Ripheus23

Here's a nifty way to explain how the Dedekind knife maps to higher and higher cardinalities under the auspices of the continuum:

Suppose the erotetic powerset of aleph-zero encodes the first set of functions that can approximate the epistemic decimals of the continuum. Now, this will be the smallest approximation, so it will map to more numbers than are in countable infinity, but strictly less (infinitely less) than in the continuum. [An "epistemic decimal" is just the abstraction over the concept of using decimal/n representations to identify the countable and continuous cardinalities as such.] By aleph-aleph-zero, we will have all of these serrations of the knife at once, in actuated infinity as such. However, these are still only the maps into the real number line, not the real number line itself, so there is still an infinite subset of the real number line that is not contained in the map through the knife, which is additionally constitutive of the continuum as such. I don't know what that subset actually is [unless the axiom of modality and its analogies make sense?], to be sure, if the premises even are to be granted, for that matter...

Anyway, you can also say that each powerset of aleph-zero through -n expresses an increasing implexion in the complete continuum. That is, assign to the continuum a countably infinite order of implexion, where different infinite sets within it are ordered together upwards, towards the completed infinity. By aleph-aleph-zero, we have all the parts in at "adjacent" state, and aleph-aleph-1 is their complete unity in the continuum. So again, c = aleph-aleph-1.