How can I create an infinite space bound by another infinite space (or contained within it)?

Question

First let me preface by saying this question will be very math heavy, and might be as equally suited to math stack exchange as world building. For that I apologize, but since the reason for my question is world building, I've placed it here.

In the Universe I'm trying to build, there is the Mundane world (where humans live and things more or less follow the set, natural laws of physics), and the Divine world (where Gods live, and things are slightly less rigid). Both worlds are "infinite," but the Mundane world is still "bounded" by the Divine world.

This is where the problem with my definition begins. If both worlds are infinite, how can one be "bound" by the other (as in contained or embedded within)?

In math, there is the concept of countable and uncountable infinite sets, and, quite non-intuitively, one infinity can be "greater" than another infinity, or can even contain that infinity entirely. I'm looking for a similar concept for my Mundane and Divine spaces, in other words, the Mundane, physical universe is infinite, and yet, still contained within the larger, "more infinite" Divine universe.

The first idea that I thought of, and which I'm sure someone will offer as an answer, is to simply embed our 3-dimensional physical universe (Mundane world) into a "higher dimensional" spiritual universe (Divine world). But this feels like side-stepping the real question and seems very much like a cheap way out. As such, I won't be giving points for such an answer. I'm not fully ruling out a 4 or 5 or whatever dimensional universe, but such a universe needs to allow a way for a 3-dimensional and infinite "Divine Space" to enclose a 3-dimensional and infinite "Mundane Space."

The Universe we know does all kinds of things that seem to "break common sense math," such as Quantum Renormalization and String Theory relying on the axiom that the sum of all natural numbers is -1/12, or Gabriel's Horn, which has infinite surface area in a finite volume. Yet all of these strange things are mathematically valid, and in some cases must be true given experimental evidence for how our physical Universe works, and I'm looking for a similar approach here.

So my question is:

Without simply embedding it in a higher dimensional space, is there a way, mathematically, to describe an infinite space bounded by another infinite space?

Edit: Here is a qualitative, though certainly not quantitative (or mathematically rigorous) explanation of a partial approach:

Imagine we somehow have an infinite hypersphere, one which contains an infinite 3d volume, and we look at it as a projection into 3d space. When you project a hypersphere into 3d space, you get something resembling two spheres inside each other. In this case, the "outer sphere" is the divine world, with its surface facing inwards towards the inner sphere, and the "inner sphere" is the mundane world, with its surface facing outwards towards the inner sphere. Both surfaces are actually 3-dimensional spaces, and both are infinite, yet in a higher spatial dimension, the outer surface "contains and bounds" the inner surface, with the extra dimensionality of the hypersphere being used to "fold" the infinite Mundane Universe so that it is contained inside the infinite Divine Universe.

As an aside, it is trivial to generalize and say a hypersphere contains an "infinite number" of 3d spheres, in the same way a plane contains an infinite number of lines or a cube an infinite number of planes, but this is the "just throw away a dimension" approach I'm trying to avoid, plus in this generalization the 3-d sphere is finite in extent.

What we need is, like the Gabriel's horn approach, to come up with a finite space containing an infinite smaller 3-dimensional space, and that finite space itself being able to be embedded an infinite space of the same dimensionality.

I'm not sure this makes sense, but then higher dimensional thinking never really does to brains evolved to watch out for lions in 3 spatial dimensions + 1 of time... ^^; Is there a mathematical definition closely resembling what I've described?

Edit2: As another aside, perhaps to give some background which may help in coming up with a more satisfying answer, the thought experiment which triggered my asking this question in the first place is this:

In my story, the Divine world and Mundane world cannot interact directly, and must do so through an intermediary "quasi-world" which allows one to pass between them.

This lead me to asking the question that, in this quasi-world (which resembles a forest but has a sky and sun), if you took a rocket and flew straight up, how far would you go and what would you see? Assuming you found you were on a planet, how far would space in that quasi-world go?

This further lead me to asking, if the Mundane World necessarily had to be contained within the Divine World, how could you ensure the Mundane World is infinite, yet still contained within the Divine? Presumably the point at which they join would be the quasi-world, and this would be finite, but the two worlds themselves would also be infinite, with the Mundane contained within the Divine.

Answer 1

Mathematically you are indeed solid, take an infinite set from an infinite set and it remains infinite, counterintuitive but that's because we deal with concrete numbers of stuff in our everyday physical existence, infinity is hard to grasp from that perspective.

In terms of three dimensional spaces your initial infinite space (the Mundane world) must be a folded space like Gabriel's Horn. The second infinite space (the Divine) can however be described relatively simply, it's an infinite plain surrounding the mouth of that Horn. The one space is infinite in living space while still being finite enough in volume to be surrounded by a space that is simply and truly infinite in volume.

Answer 2

In math one infinite set of numbers is larger than another infinite set of numbers, when you can propose a way to count every number of the second set, without using all numbers of the first. Or proving that even when every number of the second set is used, some numbers of the first set are still unaccounted for. For example say you would like to show that all rational numbers are larger than all natural numbers. You'd count: 1 for 1, 2 for 1/2, 3 for 1/3, 4 for 1/4 etc. As you could use all natural numbers and still stay between 0 and 1, you can fit a one dimensional infinity into another one dimensional infinity.

For three dimensions, it could get tricky to make sense of it. Maybe the divine space has a bigger Planck length?

EDIT: So you would spin the analogy further by using 3D coordinates. So the coordinate (0,0,0) in the mundane universe would have the coordinate (0,0,0) in the divine universe. And (2,1,1) has (1/2,1,1) etc. In general (x,y,z) of the mundane universe has the coordinate (1/x, 1/y, 1/z) in the divine world. Simply the mundane world is quantified, which means it has a minimum spacial length, the plank length, and the divine universe is a continuum, like the irrational numbers. So in between (0,0,0) and (1,1,1) of the divine world, the entire mudane universe, no matter of its size, fits. (Maybe the first three paragraphs here make it a little more clear on how to image it: https://en.m.wikipedia.org/wiki/Loop_quantum_gravity This would describe the mundane, our, universe)

Answer 3

It doesn't really impact the story that much, at least what's written so far, but depending on the answer it could guide future developments in the story (such is the point of world-building after all). The closest thing that's happened so far is that the god explains to her divine human counterpart that the Divine and Mundane realms can't interact directly, and must do so through an intermediary "quasi-world" that allows one to pass into the other. Presumably the quasi world is finite somehow. Really what triggered this question was a thought experiment in that quasi-world (which resembles a forest but has a sky and sun), if you took a rocket and flew straight up, how far would you go and what would you see? Assuming you found you were on a planet, how far would space in that quasi-world go? This lead me to asking, if the Mundane World necessarily had to be contained within the Divine World, how could you ensure the Mundane World is infinite, yet still contained within the Divine

So, I'm no mathematician, but I am a philosopher so I'm going to try and answer the question strictly from that perspective, in terms of how it's going to impact reality for your characters.

If I'm following the above description, then yeah, the Gabriel's Horn is the simplest way to describe it; e.g. infinite in some dimensions, but not in others. In this case your Gabriel's Horn would be a four-dimensional shape, with 'volume' being replaced with the fourth dimension.

In a PRACTICAL sense, the result of this containment (I think) would be that from anywhere in the Mundane world, you could pass through the Quasi-world into the Divine, but the reverse would NOT be true.

E.g. you would have to be in certain specific places in the Divine world to pass through the Quasi-world into the Mundane, and passing from the Mundane into the Divine could only result in you arriving in those same specific places.

Answer 4

So there's countless ways to do this in mathematics, but my personal preference is to build up a world as discussed by Dan Willard. I offer it as an answer here because it has what is, in my opinion, the most interesting way of embedding infinities I have run into.

In his paper, Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles he explored self referential systems. These are systems which can describe their own behavior. This is a really desirable trait, because it means the universe can be truly understood by an individual within it. Unfortunately, there's a problem if you want to prove everything in the universe. A peksy little set of theorems known as Gödel's incompleteness theorems raise their ugly little head. They show that if said universe can prove all of the truths in arithmetic (i.e. can prove that 2+2=4), and are self referential (you can use the system to prove the rules you used to prove 2+2=4), that the system has to be inconsistent.^†

This has been a bit of a pest for philosophers and mathematicians. It's desirable to be able to prove everything and have all of arithmetic be true. Gödel kind of rained in that parade.

Dan Willard explored systems which contained all of the truths in arithmetic as we know it except for totality of multiplication. Normally we assume that if I have two numbers a and b, then a*b is also a number. He explored what happens if you remove this assumption. It turns out to be enough to let you build a universe which starts from an infinite set, and divides and subtracts down towards 0 and 1. You can then use that to prove everything in his arithmetic, and indeed prove everything in the system. By relaxing that one rule, he sidestepped a particular step in Gödel's theorems (diagonalization, if you're curious), and made a self-referential system which could prove its own consistency and contain all of arithmetic (except totality of multiplication).

So why do I bring this one up? Well, there's a really interesting curiosity that comes up when you explore these systems. They can be created within an existing proof system. For example, we can take a system which proves arithmetic as we know it, and construct a "Willard World" inside of it --- a self-referential system which can prove all of its own statements, and all of this relaxed arithmetic.

To do this, you start by constructing a countably infinite set. You then use this set to construct the system. The funny thing is that there are some countably infinite sets which, when you are done, are provably uncountable within the self-referential system.

So you have a construct which is provably uncountable to entities defined within the system, but which is provably countable to entities which are above the system, even though the construct is the exact same construct.

This means that there are cases where an entity inside the system sees something which cannot be attacked via mathematical induction, but a "deity" on the outside can see the mathematical induction and prove it without trouble! (In particular, these strange patterns show up when one tries to prove whether you can construct two numbers which, when multiplied do not yield a number).

I am reminded of when the Buddha challenges the Monkey King in Journey To the West:

The Buddha said:

"I will make a deal with you. If you can somersault out of my right palm, then I will let the Jade Emperor give you his power; otherwise, you will have to cultivate for thousands of years on Earth."

Looking at the Buddha’s palm, which was no more than a foot in length, the Monkey King smiled to himself and hastily said: “Are you sure you can handle this?” The Buddha said: “Yes.”

So the Monkey King stood in the middle of the Tathagata’s right palm, feeling that the palm was no bigger than a lotus leaf. He did one somersault and kept moving forward until he saw five huge pillars.

He had surmised that he had reached the end of the Heavens and to prove his trail, he urinated at the bottom of the first pillar, pulled out one of his hairs and said: “Change!” He then changed the hair into a big brush and wrote on the middle pillar the words: “The great Sage as high as Heaven visited here.” Th Monkey King returned to the centre of the Buddha’s right palm with another somersault and shouted to the Buddha: “I left and returned; you should now let the Jade Emperor give me his power.” The Buddha said: “You monkey, do you know that you are still in my palm?”

The Monkey King said: “You just don’t know that I went to the end of the Heavens and found five red pillars. I left a sign there. Do you dare to go with me to check?”

The Buddha said: “There is no need for me to go and check; you just look down and you will see.” The Monkey King looked and found that on the middle finger of the Buddha’s right hand, there was a line of words: “The great Sage as high as Heaven visited here.” And there was also a strong smell of urine in the Buddha’s hand.

The Monkey King was very surprised and said: “How can this be. I wrote these words on a pillar that supports Heaven, but how can it be on your finger? No, I don’t believe it. It is impossible.” The Monkey King tried to escape from the Buddha’s hand, but he turned over his palm and changed his five fingers into a mountain of the five elements of gold, wood, water, fire, and soil, and suppressed the Monkey King under the mountain where he remained imprisoned for five centuries.

(abridged source)

^† This is a very informal wording. For the airtight wording, please refer to the explicit mathematical phrasing Gödel used.

Answer 5

As a couple of the comments hint at, it's difficult to give a precise answer to your question because it's using vague terminology. Mathematics is a stickler for precision, so I'll give a couple of possible interpretations to your question:

Embeddings

I know that you were looking for answers other than embeddings, but I believe you likely have a somewhat incomplete knowledge of what the word means in a rigorous mathematical sense, so I think it would be illuminating to delve into the subject in more detail.

Embeddings, as wikipedia assures us, are structure preserving injections. What the hell does that mean? Well, in the modern formulation of mathematics, everybody loves to build stuff out of sets, which for the purposes of this discussion are just collections of objects (although it can get considerably more complicated if you delve into the details). They like to do this because they're powerful, yet have a remarkably intuitive feel (a set is just like a bag of mathematical objects). Now, sets are all well and good, but what we'd really like to do is establish relationships between sets! For instance, it certainly seems like the sets {0, 1} and {a, b} have something in common, doesn't it? But neither of them have any elements in common, so how could they be anything alike? This is where functions come in!

Without getting bogged down in the actual formal construction of a function, they can simply be thought of as little machines where you throw in an element from one set, and a unique element from the other set pops out. This uniqueness is important-- if we call our function $f$ and denote the value that $f$ spits out when $0$ is thrown in by $f(0)$, then $f(0)=a$ and $f(0)=b$ are both valid function values, but we can't have $f(0)$ be both $a$ and $b$.

Now, an injection is simply fancy math speak for a function where every different input spits out a different output. For instance, if our function $f$ goes from $\{0,1\}$ to $\{a,b,c\}$, denoted by $f:\{0,1\}\rightarrow\{a,b,c\}$, then $f$ defined by $f(0)=c$, $f(1)=a$ would be an injection, while that defined by $f(0)=f(1)=b$ would not. From this example, it should already be clear how injection provides a sense of a space fitting inside another one. A related object is called a surjection, which instead refers to a function where every member of the target space pops out of the function given some input (it could happen for multiple inputs). Astute readers may note that it is impossible to construct a surjective $f$ with the input and output spaces described above. As a final piece of terminology, a bijection is a function that is both surjective and injective. Bijections determine what sets are "equivalent" to each other in a precise way-- sets that have a bijection between them are said to have the same cardinality, which is a fancy word for size. This is the precise mathematical description for how {0,1} and {a,b} from above are similar to each other.

As an important aside, for finite sets, we have the nice property that if there is a non-surjective injection from one set to the other, there can't possibly be an injection going the opposite way (this is much less surprising than it might sound, just look at the example from before). But this isn't true for infinite sets, which is why you hear all sorts of weird results like the rationals being the same size as the natural numbers. So simply having a non-surjective injection between the two states is no longer enough to guarantee a notion of one space "fitting inside" another space. Now, there are different cardinalities of infinite sets, but most of the time when physicists talk about one space fitting inside of another space they're talking about spaces of the same cardinality.

The distinction we're missing here is the structure-preserving part of the definition. Let me level with you-- sets are kinda bland. I mean sure, they're useful as building blocks, but by themselves they just kind of sit there. When physicists model the world, they like to have lots of structure to the spaces they use, because the world has lots of structure. These spaces can have all sorts of stuff defined on them, from notions of closeness, to an order, to operations that take two elements and spit out a third. When you require that an injection preserve the structure of these spaces, it severely limits the number of functions that can be created, and gives you a much more meaningful sense of a space "contained in" another space. For instance, there are plenty of injections from 3-d space ($\mathbb{R}^3$) to the plane ($\mathbb{R}^2$), but there are none that preserve the linear structure of euclidean space.

The important thing to note is that this concept of embedding need not be applied only to vector spaces, like you seem to implicitly be assuming. To name a few possible types of spaces: we could have metric spaces, topological spaces, groups, or smooth manifolds, which give rise to the structure preserving maps known as isometries, homeomorphisms, homomorphisms, and diffeomorphisms, respectively (nitpick alert-- technically most of those are the names for bijective structure-preserving maps, but embeddings are bijective if you only look at the image instead of the whole co-domain). Don't worry if you don't know any of those words-- I just wanted to give you a sense of the incredible richness of embeddings.

Boundaries

To be honest, I find embeddings to be a much better sense of a space "containing" another, but I figured I'd include a brief discussion of boundaries since you mentioned gabriel's horn. Boundaries hail from the world of topology, which is essentially the study of having a sense of how close stuff is to each other but without being able to measure actual distances between points. More or less, the boundary of an object is the collection of points that you get if you take a bunch of spheres and shrink them down as small as possible and then only keep the ones that never fully resided inside the object or outside it. This should be taken with a grain of salt, since a sphere isn't a concept even defined in general topological spaces, but I believe it gets the point across.

I suppose you could say a boundary "contains" an object, but it'd be more in the sense that the crust of a loaf of bread contains the bread, as opposed to the sense that the universe contains a loaf of bread.

Answer 6

Two points:

1. Why can't they just border each other, each infinite in one direction, but not the other? If you split a line into two days, each is still infinite, but they do connect. The downside of this is that there would be one giant wall in space where people could pass through, but nowhere else.

2. What about, rather than having the human world embedded in the divine world, have both of them embedded in a 4D world, one lying "atop" the other in the extra dimension. Imagine laying one sheet of paper on top of another. Would this work?

3. One final solution. Just say "F*ck it, we're dealing with a deity, presumably the entity from whom all logic and mathematics originates. They can screw with the rules of math as much as they want."

Answer 7

The mundane is one facet of the infinite facets of the divine.

1: I walk down the street. I enter the store. I buy a cup of coffee.

2: I walk down the street. I hear my footsteps. I smell something good; a tree in flower. I enter the store. A young man looks up. I buy a cup of coffee. I note the barista gives me the second cup down from the stack.

3: I walk down the street. One sock is not on straight. I hear my footsteps, and a lawnmower in the distance. A bird is singing above me. I see it and smell the flowers of the tree it is in. I enter the store. The door creaks. I see a childs handprints on the glass below me. A young man looks up, and looks back at his phone, disappointed. I buy a cup of coffee. The barista gives me the second cup. She has a piercing in her ear.

4: I walk down the street. One sock is not on straight, or maybe there is something in my shoe. I hear my footsteps and a lawnmower in the distance; it is my own lawnmower that I lent Boris last week. A bird is singing above me; no - it is a baby bird and it is begging. A parent bird arrives but is slow to feed it. Eventually it does. The tree is in flower. I see the flowers on the grass and the sidewalk; they are falling. I enter the store through the creaking door. I see a child's handprints on the door and I think I know that child; her handprint is from an ice cream drink I saw her get last weak. The young man looks up and then back at his phone. I am not the one he is waiting for. The barista gives me the second cup, full of coffee. She smiles because I come in all the time, and I tip. She has a piercing in her ear that she did not have last week.

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Our mundane world is encompassed in the divine. The space of the mundane is infinite, and the space between your front door and the coffee shop is infinite. The difference between the mundane and the divine is perception of detail. The mundane is one facet of the infinite facets of the divine. No-one, not even gods, can perceive the whole of it.

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An addendum for an OP that wants math. The mundane is unbounded and so has no shape - it extends infinitely in all directions. The divine is within and around it.

The mundane is the infinite set of real numbers. The divine is the infinite set of real numbers and the infinite set of decimal numbers between each real number.

Answer 8

Math does this all the time, you don't even have to go to great lengths

The set of rational numbers "Q" is contained within the set of real numbers "R". Both sets are infinite, but "Q" is countably infinite an "R" is uncountably infinite (therefore "larger").

Some of the neat properties of this relationship are:

1. "Q" doesn't need the real numbers to do basic math operations +,-,*,/. It's "closed" under these operations. It's only when you introduce algebra that you need the real numbers. This is akin to the normal world being self consistent, but the divine offers something more powerful.
2. The real numbers "R" include all the rationals "Q" but have all the points in between the rational numbers. It's like the rational points are peaks on a mountain range, but the real numbers are all of the mountain range, the valleys, the slopes, everything. And there's a giant who's dancing on the mountain range, who's feet only ever touch the peaks, he doesn't know anything about the rest of what makes up the mountain range, because the peaks are all he interacts with, but really there's much more to it if you were to look deeper.

Answer 9

I do not like all this questionable "math magic" (you know, Gödel and all that stuff). From phisical point of view:

0) Our "mundate space" is not infinite by current scientific consensus. But lets asume plain old infinite Galileo-Newtone universe (with tired light to make skyes dark).

1) Dark matter/energy/sub particles/some exoctic particles It IS the divine space and it is embeded or bounded by any phisical partical of our mundate space.

2) Virtal particles/vacuum quantum fluctuations. Since there are much more vacuum, then particles volume, it a good place for divine space. And mundate space is truly embeded into divine one.

3) Quantum time phasing. Just like single core computer multiprocessing works. You have 1 phase to "execute" mundane world phisics, and 10^N phases for all the divine stuff (or vice versa). Embedment happens in time dimention. It is a good way to justify 1 day = 1 milloin years time pace difference

4) Virtual reality. This one is harder and closer to mathematics. One of the universes is just sort of turing mashine and other is a memory contence (and a programm) of this machine. Since this "mashine" is infinite and have an infinite memory, second one can be effectivly infinite too. Wich space is wich - is up for you to decide.

I ommit vulgar multuverses and multu dimentions, but they are valid variants.