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Suppose an infinite plane which is similar to Earth's surface and has uniform Earth's gravity field.

The plane is devided into two infinite parts by a slightly curved line.

On the both sides of the line there are two empires in constant war with each other.

The empires employ all possible technology to eliminate each other and advance the front line. They are not restricted in the choice of weapons to use, including not only weapons of destruction, but also infiltration, espionage, political subversion etc.

The both empires are infinite and the further the point is from the front line the more stronger the power and the technology is more advanced.

The empires not only fight but conduct research, study each other etc so to invent newer weapons. The learned findings and information is transferred back to the higher command and orders are transferred to the front line. But the only purpose of any research and activity is the war.

Each command level has even higher command level above them ad infinitum, which controls operations on even greater area. The low-levels do not know who is the supreme command and whether it exists at all (they suppose it is infinitely far).

What are the possible hurdles in existence of such world? Could it be stable? How the warfare can be resembling the warfare in real world?

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    $\begingroup$ Hm... infinity always has some problems. How do you build and control an infinite empire? It would take an infinite time to build such an empire. It would also take an infinite time for orders to go down an infinite number of command levels. $\endgroup$ – drat Nov 5 '14 at 10:14
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    $\begingroup$ Why would they go to war in the first place? They both already have infinite land and resources. There is nothing to gain from conquest. $\endgroup$ – Philipp Nov 5 '14 at 12:02
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    $\begingroup$ Not quite the same as what you postulate here, but do read “Traveller's Rest” by David Masson (somewhat spoilery summary). $\endgroup$ – Gilles 'SO- stop being evil' Nov 5 '14 at 14:12
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    $\begingroup$ @Gilles Religion is rarely a reason for war. Religion is usually used as a propaganda tool to justify wars which are usually fought for much more worldly reasons. $\endgroup$ – Philipp Nov 5 '14 at 14:30
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    $\begingroup$ Voted to close as unclear because this, logically, is not a valid question. How do you define an empire if, for all times and communication or processing speeds, its leaders' commands can reach no fraction of the area they control? Infinity is not a number and shouldn't be used as such. $\endgroup$ – Vandroiy Nov 5 '14 at 14:43
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The thing that I think would be most relevant would be the fact that anything that is possible, regardless of how unlikely it is, will have happened somewhere.

If it's possible for a rebellion to successfully happen, it would; there would simultaneously be an infinite number of new rebellion factions forming at all levels of each empire, from the front lines backwards towards the limits of each empire. There would also be an infinite number of these rebellions being defeated every moment. Assuming that it's possible, some of them would survive and become more powerful. None of them would be able to rival the two infinite ones (since they'd have to be finite), but since there's an infinite number of them then it's not unreasonable to think that some would grow arbitrarily large.

If it's possible to stumble onto some sort of superweapon, then it will happen. This means you'll have spots where one side has a massive advantage over the other, because of some incredibly unlikely technological breakthrough. Of course the war won't be changed by this because it will take an infinite amount of time for the entire empire to hear about it. You'll have spots where two of these superweapon discoveries meet for the first time, and have to face each other. One side could create a perfect bioweapon which kill both sides, spreading outwards from the starting point.

You'd also see differences in the people. From the start of this world, people would begin diverging in appearance due to random genetic mutations. Features would get shared, but since everything is infinite then there will always be points that have had no interbreeding. Given enough time, people from different locations would look nothing alike. A few thousand years would probably be enough to generate some superficial differences. Give it millions/billions of years, and you'll find groups that aren't even the same species any more.

In an infinite world, everything that is possible will happen. It's hard to describe what this world would be like, because it would be everything you could conceive of.

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  • $\begingroup$ If time travel is possible in this world you could have the first side that discovers it win. It could infinitely loops back into the past and use infinite time to create unbeatable weapons. It could then set out to infinitely kill people. Of course they would be the winners only after infinite loops. But they would still win. $\endgroup$ – Black Nov 6 '14 at 12:35
  • $\begingroup$ Don't forget about the possibility of Starcrossed Lovers and all it entails! $\endgroup$ – M.Herzkamp Mar 27 '15 at 15:23
  • $\begingroup$ Worth noting that 'anything that is possible' does not have to equal everything that can possibly happen in accordance with natural laws (i.e. physics etc). Just because everyone in both empires could simultanously decide and go through with, say, playing volleyball for ten minutes at some arbitrary point in time does not imply there is a guarantee for it to happen, even with infinite time. $\endgroup$ – vruvre Nov 25 '15 at 19:39
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You use "infinite" a lot. Seemingly without fully realizing the consequences.

If an area is of infinite size, then there is no way to move outside of it. Weapons can't reach into the neighbor's part of the greater world, because absent actually travelling at infinite velocity (which, as aptly noted in Star Trek, means not that you are going very fast, but rather that you are occupying every point simultaneously), it will take infinite time for anything to reach outside the infinite-size empire. Since matter cannot travel at infinite speed (even light doesn't!), that becomes an impossibility.

A missile will take infinite time to reach its target. Assuming that it is travelling over ground, it requires infinite amounts of fuel to propel itself, leading to infinite weight. Infinite weight in a finite volume leads to a singularity (or you may think a black hole).

The people in your empires would not know about the respective other empire, because that other empire would be an infinite distance away from them. There can exist no "front line" in such a world, because at any arbitrary point, you are still an infinite distance away from your opponent. Since you are an infinite distance away from your opponent, communications (of any kind) takes infinite power and infinite time.

And so on.

There is a reason why infinity symbols is usually a bad thing to have pop up in equations. Real things are not infinite and cannot span infinity.

If you want to put reasonably realistic hurdles in place in your world, avoid terms like "infinite" and look into alternative approaches.

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    $\begingroup$ @Anixx The problem is that if the empire truly is "infinitely" large, there can logically be no place where the distance to the neighboring empire is less than infinity, because if there was, then the size of the empire from which such a measurement is taken would be finite. $\endgroup$ – user Nov 5 '14 at 10:29
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    $\begingroup$ "If there is an infinitely large surface, then at any point X on that surface, the distance to the edge of the surface must be infinite" - wrong. Devide a plane into two halves, both will be infinite, but the distance from each point of a half to the other half will be finite. $\endgroup$ – Anixx Nov 5 '14 at 10:38
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    $\begingroup$ @Devide a plane into two halves, both will be infinite, but the distance from each point of a half to the other half will be finite. $\endgroup$ – Anixx Nov 5 '14 at 10:39
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    $\begingroup$ While I agree with the general sentiment that @Anixx doesn't realize the full implications of an infinite universe, you, Michael, are making incorrect statements regarding its geometry. The question postulates that the universe is a plane and that each faction holds one of the two connected areas divided by a line. The distance from any point to the line is finite. Crossing the line can take an arbitrarily small time depending on how big a step you want to take. $\endgroup$ – Gilles 'SO- stop being evil' Nov 5 '14 at 14:08
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    $\begingroup$ This is wrong. Simplify it down into one dimension, and imagine the two sides are the positive and negative numbers on a number line. Both "empires" extend out infinitely towards ∞ and -∞, but there is still a place where they meet, and where the conflict could happen. $\endgroup$ – resueman Nov 5 '14 at 15:27
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What is the motivation?

The big point is that on such a world victory or defeat is impossible by definition. If one empire would manage to advance a million miles forward, it would have the following consequences (assuming that the empire "backside" is homogenous):

1) The winning empire would be exactly as large as before.

2) The winning and losing empire would be able to manufacture and contribute the exact same amount of resources to fighting on the [new] frontline as before.

3) The distance from factories to the frontline (on both sides) would be exactly the same as before.

4) None of the empires can ever be defeated - for any finite changes in front line, there would still be exactly as much of the enemy to conquer. The war can't end that way.

Actually, the war can't end ever - if the "headquarters" that you describe would agree to a cease-fire, it would never reach all the frontline, at any point of time there would be still an infinitely long frontline that has not gotten the message yet.

Thus, any local changes/victories or losses can only have zero total effect. Not a small effect, not an infinitely small lambda, but an exact mathematical 0. Which may fit the desired feeling of such a story.

Read up on the classic Hilbert's paradox (http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel) for some inspiration.

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  • $\begingroup$ The motivation for local participants is to survive, push the frontline further and to get promotion. They also could be controlled by other means besides motivation (implants, software, fear, religion, hostages, genetic modification etc). There is no motivation for the empires as a whole because there is no supreme command. $\endgroup$ – Anixx Nov 5 '14 at 16:08
  • $\begingroup$ @Anixx There is a point where the people can no longer have any effect on the war. This point would be so far away that by the time any technological advances are made and transferred to the frontline, the frontline has already discovered it for itself. People can't survive long enough to get there to help and even supplies break and degrade before they even get there. What would they do to help the war? In a world focused solely on the war, these people are useless, and there is an infinite number of them. $\endgroup$ – DoubleDouble Nov 5 '14 at 17:01
  • $\begingroup$ @DoubleDouble first of all, it is not necessary people who lives far away, so their lifetime may be any long. Second, the further away the more global military planning is conducted that would affect greater theater. Also, deep in the rear reserves may be prepared to be enacted in a greater offensive. Also given that the enemy also has unlimited time to collect reserves, an enemy offensive of any scale can be anticipated so no matter how far the rear is it should be prepared for defense. $\endgroup$ – Anixx Nov 5 '14 at 17:28
  • $\begingroup$ @Anixx I feel like your first point is the most important, but I can't fully understand it with the way it's worded. Can you clarify? $\endgroup$ – DoubleDouble Nov 5 '14 at 17:56
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The main struggle would be a psychological one: if everything exists in infinite quantities (including humans), then everything becomes meaningless. If a battle kills 10,000 people, send in another 10,000, because you got an infinite amount of soldiers. A single life becomes irrelevant, as you have an unlimited number of replacements. And therefore each individual person would feel insignificant and irrelevant.

With a never ending chain of command, unlimited food, and no perspective in life, individuals have no goal and would fall back to just satisfy their basic urges. After all, most individuals would just be human "garbage", wasting space, and being eventually used up in a pointless war, that can never be won or lost.

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    $\begingroup$ If space is unlimited (infinite), can it be "wasted"? (I suppose that's a question for Philosophy, but...) $\endgroup$ – user Nov 5 '14 at 13:28
  • $\begingroup$ Living beings would still occupy a certain amount of space, and "waste" that space. It would be possible to move those individuals to less crowded areas, but then those would "waste" resources on transportation. In both cases, those people would just be an annoyance to society. And with infinite amount of people, almost all of them would be an annoyance. $\endgroup$ – TwoThe Nov 5 '14 at 13:30
  • $\begingroup$ Food is not unlimited on the local level, and each individual can make a carrier by being promoted to a higher position. And if humans are unreliable, the higher command may design measures to motivate their better or control by other means like implants or replace humans with software etc. $\endgroup$ – Anixx Nov 5 '14 at 13:40
  • $\begingroup$ It would require a strong motivation like a deep faith or similar to give those people's lives a meaning. $\endgroup$ – TwoThe Nov 5 '14 at 15:58
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After long session on chat, I decided to put answer together to hopefully satisfy you:

Local point, local time

would remind my favourite comic Gone with the Blastwave: You are on incredibly big war field, you dont necessarily remember why you fight the enemy, but you are red, they are blue, which makes them enemy, so you kill any blue solider you can see.

If you allow feelings to your creatures, it is possible that after some time, they will consider themselves pointless and happily kill self for some war advantage

Global point global time

It is pointless: after some level of "focusing out" the border never changes, no matter what opposing sides try to get upper hand. The advance will be only "marginal" considering the whole war field is infinite. The "border" would probably look like fractal, which would be changing in time, but overall staying almost on the same place when zooming far enough

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A strange game. The only winning move is not to play.

In this scenario, if one empire has not already won, then neither of them are ever going to win (barring outside influence). They are perfectly balanced in all aspects, since even a infinitesimal difference would lead to victory once an infinite amount of resources are put behind it.

With an infinite amount of resources/knowledge, an empire would realize the futility of future combat (especially since they have all the resources they could ever want).

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  • $\begingroup$ There is no supreme command of each empire, and each local commander has superiors. And his resources are limited. Even if do not want to fight, this does not change the entire scene. $\endgroup$ – Anixx Nov 5 '14 at 15:22
  • $\begingroup$ @Anixx - And who are their superiors' superiors (and so on)? Someone is running the show, or else it's not two empires fighting but a mishmash of different smaller states that (for some inexplicable reason) never change alliances. $\endgroup$ – Telastyn Nov 5 '14 at 15:46
  • $\begingroup$ they are hierarchical. But the command chain is infinite, each commander has a superior. $\endgroup$ – Anixx Nov 5 '14 at 15:48

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