Can there be two infinite beings?

Question

Is the existence of two —or more— infinite beings logically possible?

What are the arguments for / against the possibility of the existence of two infinitely powerful beings?

Answer 1

What do you mean by "infinitely powerful", and what does it mean to be infinitely powerful in an intelligible universe?

Suppose you mean an omnipotent being. Is it possible to have two omnipotent beings? As I've noted elsewhere (see my remarks on the notion of 'unmakability'), this depends on what you intend "omnipotent" to mean — what the range of possible powers there is in the world.

For instance, an omnipotent being could presumably destroy any object, including another omnipotent being. Is this a contradiction? Not really, though it depends on the details. If the omnipotent beings both have the power to make each other cease to exist instantaneously without reaction, then it would simply happen as soon as one (or both) of them willed it. But can't either of them make themselves impervious? That depends on whether there's any particular mechanism to destroying objects — resistance against destruction only means that it can prevent that which brings about destruction from succeeding. So how one of the beings could possibly resist destruction now requires us to consider that there be a mechanism for the destruction: that is, that the world is one of mechanical principles, and the omnipotence of these beings consists in unfettered access to the resources (such as energy) by which the mechanism operates. Pretty much any impasse that can arise from two omnipotent beings can be characterized in this way: if one wants to make X happen and the other wants to prevent X from happening, then we must consider the mechanism by which X is possible — or posit a world in which the notion of "something happening" is much more complicated and nebulous than it is in our world.

Let us continue to consider the possibility of being A destroying being B. With normal physical objects, to see if one would be able to destroy (for instance, penetrate into) another, we would have to consider a balancing of forces: to see which force is greater, and consider the net effect, which consists of the difference between one force and another. With this sort of situation, involving what amounts to an immovable object and an unstoppable object, the corresponding difference of forces would be of the form ∞ − ∞ , which is a classic indeterminate form. When this occurs in mathematics or in physics, the message that one obtains is either that not enough information has been provided to obtain an answer, or there is no well-defined answer.

1. What extra information could we be missing? Well, to be omnipotent means that the being could presumably marshal an unbounded amount of power at any given point in time — but that does not mean that it always does marshal an infinite amount of power. If one of A or B does exert more power than the other — either to destroy or to preserve itself — then it will prevail, if only because the other being is holding itself back. This is concievably true even if both of the beings are using an infinite amount of power: depending on how we model the infinite amounts of energy they have at their disposal (either by asymptotic analysis or by transfinite cardinals, to give just two examples of mathematical models of the infinite), both A and B could still use an infinite amount of power, in such a way that one is using a more infinite amount of power than the other.

2. If both A and B are "omnipotent", this presumably means that neither is more powerful than the other. So such restraint as I describe here is only possible if either A or B deliberately, or through its particular mood, expends less energy in its destructive or defensive efforts.
   
   If the two beings A and B expend exactly the same energy as one another, then again what happens depends on whether they are actually using an infinite amount of power at that moment, on the initial conditions of the mechanism of destruction, and (if they are using an infinite amount of power) again on the model of infinity that we consider. Suppose, for the sake of definiteness, that A and B have taken human form momentarily at a whim. To destroy B, the being A must choose some particular mechanism: for instance, perhaps A chooses to fire an fast and hard bullet at B. To defend themself, B has to use some mechanism to stop the bullet: for instance, erecting a strong barrier between them and the bullet. (We ignore the fact that these conditions violate special relativity and known material science.) What happens?

   • If the bullet and wall have the same finite strength, and are fired/erected with the same finite speed, it depends on whether or not B erects the wall in time to prevent the bullet from passing completely through the wall. If not, the bullet makes it through, and without further action B dies; otherwise the bullet is blocked.

   • If the bullet and wall have the same infinite strength, and are fired/erected with the same instantaneous speed — but the power used by the two beings can somehow be described in terms such that (despite being infinite) they cancel to give a definite positive or negative outcome — then the same observation applies: that definite value implies either than A succeeds, or that B succeeds.

   • Suppose that we require that the puzzle take the classic form of the unstoppable force and the unmovable object: that we require both of the infinite efforts of A and B to be absolute and unreconcilable, in order to force a paradox. In this case, there can be no outcome: by the very fact that the two forces cannot be evaluated relative to one another, to see whether either A or B prevails, there is simply no outcome. What does this mean? Essentially the end of time; reality ceases to exist, because there can be no events afterwards to determine whether A or B prevail. At the very least, everything which would ever possibly interact with the future causal cone of the battle between A and B must cease to exist.
     
     — But can't we consider the possibility that A and B co-operate to make sure that the universe continues to exist, while trying to destroy one another? That comes down again to what mechanism there could be to prevent the universe from ceasing to exist when they are doing something which would cause the universe to cease to exist. Again it comes down to how these efforts cancel with one another; and if there is no way to reconcile the efforts, then there is no outcome and no future.

In the above, I have made one key assumption: that whatever powers we assume that these omnipotent beings have, that these powers are somehow intelligible. In essence, that despite the existence of these beings, that the universe is a world of laws, and that the power of these beings to do what they please stems from complete mastery within those laws. I see no way to answer this question — or indeed any other question whatsoever — if we assume that the universe is unintelligible even in principle; so I think this assumption is quite reasonable if we're going to entertain the question at all. But an important consequence of this assumption is that the question attains a definiteness which is not typical when such problems have been considered historically, and how precisely we choose to make the question definite enough to be answerable — what specific conditions we suppose in terms of how A and B act — will change the nature of the answer.

Answer 2

If there is a pair of coexistent necessarily omniscient and necessarily morally perfect omnipotent agents, then there is a pair of incompatible contingent states of affairs each of which is morally optional for these agents, that is, neither morally prohibited nor morally required for them.

If a plurality of coexistent omnipotent agents were even possible, then possibly, at a time, t, some omnipotent agent, x, while retaining its omnipotence, endeavors to move a feather, and at t, another omnipotent agent, y, while retaining its omnipotence, endeavors to keep that feather motionless.

Intuitively, in this case, neither x nor y would affect the feather as to its motion or rest. Thus, in this case, at t, x would be powerless to move the feather, and at t, y would be powerless to keep the feather motionless!

But it is absurd to suppose that an omnipotent agent could lack the power to move a feather or the power to keep it motionless. Therefore, neither x nor y is omnipotent. This line of reasoning appears to reduce the notion of a plurality of coexistent omnipotent agents to absurdity.

Answer 3

Aleph number.

If two beings are infinite in sense of ℵ₀ there's nothing to keep them from coexisting in an ℵ₁ universe.

A human is infinite in that a body can be subdivided infinitely (only conceptually beyond Planck size but still.) Yet there are many humans.

Or do you mean something else by infinity? Omnipotence?

Omnipotence only implies a specific potential at any given point of time: an omnipotent being may choose for anything to happen at any given point of time [not necessarily current]. Still, while the moment of effect is arbitrary, the moment of choice is specific and a subject of normal laws of casualty.

One of two omnipotent beings coexisting in a single universe may wish the other removed from existence, back since their beginning, or limit their omnipotence, say, by making self immune to such effect. Still, the moment of such action is a specific point of time. If one has been deprived of means of harming the other, including going back in time to undo that wish, they are no longer infinitely omnipotent. In that case two such cannot coexist. But then, if neither makes that choice, they are still both omnipotent - conditionally; until one choose to deprive the other of it.

Answer 4

Any one of any number of omnipotent beings would be able to completely control all others, since that is an ability, and the each would necessarily have it.

So, quite quickly, any set of omnipotent beings that disagreed about anything would cease to do so. (Standards of mutual respect that forbid directly controlling one another would constitute limitations on their individual omnipotence. Even interacting in time is a limitation that no omnipotent being would have to subject itself to.)

At that point, it is ambiguous whether they are, in fact, different beings, or simply components of a single composite being with an internally complex will. What is a being beyond a singular concordance of will?

So I would say no.

Answer 5

Two Von-Neumann Agents in a box can be modeled as one Von-Neumann Agent, if all you can ever see is the outside of the box. If the agents are using an advanced form of decision theory, they can even make meaningful decisions inside this box (collaborating on prisoner's dilemma), despite differing utility functions.

Von-Neumann Agents by mathematical definition can carry out infinite amounts of perfect reasoning in finite time, so they are in a sense infinitely powerful.

ETA: Further information and loose sources:
Von-Neumann-Morgenstein Utility Theorem on Wikipedia gives a good outline of the mathematical axioms.
LessWrong: Newcomb's Problem and Regret of Rationality
LessWrong: Decision Theories Primer

Answer 6

Yes, power on one axis/being would simply be distributed in time on the other. Given that they are eternal as well as infinite, the more likely scenario is that they're wise enough to negotiate the boundaries are. Since there is a documented story of one such infinite being gaving rise to another being, that is the likely trajectory of Man if and when He/She completes their journey of Knowledge.

Answer 7

If you really want to ask deeps question you should even think about this. Its the same question like:

What happens if an unstoppable force hits an immovable object?

There never will be a good answer because there is no such thing as immovable/unstoppable or infinite. These are just theoretical concepts.