Ultrainfinitism, or a step beyond the transfinite

Question

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):

1. (set-theory) has anything like the above be attempted?

2. (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).

1. (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.

Answer 1

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.

Answer 2

In an (I hope) temporary bout of megalomania, I answer as follows. What you and Cantor and others regard as the absolute infinite, $V$, is really only a level $V_\kappa$ of the cumulative hierarchy, corresponding to an inaccessible cardinal $\kappa$ below which there are cardinals that are, in the sense of $V_\kappa$ (but not in the sense of my whole universe), large in all the ways you mentioned. My universe has lots of far larger cardinals, including larger $\kappa'$'s that share the properties I just stated for $\kappa$. All these hyper-transfinite things, which the rest of humanity can't see, make me feel wonderful, until I realize that my hyper-universe does't seem to have any essentially new properties, compared with your tiny $V_\kappa$; indeed, my universe seems to be adequately described by ZFC plus the axiom () that there is a proper class of inaccessible cardinals $\mu$ each of which is the supremum of the smaller cardinals that satisfy, in $V_\mu$, the large-cardinal axiom $I_0$ (or whatever is currently at the top of the large-cardinal chart). My () is a bit stronger than $I_0$, but not enough stronger to impress any set-theorists. So I guess I'll go take the medication to cure my megalomania, and rejoin the rest of the world in ZFC plus (not entirely specified) large cardinals.

To summarize: The intuitive idea of $V$ is that it contains all the sets. If the cumulative hierarchy can be continued hyper-transfinitely beyond your $V$, then your $V$ isn't the genuine $V$.

Answer 3

Perhaps you could take a look at William Reinhardt's paper 'Remarks on reflection principles, large cardinals, and elementary embeddings' (1974). Reinhardt suggests extending the set-theoretic universe beyond $V_\Omega$ (where $\Omega$ denotes the class of ordinals) to some "virtual realm" of larger objects: $V_{\Omega + 1}$, $V_{\Omega + \Omega}$, and so on.

Answer 4

Not really a complete answer, but too long for a comment:

If I understand you correctly, the answer is yes, this idea exists in multiple different forms. The one I find most intriguing currently is Joel David Hamkin's work on the set-theoretic multiverse (http://jdh.hamkins.org/themultiverse/); I'm not sure this is the sort of thing you're looking for, but I think it might be.

As for considering models of set theory which are taller than one another, there is extensive work on end-extensions of models of ZFC, both well- and ill-founded extensions. For example, there is a nice result (due to Barwise, I think) that says that every model of ZFC has an (ill-founded) end extension satisfying $V=L$. And if you don't demand that $M_1$ have ordinals that $M_0$ doesn't (so $M_1$ might just be ``wider" than $M_0$) then inner model theory has quite a lot to say.

Your cofinal tallness axiom sounds very much like the Axiom of Universes (http://en.wikipedia.org/wiki/Universe_%28mathematics%29). Actually, I think that what you propose is much weaker: the axiom of universes doesn't just demand that the universes in question be models of ZFC, it demands some nice reflection properties be satisfied as well.

There are also set theories like NGB or MK which directly treat proper classes; in these set theories, we can directly talk about well-orders of proper class length, so that seems to be close to what you're mentioning. These set theories have a wide range of complexity: NGB is conservative over ZFC, meaning that it doesn't prove any new facts about mere sets, even though it does say things about proper classes.

Answer 5

I wanted to give a minor (but important) point which I think is amiss in this page.

We can classify notions of how big is a collection. In an over-simplification we begin with set (sets exist) to class (classes are definable) to $2$-classes, and so on.

Note that we have to improve our theory's capacity to discuss these things. Namely, we can talk about some collections of things which exist, but not on all collections and certainly not on collections of collections ($2$-classes in ZFC).

Russell's paradox is not really about sets. It is about collections. It tells us that some collections are too big to exist. We can easily replace the "set of all sets which do not contain themselves" with "the collection of all $2$-classes which are not members of themselves", which will prove that this collection is a $3$-class.

This goes on and on, and it proves that there is always a "bigger notion of size". While that for itself is important, and I will get to it in a moment, one can think it through and realize that this is really just assuming there is a $2$-inaccessible cardinal, and saying that everything below the $\alpha$-th inaccessible is an $\alpha$-class (where $0$-class is a set, of course). Then one can continue, on and on and on, until one gets to Mahlo cardinals and so on (as Joel and Andreas have indicated). However, we usually end up back with sets+strong infinity axioms. Not with some crazy new concept of collections.

For this reason, I believe, it is important to actually fix some background universe from which there is no escape. If we assume that this universe contains sets and those sets obey the axioms of ZFC then this universe is not a set, of course. This universe is the absolute infinite and there is no classes beyond it.

Of course we are free to choose for different proofs and needs different "degree absolute" and Number Two to accommodate it with. However this is like deciding to live on a certain planet, for a while, then choosing another planet. We still have to stay in our universe; or dimension; or so.

Let me finish with my philosophical bent (which I have to admit has not yet been fully baked yet): there is such incredible universe which are are not privy to understand or see in fullness (or even know whether or not its axioms include ZFC), inside this universe there is a plethora of smaller universes of all sort of theories (ZFC+large cardinals, for example) which we can skip between whenever we need them.

Being strongly agnostic, however, I do not mean this existence in a Platonist way. I mean, at least for now, inside my head.

Answer 6

You might like the peculiar set theory NFU (Quine New Foundation, but with atoms) extended with axioms which are quite natural in the NFU context and that turn out to be equiconsistent with ZFC plus suitable large cardinal axioms. You can follow the wikipedia page about NF to reach the references. In this world, the strongly cantorian sets make a natural model for ZFC plus large cardinals, and there is a universal set (yes, set, not a proper class).

However, I sometimes dream of something even stronger. Extend ZF (without choice) with something like Reinhardt cardinals, the ones that are incompatible with choice by Kuhnen's proof (even recent attempts were unable to show incompatibility with ZF, even if incompatibility seems not far away). Then this should correspond, in the NFU world, to something where the atoms this time are not much more than sets (something that must happen in NFU with choice, by Specker's refutation of choice in NF), so that suitable models of ZF with suitable Reinhardt cardinals should correspond to suitable models of NFU with so many sets (in comparation to atoms) that a model of NF (without atoms!) should be possible (consistency of NF without atoms relative to some standard set theory is an open problem).

This would be a world where extremely large sets exist, so large that choice functions in the largest collections cannot exists (in italian I would say "assioma dell'imbarazzo della scelta", I have no idea of a proper english translation). A world where Specker's refutation of AC in NF corresponds to Kunen's refutation of AC in ZF plus Reinhardt cardinals (despite the fact that the sequences of cardinals which the two proofs use go in opposite directions). A world that actual set theorists do not consider as "real" (they like choice too universally to restrict it to only to an initial segment of the universe; to model failures of AC they prefer inner models rather than extensions), a world whose consistency is infact unknown. But you asked for people with strong faith in the strong infinity ... [incidentally, bishop Berkley would have been happy with Soloway - Shelah theorem about Lebesgue measurability of every set of reals: he probably would have said that an Analyst can chose to live in a choiceless world, if he like so, but can do this reasonably iff he has faith in the inaccessible infinity]

Answer 7

I would go so far as to say one has to not think of things larger than the collection of classes (in so far as such a thing is even defined) not as just 'bigger sets' but something else. I posit that that something else is ... categories! In particular, not just categories in the fairly vanilla sense as a class of objects (which may be a set) and a set of arrows (or perhaps a class) between any two pairs of objects, but using the first order definition of a category without an equality predicate on objects, and a dependent-type version of equality on objects (can only compare arrows if they are in the same hom-collection).

A small category with an equality predicate on its objects admits a(n essentially) surjective functor from a discrete category if we assume enough choice. In ordinary foundations (such as ZF(C), NBG or variants), Vopěnka's principle is a large cardinal axiom equivalent to the assertion that there are no subcategories of a locally presentable category (e.g. $Set$) which are simultaneously large (have a class of objects) and discrete. The principle can be seen a shadow in ordinary foundations of the idea that there should be categories which are really just too big to have a collection of objects that behaves like a set. Notice that one can form the posetal coreflection $Pos(C)$ of a category $C$ (it has the same objects and there is a unique arrow in $Pos(C)$ between any two objects if and only if there is any arrow between the analogous objects in $C$), and even take the core (the largest subgroupoid) of this. But we cannot get a category with an equality predicate on its objects unless we are happy to form some sort of quotient of $Core(Pos(C))$ to get a discrete category, and then it requires serious use of global choice on these super-large 'collections' to turn the canonical functor $Core(C)\to Core(Pos(C))/\sim$ into a functor $Core(Pos(C))/\sim \to Core(C) \to C$ to get an essentially surjective functor from a discrete category.

As a sort of half-way between this notion of category which is too large to be a class, we have the first-order characterisation of the category of classes, otherwise known as algebraic set theory. One could apply the more philosophical ideas from the above paragraphs to the very concrete definitions of algebraic set theory (see for instance section 3.1 of this introduction). One would then have a category of classes which is itself genuinely not a meta-class, nor some sort of collection which behaves like a class, nor just a 'class' corresponding to a large cardinal in model/universe of set theory containing the current model/universe. This would probably require playing around with the axiom (US) (section 3.1 here).

Mike Shulman has some good comments on similar (though less extreme) ideas in this answer.

If one complains that this is just a first step, and really we want a whole hierarchy of notions of 'bigger than anything we can come up with so far', then Michael Makkai has considered foundations (a sort of type theory, called by him FOLDS) in which it is impossible to consider equality (as above), isomorphism (as might be considered natural in 2-categories, for example), equivalence,... so that we can really only talk about each of these notions if we are working in an $n$-category for some finite $n$, and in general the only available generalised notion of equivalence is full-blown $\omega$-equivalence of $\omega$-categories. But this sort of approach has not been thought of in the sense of making larger and larger hierarchies of objects. (It has come up in Voevodsky's univalent foundations, but only from a homotopy point of view.)

Answer 8

As a sometime student of mathematical logic, I would say that the spirit of your endeavour is as old as philosophy itself. Your recasting and limiting the exercise to use work of Cantor and his formalist successors will put a perspective on the endeavour that will lead many to say that not much new will be obtained. Let me suggest some ideas to help refine or direct your considerations.

What are particular goals for such a research activity? Is a new system of numbers really needed? Suppose that such a system were created as a metric used for some property of classes of a theory expressed in second order logic. Even if you were to enhance the language with a set sized collection of symbols, the multitude of classes so described would be set sized. Even if you decide to start with some ultrainfinite class (much like one has an Infinity Axiom) and produce a large enough language, how many ways can you act on that language to define/produce new ginormous classes which would require you to invent a new system of enumeration? Unless you adopt a language and a perspective and a behaviour where everything you do is of an ultrainfinite nature, you will wind up using subscripts like 0,1,2 to describe the sequence of actions one performs to derive one class from another, and you will end up talking about set many things.

I think you will be more successful in developing a theory of ultrainfinitism if you put on the back burner any notions of relating it to the infinities of set theory, and focus on what it would be like to do unimaginably many things at the same time. For example, consider functions or relations of class-sized arity, and how they can be combined, or consider composition of a ginormous quantity of arrows in some system which bears only a mild resemblance to category theory. It is hard for me to think of doing such things and iterating them on anything away from a set-sized level, but when one has such a system or systems of multitudes in which you can do mathematics, then you or someone else can try to relate it to set-sized systems.

Gerhard "Likes Avoiding Really Big Headaches" Paseman, 2012.06.30

Answer 9

For about two years I have been in contact with a splinter group of amateur enthusiasts who have been striving towards this endeavor. One of their considered objects is "an infinity so large, that it cannot be realized as a von-Neumann-style collection of smaller objects of a single sort". I will attempt to justify that the concept of set is flexible enough that (1) not only is multi-sortedness not necessary for unlocking any novel structure, but also that, post-adjoinment, such objects cannot even be considered "too large to be sets" (2).

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Edit: This section was edited in June 2023 following a comment by user Holo.

The main motive behind the endeavor of defining infinities past $\mathsf{Ord}$ seems to often be the idea that set-hood is a restrictive property, since realizability as a collection of objects of a single sort is something that has to be shed when considering objects built from proper classes like $\textrm{Ord}$. However, behavior of two-sorted models can be emulated by a one-sorted model via a simple translation based on, e.g. $(V,W,\in)\mapsto (\{0\}\times V\cup\{1\}\times W,\in^*)$, where $a\in^*b$ iff $\exists x,y(a=(0,x)\land x\in y)$. Since behavior of proper classes can be emulated via sets, we shouldn't expect novel behavior to appear when considering objects larger than $\textrm{V}$, e.g. the behavior of proper classes can be couched in terms of a "restructuring" of $(V,\in)$, and no special two-sortedness is needed.

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So there is no novel behavior whose existence necessitates abandoning one-sortedness of the universe, for example if we were to adjoin a larger-than-Ord object $\Omega$ to the universe, the apparent behavior of the new $\mathsf{Ord}$ can be couched in terms of a "reordering" of the original $\mathsf{Ord}$. But can we even call the adjoined objects "large"? For example, if we have a model $(M,\in)$ of ZFC, and we adjoin an ordinal $\Omega\notin M$ to $M$, post-adjoinment can we even say that $\Omega$ is "larger than $\mathsf{Ord}$"? $\Omega$ was indeed not a member of Ord^M, the collection $\{\alpha\in M\mid(M,\in)\vDash``\alpha\textrm{ is an ordinal}"\}$. However, after adjoining $\Omega$, even in a structure as small as $(M\cup\{\Omega\},\in)$ we will have $(M\cup\{\Omega\},\in)\vDash``\Omega\textrm{ is an ordinal}"$, therefore $\Omega$ is a member of $\mathsf{Ord}^{M\cup\{\Omega\}}$ and in this extension $\Omega$ is not larger than $\mathsf{Ord}$. The same argument holds for adjoinment of non-ordinal $\Omega$, if all preceding instances of "$\mathsf{Ord}$" are replaced with "$V$".