(SIDE NOTE ON NOTATION: The standard aleph series would now be $\aleph_{0,0}$ , $\aleph_{0,1}$$\aleph_{1,0}$, .... The second subindex controls the degree of hyperfiniteness, much like degrees of unsolvability. I could have put it on top, but then it would cause troubles with cardinal exponentiations ).
(set-theory) has anything like the above be attempted?
(algebra) can we create a system of "numbers" which strictly contains cardinals plus other numbers strictly greater than them? And if yes, what is their arithmetics?
(philosophy) is there any speculation as to a radically NEW notion of infinity, which makes all large cardinals small?
NOTE: by 2 I mean: axiomatize directly the class CARDINALS. Then find a new class of numbers, say HYPERCARDINALS, which contains CARDINALS as an initial segment, and moreover such that the numbers in HYPERCARDINALS - CARDINALS has some arithmetical property that ordinary cardinals, no matter how large, have not (this will rule out simply having copies of cardinals appended after one another).
- (philosophy) is there any speculation as to a radically NEW notion of infinity, which makes all large cardinals small?
NOTE: this is of course connected to 2 above, but would interpret the new arithmetical/algebraic characteristics of the hyper-cardinals as speaking of new properties of hyper-infinite classes. Essentially this interpretation would unravel new conceptualizations of the informal notion "being infinite" . Of course, the challenge here is to steer away from blatant inconsistencies, such as the ones discovered in the early history of Set Theory, and which were eliminated in the formalized ZF approach.