How Big is Heaven, Revisited

[sdi]

When I last looked at the sizes of the various planes, I started from the top, assuming that the One is infinite, and working my way down. But I think I made an error in so doing: the One isn't infinite, or even trans-infinite: It's totally ineffable and incomprehensible in ways that are wholly beyond the reach of even the Gods.

I'd like to borrow a concept from Computer Science—sorry, but I am a programmer, after all—and that is the concept of a Turing machine. A Turing machine is a sort of abstract, idealized computer: the specifics of the way in which it is abstract and idealized don't actually matter, because it turns out that every computer we've ever conceived of is exactly equivalent to it (that is to say, can carry out exactly the same set of algorithms). (There are, in fact, hundreds or thousands of other kinds of abstract computers; my personal favorite is concatenative combinatory logic, but that's neither here nor there.)

What's funny, though, is that we've never built a Turing Machine, or indeed anything equivalent to it, and the reason is very simple: Turing Machines require unbounded amounts of memory (or an equivalent, such as the natural numbers, which can themselves grow without bound) to work with. Even the largest computers, wastefully extravagant or not, still must work within the bounds of the finite. In this sense, realized machines are no more powerful than finite state machines (very simple, even "toy," computers that programmers play with in their first year of university), albeit incredibly complicated versions of these.

So a Turing machine is a strictly abstract, conceptual kind of computer, the sort of thing that's limited to the minds of computer programmers; and here in the material world the best we have to work with is finite state machines. This is suggestive of the beginnings of our conceptual hierarchy, isn't it?

World	Size	Computational Model
material	finite and bounded	finite state machine
ætherial	finite and unbounded	Turing machine
	⋮

But what's above that? Hypercomputation. This is effectively what a Turing machine can compute if it isn't limited by the number of operations it's allowed to complete (e.g. is allowed to "run" forever and still give an answer, which is not a thing that makes sense outside of eternity); that is, a Turing machine that's actually infinite (and not merely unbounded-but-finite). Equivalently, Turing posited the notion of an "oracle machine," which is a normal Turing machine that has a magical, always-correct "oracle" that it can ask for answers and get an instant response. Both of these very naturally relate to notions of "heaven" and "God"—most visionary experiences are accompanied with a sense of "knowing" the answer to questions as soon as one asks them, and haven't we all experienced apparently God-given flashes of insight?—and I think that lets us pencil in another row of our table.

But what's beyond? It's hard to say, and I rather doubt we can know. So perhaps the cosmos looks more like this:

World	Size	Computational Model
material	finite and bounded	finite state machine
ætherial	finite and unbounded	Turing machine
empyrean	infinite	oracle machine
	⋮
	trans-infinite
	⋮
the One	incomprehensible	the One

(Is the Empyrean equivalent to Soul? Does that make the trans-infinite equivalent to Nous? If it is, then we have our Neoplatonic cosmos, but if it isn't?)