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):
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(set-theory) has anything like the above be attempted?
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(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.
Best Answer
My view is that the large cardinal hierarchy already has all the principal features of the project you are proposing.
Each of the large cardinal concepts can be regarded as corresponding to a certain conception of the set-theoretic universe, if one should entertain the von Neuman hierarchy up to such a cardinal, and this makes a perfectly good universe. Every inaccessible cardinal $\kappa$, for example, gives rise $V_\kappa$, a transitive model of ZFC and a Grothendieck universe in fact. Every Mahlo cardinal $\lambda$ is a limit of many inaccessible cardinals $\kappa\lt\lambda$, and the models $V_\kappa\subset V_\lambda$ have much the same relation as what you describe in your question. If $\lambda$ is Mahlo, then the smaller models $V_\kappa$ for inaccessible $\kappa\lt\lambda$, which are perfectly good set theoretic universes, each extend up to $V_\lambda$, a larger universe having what it thinks is a proper class of inaccessible cardinals (and hence also the Universe Axiom). Indeed, when $\lambda$ is Mahlo then the collection of inaccessible cardinals is not merely unbounded in $\lambda$, as you request, but also forms what is known as a stationary class in $\lambda$, meaning that it intersects nontrivially with every closed unbounded set. This seems to extend and refine the idea of your cofinal tallness. Similarly, every weakly compact cardinal is a stationary limit of Mahlo cardinals, and if $\delta$ is a measurable cardinal, then not only are the weakly compact cardinals below $\delta$ stationary in $\delta$, but they form a set of normal measure one, a much stronger notion. This reflection phenomenon is nearly universal in the large cardinal hierarchy, where properties of the larger large cardinals reflect down to robust classes of the smaller cardinals. The strong cardinals reflect in this way down to the measurable cardinals, and the Mitchell order carries this idea still further. Supercompactness reflects down to superstrongness. It is an intensely studied phenomenon.
In this sense, the subject of large cardinal set theory is already undertaking your project. What we are studying is precisely how all the various large cardinals can be construed as smaller universes extending into larger ones. For the large cardinals that are axiomatized in terms of the existence of certain embeddings $j:V\to M$, this extension process proceeds in two ways: $M$ is larger than $V$ in the sense that $\text{ran}(j)\subset M$, and $M$ is smaller than $V$ in the sense that $M\subset V$. It is the interplay and tension between these two sense that gives rise to much of the power of these axioms.
I would say that this includes elements of algebra, broadly construed, if one regards the direct limits and large systems of large cardinal embeddings that arise in the theory as having an essentially algebraic aspect. Surely the extender embeddings concepts developed in the theory of canonical inner models exhibit a fundamentally algebraic character.
And the subject is hugely involved with philosophical considerations, which guide the choice of new large cardinal axioms as well as motivate or attempting to explain why we should believe that they are consistent or true. One can say infinitely more about this.