Every so often Marvel attempts to explain some of the weirder aspects of the fictional universe described in their comics by invoking some more-or-less obscure bit of science, technology, engineering, or mathematics. I am concerned with the mathematics today, and in particular the way that the mathematics of the transfinite has been used (abused?) to explain the relative power status of various ‘cosmic’ beings (e.g. how can one omnipotent being be more powerful than another?)

One such incident occurs in *Doctor Strange: Sorcerer Supreme* #21, where it is implied that the Marvel multiverse is made up of the transfinite numbers. *Transfinite cardinal numbers* – presumably the authors don’t have ordinals in mind – are the numbers that measure the size of infinite sets (the *finite cardinals* are just the numbers that measure the size of finite sets – that is, 0, 1, 2, and so on). Since there is an infinite series of larger and larger infinite sets, there is an infinite sequence of transfinite cardinal numbers. Georg Cantor, who was the first mathematician to study transfinite numbers, himself makes an appearance on this page, although for some reason he is transformed from a German mathematics professor to a Russian monk (I doubt Cantor ever wore a cool hood like the one in the art – Cantor did in fact go mad, however). But the page makes a deeper mistake: a tranfinite number is not a (cardinal) number greater than infinity, but rather a (cardinal) number greater than any finite cardinal number (thinking about the etymology of the word should have made this clear: “transfinite” is “beyond the finite”, not “beyond the infinite” – presumably they meant something like Cantor’s notion of the *absolute infinite,* which is beyond all the transfinite numbers, but whose existence is highly controversial amongst mathematicians and philosophers who work in this area).

Things get a good bit more confusing in other, later comics, however. In the page reproduced on the right, Kubik is explaining the nature of infinitely powerful cosmic beings to the newly formed Kosmos (I think it’s from *Fantastic Four* somewhere – please school me on the exact reference in the comments!) First off, Marvel seems to equate omnipotence with being infinitely powerful (already problematic). But then, since on this reading of “omniscient” there are lots of omniscient beings in the Marvel multiverse, and some of these are clearly more powerful than others, an explanation of this discrepancy is required. Cleverly, Cantor’s hierarchy of transfinite numbers is wheeled in to do the job.

So far, so good. But the problem is this: they get the math wrong – *badly*. Here is the relevant bit of the conversation between Kubik and Kosmos:

Kubik: Consider then, the set called whole numbers ~ 1, 2, 3, 4, and so on. Is it not infinite?

Kosmos: Obviously.

Kubik: Then consider the set called the even numbers ~ 2, 4, 6, 8, and so on – how long is it?

Kosmos: Why, infinite, of course.

Kubik: Half of infinity is still infinity. And the same would be true of the set of odd numbers?

Kosmos: Of course.

Kubuk: Both sets are infinite, and yet the set of whole numbers contains both subsets, and is therefore twice as large as either subset alone.

Kosmos: …? … !

Kubik: Thus are demonstrated two levels of infinity. There are, of course, an infinite number more.

[*Admission*: I like to interpret the fourth panel – the close-up of a shocked Kosmos where she utters “…? ….!” – as depicting Kosmos’ dismay at what a crappy mathematician Kubik is, especially for an infinitely powerful cosmic being. But I suspect that is not the reading the creators intended – see below!]

Basically, the infinitely powerful Kubik has achieved a cutting edge level of mathematical expertise – or would have, were it about 1600 AD. The confusion here is pretty much what has come to be called *Galileo’s paradox*: How can the collection of even numbers be both half the size of the collection of whole numbers, since we obtain the evens by taking every other whole number, and also the same size, since we can match up each even number to exactly one whole number (and vice versa) as follows:

1 <–> 2

2 <–> 4

3 <–> 6

4 <–> 8

5 <–> 10

and so on.

Roughly three hundred years later, Cantor cleared all this up. Basically, his idea was that two sets are the same size if and only if the elements in the sets can be matched up one-one like the wholes and evens are above, and are different sizes if they can’t. He also proved that for any set (including any infinite set), there is another set that is strictly speaking bigger than the first set. As a result of Cantor’s insights (along with those of other mathematicians including Dedekind, Zermelo, and others) set theory is one of the richest and most productive areas of modern mathematics. But here’s the kicker: the set of all whole numbers, and the set of even numbers, aren’t an example of this phenomenon: they are the same size, or, in more technical jargon, have the same (transfinite) cardinal number, since they can be mapped one-one to one another as shown above (self-serving plug: for a good, rigorous yet readable introduction to all this, see Chapter 4 of this excellent book!)

Now, the obvious explanation of all of this is that the writers at Marvel recalled hearing something about different sizes of infinity in a philosophy course at some point during their alcohol-soaked college years, but couldn’t remember the details (or misremembered them, etc.), and so they just made some shit up. Fine and dandy. But the really interesting question is this: How should we interpret passages such as the one above, where cosmic beings seem to be sincerely explaining the nature of the multiverse in terms of transfinite cardinal numbers, but where they get the mathematics horribly wrong?

The first option is to interpret the incident as one where Kubik just makes a mistake. On this reading, everyone, including infinitely powerful cosmic beings, are fallible, and Kubik just didn’t pay enough attention in his “advanced mathematics of the infinite” course at Cosmic Beings College. But this seems a stretch. After all, setting the mathematics aside, it seems clear that we are meant to take this page to be a sincere and correct explanation of the nature of infinite powers within the Marvel universe (i.e. the creators intended us to interpret it that way and, barring ret-conning or other overt manipulation of the content, it seems that is enough to justify the claim that we ought to understand the page that way).

The second option is to interpret the incident as one where Kubik gets it exactly right. On this reading, the mathematics of the Marvel multiverse is just different than the mathematics that holds in our actual world. Of course, most philosophers and mathematicians think that mathematics is necessary – that is, that the truths of mathematics are the same in any way that the world could have possibly been (further, many think that even in ‘possible worlds’ where the laws of physics might be different, the laws of mathematics would remain the same!) In short, there is a way the world could have been where I was a mathematician and not a philosopher, but there is no way that the world could have been where there are more whole numbers than natural numbers. As a result, Marvel comics are describing a ‘reality’ that is not even possible in a basic, logical sense.

As a result, if we opt for the second reading, then we find ourselves in a conundrum: If Marvel’s comics, and the description of the multiverse contained in them, is a *correct* description of that multiverse, but is also (from the perspective of the real world) completely *impossible* – that is, the world couldn’t possibly have been that way – then it is not clear that, in some deep sense, we can understand these comics in the first place. The characters depicted in these comics live in a reality that is so different from our own that it is not clear what it might mean to say that we understand what living in such a world might be like. But of course we do *seem* to understand what living in such a world might be like, and presumably we think that we get more information about what it would be like every time we read a new Marvel comic. Hence, the conundrum.

Of course, DC might have even bigger problems along these lines – see the Superman panel above.

Alright – discuss!

## 12 Comments

Some thoughts, not particularly structured.

1. What if Kubik is deliberately false example in order to get the point across quickly in (relatively) few words (possibly knowing in his omniscience that he exists as a series of two dimensional images and his speech must fit into word balloons).

2. This isn’t directly relevant because we’re talking about a cosmic entity here, but even if “the laws of mathematics” are the same in all possible worlds, knowledge of correct mathematics isn’t a universal constant. How much would a lack of solid understanding of set theory change the world?

3. Over in pre-Crisis DC there was a “mathematical progression” that was a member of the Green Lantern Corps referred to briefly in the famous story “Mogo Doesn’t Socialize”.

Oops! That’s one too many zeroes, Superman. Remedial Super-Mathematics for you!

I’m going to have to find it (I may have sold off the issue in one of my purges), but there is a relatively recent issue of Spider-Man where he takes a trip into the beyond with the FF and Reed Richards explains something akin to option #2 above – he calls it something like Macro-Science (that isn’t right, but I don’t recall), anyway he claims that at a certain point the complexity of math and physics in their universe is such that it warps perception into understandable (if patently absurd) representations. Thus, a planet eating force appears like a giant man in a horned helmet with a “G” on his chest.

Yeah, I sold it off, but based on my spreadsheet it should be ASM 659 or 660. Maybe.

Ha, I’ve wondered about this too — not specifically Cantor on infinity (although Cantor’s work is, apart from Godel and Church-Turing, just about the only bit of theoretical maths that I know) — but how to understood stories that, prima facie, take place in logically/mathematically impossible worlds. It is striking that the writer here gets the idea of differently sized infinities

exactlywrong. (But what really struck me about that transcript of the dialogue is how Socratic it is, with Kosmos in the role of Meno’s slave, naturally).It seems like part of the background here is a certain picture of how fiction works, viz. that a fiction describes a possible world . Or, rather, since no fiction completely specifies everything in that world, fictions pick out a (potentially infinite, depending on the details) set of possible worlds, constrained by the need to be compatible with the facts detailed in the fiction. So: the “Marvel Universe” refers to the set of possible worlds where there’s an Earth, with a Paris, New York, etc. more or less in the same place as our world, where physics (at least at the level of folk physics) works basically the same as ours, where there’s a guy called Peter Parker who dresses up in a costume and has superpowers, etc. etc. And the problem is that impossible fictions don’t pick out any such set, because there

ain’tany such set.Am I right that the puzzle here relies on some such account of fiction? (It’s certainly why I worry about the puzzle, because that seems to me the most natural account of fiction…but I’m super-naive here because I just don’t know any other accounts). Your statement of the conundrum, at the end, doesn’t strictly assert this picture, but it still seems like the best way to cash out in a slightly more technical way what you’re getting at with “understand[ing] what living in such a world might be like”.

There’s kind of a way out here that relies on the (in this instance, happy) fact that the story about the “Marvel Universe” is so very complicated — to pick out which worlds are being described, we have to reconcile a whole lot of information presented over the course of seven decades, some of which is internally inconsistent. Even given the existence of No-Prizeish post factum rationalisations and explanations, there’s still cases of internal inconsistency (e.g. Batman is x years old, but he’s had more than x years worth of adventures). What we do as readers is try to narrow down the set of worlds described through a kind of not-fully-determinate reflective equilibrium, which inevitably involves throwing out some of the “facts” in some of the stories. And what we need to do here is include this exchange in the facts that don’t survive the process of equilibrium.

There’s always the imperfect interpreter argument, right? That is, the Marvel universe exists, including omnipotent beings, but the only way we get to hear about it is through the offices of moderately talented hacks without much maths. So, Superman can in fact multiply; it’s just the folks writing down his words who got confused.

Nice.

There is a tension there, though, with the general tendency at Marvel and DC to do all that work diegetically rather than extra-diegetically e.g. the Hulk was grey in Hulk #1 not because of a printing fuck-up but because he actually started out grey; Namor the Submariner used to be a straight-up mass murderer when Bill Everett created him, but then he joined the Avengers because he was only suffering from super-mood swings; we keep fucking up the continuity of Hawkman because Superboy punched reality in the face…

The Beyonder also complicates the issue of what’s true in the Marvel universe, since he at least purports to be *bigger* than the multiverse. A nice thread with some relevant panels is here: http://www.comicvine.com/forums/battles-7/pre-retcon-beyonder-vs-thanos-with-the-infinity-ga-1614827/

I am gonna have to find that Spider-Man comic!

Jones: The post was already getting rather long, and I didn’t want to get too into the epistemology of fiction, but yeah, you identified the root of the puzzle, I think.

I would actually put it more broadly. If one thinks that fictions describe possible worlds (ala David Lewis’ account, and along the lines you describe), then the problem is that there are no possible worlds where mathematics works as described by Kubik.

But there are other ways to understand fictions. I prefer something more along the lines of Kendall Walton’s account (which, although he has abandoned, apparently, I haven’t!) On this sort of approach, a fiction is (in effect) a set of instructions for a game of make-believe. Again, however, it is just not clear that I can image a world where the laws of mathematics are vastly different than our own.

Even more worrisome: Cantor’s theorem may actually be not only a mathematical truth, but a logical one (i.e. it can be proven using only pure logic, without any mathematics even). This depends on some rather esoteric debates in philosophy turning out a certain way (in particular, the logicality of genuine, non-plural second-order quantification being logical). But then the world described in the comic isn’t even logically possible!

Maybe another option might be: when Kubik says “Thus are demonstrated two levels of infinity,” he did not mean to imply that these “levels” have different cardinal numbers.

Also, I went back and re-read the first bit from the Doctor Strange comic, and to me it didn’t sound like they were saying the multiverse is made up of transfinite numbers, but that the cardinality of the multiverse (regarded as a set) is a transfinite number. It might have been better if they left that comment between the dashes (about a transfinite number being greater than infinity) out, and just said “For the multiverse is literally a transfinite number of universes.”

Although a little jumbled, it would be a stretch to call it wrong. They are actually pretty spot on. All the “errors” could be interpreted as the cosmic being using slightly different terminology than us.

When they say the universe is greater than infinity, they may mean that is greater than an infinity. For example, it may be aleph_1, and therefore be greater than the infinity that is aleph_0. This makes sense: the multiverse would probably correspond to a greater infinity than to a universe.

Additionally, there is a certain sense in which 1,2,3,4,… is greater than both 2,4,6,8,… and 1,3,5,7,… . It is there *superset*. They are both infinitely large, but the one infinite set is contained in the other, like two levels of infinity.

Where its a little garbled is that: 1) Although it is twice as large in a sense, it could also be said to be three times as large, or equal sizes, depending how you look at it. 2) Saying the multiverse *is* a transfinite number is a bit sketchy. What does that exactly mean?

Anyway, minor things, that could be chalked up to translation error and slight simplifying. The only thing I would have done is made the connection between transfinite numbers and the marvel universe a little bit more explicit. Is it like infinite power levels or what?