# Limitless Space

In mathematics we have the concept of infinity, so we speak about infinitely big things, we compare them, we label and order different kind of infinities and we also have the "infinitely" small.

However for as much as we put our heads down, we can't really grasp the concept of infinity, but although it might sounds weird, we can instead easily and intuitively describe the order between different sizes of infinities.

Now let's talk about space. I remember in school there was someone who talked about the paradox of distances and how we can always walk half of the distance between two points and repeat it to infinity, without getting to the other point (the example was a turtle walking I guess).

I am thinking about the size of what exists. ALL OF WHICH EXISTS. I don't have a definition of existence, neither I want to talk about ontology, however suppose the first:

### SPACE IS FINITE

If it is, then either it has a border or not. Now, if it has a border, then there is more space after the border, or it wouldn't be a defined border right? A border is made of two sides, one inside one outside (or not? Look at the Möbius strip) so the one outside touches some more 'space' and hence we can repeat this indefinitely - paradox.

Suppose it has no border. Then the only explanation would be a limit, in the mathematical sense, i.e. we can keep on going forward and getting closer to an imaginary border, but there is no border. Of course the distance here should be affected by it and the perception of distances would be shrank down. However if so, would it actually not be a paradox as in the physical world there seem to be no way to actually infinitely approach something without finally reaching it. Also, this would show that the space is infinite inside a finite space? If this makes sense.

Conclusion: it must be infinite.

### SPACE IS INFINITE

Well if it is infinite, then there is no border and no limit! So we can always indefinitely go forward. However would this mean that the space would create itself as someone travels there? Because if it is infinite and it instantaneously exists, then how did it even start? And if it did not start, how did the things even get there in the first place? Unless it is a simulation, then something must have been moved there or created. If it is infinite, why then everything else is finite? In an infinite world, there should be infinite things!! But everything else seems finite!!

This to me looks like a contradiction. For example, why aren't there some infinitely large galaxies? It would be possible I guess in an infinite space (not according to physics, however to be entirely honest an infinite space wouldn't have much explanation either in physics!)

Hence looks like we have a paradox and therefore it must be finite!

So these things contradict themselves! So are they both true or both false? Normally two contradictive things can't have the same value, one must be true, one false or we fall in a paradox. But here only one of them cannot be correct, and so we fall again in a paradox.

What can the solution possibly be?

• "repeat it to infinity" (as if infinity were a place itself, or a number) vs "repeat it infinitely" (as in a process that can continue indefinitely) - in either case, no one lives long enough to do anything infinitely nor does anyone walk anywhere by first walking exactly half way there (like a "point" - where exactly is "halfway"??) . Purportedly Diogenes simply got up and walked away when he heard Zeno's paradox... You can concluse all sorts of stuff from any old supposition. Dec 26, 2016 at 3:27
• There are no Infinite sets. Dec 26, 2016 at 9:29
• I don't think it makes sense to speak of the properties of space without establishing some basis for which things may be predicated of it. I suspect that in practice, whatever definition that is employed presupposes an independent spatial framework within which space is thought to exist, i.e. space within space.
– user3017
Dec 26, 2016 at 9:49
• @Mr.Kennedy I apologise for the bad writing! What you say is true. However what I would like to emphazise is not one of the two options, but rather the fact that I guess only one of them should be true as they are opposite of each other,however both seem to lead to contraddictions! Unless there is some weird properties where the space can be both finite and infinite at the same time,maybe for some law of physics that work only for massively big objects like the universe itself. Dec 26, 2016 at 12:12
• Comments are not for extended discussion; this conversation has been moved to chat. Dec 26, 2016 at 20:25

In physics, infinity is usually a sign that our modelling of the physical situation has broken down; generally, the only kind of infinity that is allowed is the potentially infinite; this means that at any time a certain quantity is finite, but it may grow in size later.

There is a proviso: Infinite space, however seems a possibility on the face of it, as does infinite time: the antimony that you've pointed out is a classic one as also pointed out by Kant.

But if we instead pursue the notion that potential infinity in physics is the only possibility as axiomatic we can see that elapsed time must be finite; we can also say that physically, an infinite elapsed time is not credible, for how would we have reached the present moment? The error lies in viewing the past as like future, in being potentially infinite; however, the past is the past because that moment was lived through, and for that reason is finite. The future does not have that character, it is yet to be lived though, and so can remain potentially infinite.

Now, this means that the universe began somehow, and in some place with some extent; it being the universe, it must all of the place and to the extent of that place.

Further, we can ask the question, did it begin all at once in a particular finitely bounded place, or over an infinitely bounded region? The former seems the more likely, for if it were to begin over a whole infinite region this seems unreasonable as we know that cause and effect in the universe is local in character - there are no effects at a distance - creation must have been coordinated somehow; so we take it as reasonable that the universe began, and when it began it was finite in extent.

When we further suppose change, be it growth or diminution, is continuous, then the universe can only change continuously in the finitely elapsed time since it's 'creation'; and finite change from a finite beginning means finiteness at present, and hence a finite universe at present.

Thus, from a reasonable set of physical assumptions we have shown that the universe ought to be finite in scale, temporally & extensively - and this is also consonant with currently well established scientific consensus (excluded speculative notions of multiverses and their like).

• The appearance of infinity is only a sign that our modelling as broken down when it appears in a place where the model should have, for example, a real number.
– user6559
Oct 2, 2017 at 21:16
• @Hurkyl: infinite precision is different from the infinitely large; however, in what I've written above, I'm taking for granted that all procedures for extracting or placing a finite number have been tried at the time the model was constructed - hindsight is a wonderful thing... Oct 5, 2017 at 8:26
• Maybe I should give an example of what I mean. For a question like "How far is it from here to there?" an infinite answer would suggest something has broken down. For a question like "How many distinct places are there in the universe?" or "How much precision does the real world have?", an infinite answer is among the expected possibilities and doesn't suggest a breakdown. (and, of course, questions like "how many integers are there?")
– user6559
Oct 5, 2017 at 8:39
• @Hurkyl: 'infinite precision' is an expected answer if you subscribe to a continuum picture of infinite divisibility, I don't think that's been settled yet. Oct 5, 2017 at 8:43

Let's start by defining our space as (3D + 1T). As each of its components grows, our universe "grows." However, at any instant of time, our universe is finite. But, even though it is finite, we can not detect or reach its "border" because we can't see or move in the "fourth dimensional' direction.
The best analogy I can give you is: a very thin, transparent balloon, with a virus on its surface. The virus can only see and move North, South, East, and West. So, if you tell it to move Up, it will say, "you mean North?" And no matter how much you explain, it will never comprehend the "Up" direction.
In conclusion, although our universe appears infinite, it is finite, but it is also expanding.

• someone brave enough to attempt an answer, thank you! Although, I can easily recall some talks given by Neil deGrasse Tyson explaining that all we know so far is due to (of course) our observatory tools, our time and the fixed (and finite) speed of light. We can receive information up to 13.7 billions light years (tbh I dont remember the exact number, its probably way bigger) but we have no idea of what is after that horizon! The expansion is proven by many data, however what we know is just that what we can see has been expanding since the big bang, we dont know more, or do we? Dec 27, 2016 at 1:27
• The one thing is that time is some function of space. I. e. if space returns to some state, then it reaches the same moment of time again. Sep 28, 2017 at 14:37
• Ask the UK answer to Tyson, Brian Cox. He comes down on the other side of this issue. By the current theory of the cosmic background radiation we do know -- there is nothing. There can be nothing beyond or behind the impression left by the uneven distribution of heat at the start of time. Everything that exists is younger, and therefore closer, even the space. Space is thus potentially and not actually infinite. If, as mathematical Intuitionists insist, mathematical infinity just represents an ongoing process, then this is a perfectly good answer. Space is young, and its growing...
– user9166
Sep 29, 2017 at 20:29
• It seems to me that if space is growing the OPs objections are still valid and remain unanswered. For Cox and Tyson space is paradoxical, and this is because they think it is real. Anybody who believes space and time are metaphysically real must face endless paradoxes.
– user20253
Oct 4, 2017 at 11:24

Perhaps this is out of place, but there seems to be an aspect of it that is sometimes neglected in the discussions about time and space. I was thinking about the role of our brains in creating our experience with space and time. Our brains are perhaps not all processing the information (conceivably coming from some sort of unseen common environment) at the same speed and in the same way. We each have our own little private reality being generated by our brains. That reality is limited by the sensory experiential capacity of our brain states. It unfolds for us at the rate our brains are processing the information, and our perception of the passage of time can change depending on our states of consciousness and the ways our brains are functioning.

Our brains are filtering out or not even registering massive amounts of information from the potential environment to create each moment of conscious experience. Our little worlds, that seem to be cohesive stable spaces with a bunch of objects behaving in regular ways, is a creation of our brains. It may correlate in some way to things happening outside of it, but some of what we observe may be effects occurring because of the way our brains are wired and the way our neurons are working. Or we could say the environment or reality we experience with our bodies and so forth seems to be a simulation created by our brains. The actual environment where our brains reside may be wholly different from the one we experience, our space and time, objects, and so forth.

A paradox that seems incomprehensible because of our intuitive understanding of spacetime from our immediate environment or experience may take an entirely different form beyond our understanding or ability to imagine in the actual world that is hidden from us. Something about these infinity paradoxes and the potential impossibility of reconciling them could be behind why we exist and why we have a sense of movement through time, perhaps because there is an infinity where there are always more possibilities that can be created endlessly, creating a growing or expanding limit. Our brains must slice out little snippets of the potential to make our stories. We need to observe the world as a single cohesive narrative unfolding with things seeming to move around at a certain pace in a space, otherwise, it seems that it would just be a chaos of meaningless disjointed experiences or we would not be conscious of it at all. Our brains are perhaps somehow putting the appearance of something finite together in our immediate field of awareness, taking limits in some probabilistic way or picking things out from a larger set of possible information states, to create an evolving moment of experience and interpretation of spacetime events.

There was something about limits of consciousness and infinity I was thinking about in response to the OP but I'm not sure if any of it makes sense, so I'll just stop here and perhaps edit it later. Sorry, I'm new to this site. I'm not sure how formal this is supposed to be. I'm not sure if this is helpful or welcome or anything like a conclusive answer to something, but there could be something in there for the OP to think about while imagining space going on forever in the mind.

I think your conclusion that space can't be finite is wrong. There can be finite space without a border or a "limit". Think of the Earth's surface. It's clearly finite but it has no border and wherever you go it doesn't involve walking towards a "limit". The same construct is possible with three dimensions (even though harder to picture, see Poincare conjecture).

• what you say is right, now comment about it. However there's the trick! you see, that happens cause there's more to earth. Of course earth is finite and if you hypothesise a finite universe they seem related. But they are really not, indeed a finite earth would just be a finite object in a bigger system, whereas the universe would be a finite system instead! So for the earth your idea works because we have the atmosphere after earth, then the outer space and the solar system and so on. So it is finite but with no boarder because there is more after earth! Dec 28, 2016 at 16:24
• However this doesn't work for the whole universe. By definition, with universe you mean everything that exists! So that something can't be finite with no boarders or limit! you see my point? the earth doesn't need a limit or a boarder, because it is inside a system. Of course objects normally have boarders etc anyway. But the real issue is with the whole system! Now immagine that the universe would consist of only one compact object, say a metal sphere. Then if it is finite, is should have a shape! But in having a shape, it has boarders or limits. Dec 28, 2016 at 16:27
• But at the same time, having a shape implies that there's something alter from it. And here is the contraddiction! I will have a read at your attachment, although I recall reading it time ago Dec 28, 2016 at 16:28
• Maybe it's just a paradox because our minds are not equipped for this kind of thinking. When I give the example of the earth's surface you probably imagine it from an astronaut's ("outside") perspective. And then you try to "zoom out" further and "look" at the universe the same way. Of course, this fails, but it doesn't necessarily mean it's a contradiction. Maybe it's just something we have a hard time thinking about. From a different perspective maybe there doesn't need to be "more space on the other side" because there is no other side. (Of course, it doesn't need to be that way!)
– oW_
Dec 28, 2016 at 17:05

Everything we think we know about the universe could be wrong. But what the hell, lets start with what we think we know. We currently believe space is a dielectric substance which is why electromagnetic waves (including photons) pass through it. We suppose space is compressible (higher density near mass and less dense away from mass) per deflection of light around planets. Space seems to determine the speed of time (slower in high dense space; faster in less dense space) per slowing of time from our perspective at event horizon of black holes.

Now take a mental leap from what we believe we know to what can be conjectured from it: If there is an edge to the universe, space would likely be so lacking in density that time would progress nearly infinitely fast from our perspective.

Ray Cummings wisely wrote "Time is what keeps everything from happening at once." So wouldn't the corollary be infinitely fast time infers everything that can happen will happen? There seems to be one hell of a lot of space in our universe so perhaps space is the most likely thing to be brought into existence when time goes infinitely fast.

If this conjecture is at all true, then space has no boundary. There are also many other conclusions that can be derived from this conjecture but I won't delve into them because they would not be germane to your question.

• That's quite the text wall. Can you make it pargraphy and explain what's philosophy about your answer? Sep 28, 2017 at 17:30

OK. Our universe is a 3-sphere expanding at the speed of light. A 3-sphere is the surface of a four dimensional ball or globe, the 4-dimensional ball consisting of spacetime. A 3-sphere is a 3-dimensional object. The earth is a 3-dimensional ball and its surface, ignoring fine detail, is a 2-sphere. It has no boundary and every point on it is the same distance from its centre, but the centre is not contained in the 2-sphere. So our 3-sphere universe is finite, but has no boundary. Every point in it is the same distance from its centre, but the centre is not in our 3-sphere universe.

This 3-sphere model explains many things extremely well. It has a Hubble constant exactly as measured and explains very simply why the reciprocal of the Hubble is equal to the age of the universe. It explains why every object in our universe is moving at the speed of light (in space or time or a combination of both) as proposed by Professor Brian Cox. In principal you could look through the 3-sphere universe to see the big bang, but it would appear infinitely far away, though in fact it is not (you need to read the blog for a full explanation of that). The 3-sphere universe has a characteristic mass which is the same as current estimates of the mass of our observable universe, and it has a characteristic vacuum energy which is very plausible value.

If you wind the 3-sphere model back to zero time, it has zero volume and infinite density, but zero mass, a little bit like the Cheshire cat in Alice in Wonderland.

• Do you have links to the blog posts you refer to? Specific quotes would be valuable. These would help strengthen your answer and make them more than an opinion. Jul 22, 2018 at 13:26

If something exists it can be concrete or incorporeal, spatiotemporal or non-spatiotemporal. It does not seem necessary to posit either incorporeal or non-spatiotemporal entities or properties. This being so, the universe, being understood as the totality of what exists, constitutes what is either within spacetime or has concrete existence, concrete existence indicating positive spatiotemporal being. If we go back to Kant's antinomies regarding space and time, we can get the paradox of space and time both being finite and infinite. It seems that existence has always existed, namely not that there is a necessary being called 'existence,' but rather that 'being' is necessary for at least one arbitrary metaphysical simple, since something cannot come from nothing, unless it somehow comes from a separate dimension, but that dimension could not have come from nothing without infinite regress. So there is no starting point for existence, or rather no point prior to existence, but space cannot be infinite, since there are no actualized infinities (save perhaps with respect to time, though even there it is dubious) but only potential ones. Mathematical objects conceived platonically don't undermine this.

• I like your answer, althought there is a theory in physics. Our system (meaning everything, the universe) is closed and has a net energy of zero. Even though it is expanding etc etc. And this is constant. If it was constant but negative, then this would be a problem as this would mean that the universe will collapse and disappear. If it was positive then something has always been existing as the total is a positive and theres no beginning. But since it is constant and zero, the universe could have started from nothing cause nothing has energy zero and our universe too! Dec 27, 2016 at 19:54
• @Euler_Salter we would need to clarify the status of anti-matter, dark matter, etc., before we can get a clear and distinct idea of what exactly a net-energy-zero universe could imply Dec 27, 2016 at 19:54
• that is true. Although the idea is not mine, I am merely refering it from a Documentary (which could be wrong, even though it is an interesting idea). However I think this approach is revolutionary. I have always been melting my brain thinking about how something could start instead of nothing or the implications of something existing forever. This is an elegant escamotage, which I reckon looks very appealing. What kind of distictions would you make Lothrop? Dec 27, 2016 at 20:03

The question doesn't have a good answer. The part of the Universe we can possibly perceive is finite. The Big Bang was something like 13.8 billion years ago, so if there was something further away at that time than 13.8 billion light-years, we can't have seen it. So what was outside that border? There's no way to tell. Space may be infinite.

For the rest, I'm going to consider the current state of cosmology to be effectively correct.

Now, say that you got on a very fast ship and traveled in one direction. Depending on your ship, and given enough energy, you could get arbitrarily close to the speed of light. If the Universe wasn't expanding, you could go anywhere, given enough time, and hence could eventually arrive at the end of the Universe if there was one. However, the Universe is expanding, and it seems to expanding increasingly faster.

The expansion of the Universe creates more space continuously. It's therefore capable of having objects moving away from us at greater than the speed of light. Nothing's moving faster than light compared to the local space, but space is expanding. At a given distance, the expansion is faster than light, and you will never get to anything currently at that distance. If there is an end of the Universe, you will never get there. You will never get close enough to detect it.

It doesn't look like it right now, but it's possible that the Universe will not expand eternally. In that case, it's a tremendously large hypersphere, and if you went far enough in one direction you'd come back to your starting point. The Universe would be finite, but without a border.

So, that's two possible reasons why you wouldn't get to a border even if the Universe is finite.

Fabulous question. It has an easy answer at the level of principles but it's not so easy to comprehend.

As you have calculated both views of space you examine are logically absurd. The solution can only be that extended space is a mental construct, not a 'thing'. The world would be absurd if space were truly and independently real. Likewise, time becomes absurd when we reify it and for the same reasons.

Thus metaphysics is a proof of the metaphysical unreality of extension. Score one for the Perennial philosophy.

The absurdity of the two view you examine is long-ago proved by the Buddhist philosopher-sage Nagarjuna as part of his explanation of metaphysics. Many philosophers stop where you have arrived, with the absurdity of two opposite positions on one or more metaphysical questions. As a mathematician you might like to move on. if so this would require studying a solution known as 'non-dualism'.

A good introduction would be Hermann Weyl. He disposes of space-time as a mental construction, pointing out that it is not an empirical phenomena. I cannot follow his maths but The Continuum or Open World will be more accessible to you.

Be careful, though. When you begin to take the results of metaphysics seriously you quickly arrive in the Land of Woo trying to decipher Lao Tsu and the Buddha. This is because only here is it claimed that your two extreme views of space-time are not just absurd, as you conclude, but wrong. It would because they are wrong that they are absurd.

The problem is only that if they are both wrong then the Perennial philosophy must be the only correct world-view, it being the only one that claims they are both wrong. Really the issue is extremely simple but we complicate it because we can't believe our own results. If you just throw away the two views you examine, as you usually would for ideas that are absurd, then most of the complication evaporates.

You are trying to thingify space. By thingifying space, you are giving thing attributes to a phenomenon we don't know and don't understand at all. By doing so, you expect it to have or lack of limits, borders and thing attributes. Perhaps such attributes don't fit to space.

Yes, a world map has limits and borders. But all borders are always subjective. What are the borders of your block, or a river? What are the borders of a rainbow? What are the borders of a dog? After thinking on all of them, think: what are the borders of a rock? How do you know which electrons on the surface belong to the rock and what are just dust or dirt? Any definition of borders, boundaries and limits is ALWAYS completely subjective. This leads to a conclusion: things are subjective. Reality is just atoms, fields and noumena we don't understand. We learn to think on things, just to interact with them in order to survive. With this, I'm saying that limits and borders, even with the concept of space we usually use, are subjective.

As you are a mathematician, remark this: numbers allow correspondences with things. But nature would be something like a long number without decimal dot. After reaching your conclusions, perhaps you found this: all math is based on things. Do you see a problem there? Personally, I do. But probably we'll find a solution. An equivalent to thingifying space happened with the planetary model of the atom: even if classical mechanics was a solid discipline, it was useless to represent the shape of the atom. Schroedinger provided the foundation for the new mechanics allowing such description.

But even if we stop thingifying space and time, we don't have conclusive answers. Kant has an interesting view of space, altough complex. Basically, (as I conclude) space would be a subjective perception of a context of interaction, which has a representation we all know, but wouldn't exist for itself. See this excellent answer from Virmaior: Why did Kant say space and time do not exist independent of us? . Perhaps the right questions could arise starting from there.

Your assumptions are wildly outdated. With the understanding of general relativity, we de eloped the picture of the universe as like the surface of a balloon being blown up, but in 4 dimensions. That would mean in principle if you went far enough in any direction you would get back to where you started - in practice, distant parts of the universe are now effectively seperated irrevocably by event horizons, just like the interiors of black holes.

So, a balloon has positive curvature - parallel lines curve towards each other. Our model now is space having a negative curvature https://en.m.wikipedia.org/wiki/Anti-de_Sitter_space almost perfectly balanced by the positive curvature from the gravity of the universes masses. But, we now also think the expansion of the universe is accelerating, leading eventually to a Big Rip, as many fractions of the universe become completely seperated by event horizons.

Our most fundamental idea of what a dimension is, is based on https://en.m.wikipedia.org/wiki/Noether%27s_theorem which gives rise to our understanding of symmetry in physics. Translational symmetries, dimensions, and conservation laws, are all intimately related.

So, next you have to think about M Theory. This seems to provide a neat explanation of the varieties of particles we see, and potentially account for the weakness of gravity, by positing 11 dimensions. Movement above 4 dimensiins would be through parallel universes, a larger landscape of all the initial conditions of all the fundamental constants of the universe https://phys.org/news/2014-12-universe-dimensions.html

In short, none of your assumptions are valid, and there is a huge variety of interesting mathematics you have not encountered, never mind physics. Don't rush to impose limits on the universe, it is consistently far more interesting than we have imagined.