Cantor has, in the immortal words of D. Hilbert, given all of us a paradise (or perhaps, I would rather say, a great vacation spot), the **TRANSFINITE**.

$\aleph_0, \aleph_1,\aleph_2\dots$

the lists goes on forever, into higher and higher ethereal realms. In his theological mind, Cantor thought of these dots as an eternal ladder, which approaches (without ever reaching it) the *Absolute Infinite*, later re-christened as $V$, the Universe of Sets, by Set Theory adepts.

Those same adepts have enriched Cantor's paradise with a great bestiary of enormous cardinals, inaccessibles, Mahlo, Vopenka, Woodin cardinals, etc. Big fellows, no doubt. Yet... In comparison with the size of $V$ they are puny, nil in fact, no more no less as Graham number, or Friedman's TREE(3) stand in comparison to (for finitists) almighty $\omega_0$.

Now, let us be brave and say: what about breaking through into the trans-transfinite?

What about , for instance, starting from $V$ itself and state *that its size is some hyperinfinte number,* say $\aleph_{0,1}$ ?

(SIDE NOTE ON NOTATION: The standard aleph series would now be $\aleph_{0,0}$ , $\aleph_{1,0}$, .... 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 ).

Wait, I hear you say loud and clear. Are you crazy?

Don't you know that there is NO SET $X$ such that $X=V$? Don't you know that there is no max ordinal?

Yes, ladies and gentlemen, I do know it. But I do reply: and so what? The objection is exactly the same as the one of the finitists vis-a'-vis $\omega$. Someone has broken through the finite, so why not the transfinite? There is no set, but who said that it must be a set?

In fact, start with a pairs of transitive countable models of ZFC, $M_0$ and $M_1$, with $M_0\leq M_1$, of different tallness (the ordinal height of the first being strictly smaller than the height of the second). From the point of view of $M_0$, IT is the full universe of sets, and the ideal ordinals of $M_1$ some unimaginable higher level of infinity. Of course, say you, $M_0$ does not see $M_1$.

True, but we do. And -I think- nothing prevents us from formalizing *their reciprocal relation* as some new theory of sets (the elements of $M_0$) and classes (the elements of $M_1$). Note that here *all sets are classes, but not viceversa*.

Also, being more reckless, we could generalize the above by stipulating an entire chain of ascending hyper-infinities, and perhaps enrich ZFC with an axiom that says that for each model there is a cofinal (in V) ascending chain of taller models, the Cofinal Tallness Axiom....

OK, now the question(s):

(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?

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.

Any reference, thought, criticism, and what not is most welcome.

badform of overloading. – Asaf Karagila Jun 30 '12 at 7:03