1)* where the star* is repetition. So N is not
just a set, but one with a sequential ordering, that of a 'closure'
with 1 generator, in short: seqdim(N) = 1.

Now we all know (I hope) another more complex seq.closure of seqdim=2
namely any finite group G_m of all m! permutations of set S of m states.
Two generators can be chosen to be a full m-cycle, and any 2-cycle.

Then notice that the concept of "subset" can be taken as the "fixpoints"
of a permutation, and its complement: the subset of "movedpoints" of the
state-set. So all 2^m subsets of set S with |S|=m can be seq_generated
by just two generators: the next level beyond Peano.

Now N is not a group, since the inverse -1 of 1 is not in N.
So take the integers Z, with two generators +1 and -1, which is an
infinite group. And all its permutations Z! are generated sequentially
by three generators: A={+1, -1, sw2} where sw2 is a 2-swap, say 0<-->1.

Then Z! = A*/Z  -- like an Integer State Machine IMS(Z,A), with
          stateset Z, and input alphabet A of only 3 generators.
And 2^Z, the Boolean Lattice of all subsets of Z, is contained in a
closure of seqdim(Z!)=3. Apparently, this is again one step beyond N
and Z, forming a nice   hierarchy: N < Z < 2^Z < Z!
                     with seq.dim: 1   2    3'   3

Here 3' is artificial, since powerset 2^Z is the only (Lattice) closure
that is not sequential, but combinational (meaning its operators are
idempotent: intersection, and union). This difference in *type* of
object between N, Z, Z! on the one hand, versus 2^Z on the other, makes
Cantor's theory so artificial. It is not only the cardinality of the
infinities involved, but more basically the *types* of object (combin.
resp sequential), and their seq_dimension, that should be considered
and clearly distinguished. Which, by the way, is already evident in
the finite -- a great advantage: Cantor's Theory can perfectly well be
'pulled-up' from the finite, via induction (Peano's sequential tool,
the basis of constructive proofs, since the Greeks of old...;-)

I hope to have convinced you that, although his theory is a bit cripple,
as it were, he discovered something very essential: higher dimensional
objects beyond Peano's naturals (of seqdim=1).      Unfortunately, the
presentation leaves something to be desired.   Moreover, the obsessive
urge to map these higher dimensional objects onto the "linear" real line
is not helping very much. Rather leave the subsets of N cq Z in their
own character, with Lattice closure as their natural habitat.

And the "sequential logic" of groups (and my hobby: semigroups & State
Machines, especially the finite ones;-), with their associative algebra,
is a much more natural context for further developments than the 'old'
and a bit stuffy (last century) type of set theory, don't you think ?-)

I made an initial attempt to describe this approach,
    from the finite and sequential generative standpoint,
in:                http://www.iae.nl/users/benschop/cantor.htm