There have been several posts on the kinds of worlds God could create. I want to argue that, odd as it might sound initially, there are *infinitely bad worlds* that God could have created. But how is that possible? Here’s how. Let the world w contain a countably infinite number of sentient beings (say, all human beings). And let each human being Hn in w be assigned a natural number, n. Let each natural number n be mapped onto an equivalent number of seconds t: number 1 to 1 second, 2 to 2 seconds, and so on upward. So we have for each human being Hn in w and number of seconds t, a pair, (Hn, t). Finally, let t be the number of seconds that Hn suffers in w. H1 suffers for 1 second in w, H2 suffers for 2 seconds in w, H3 suffers for 3 seconds in w and so on upward for all of the natural numbers.

We know that each Hn in w will suffer a finite amount of time, since each Hn = t is a finite number of seconds. Suppose it is true, for each Hn in w that he enjoys at least n^100000 seconds of joy after his suffering. The only evils in w are the instances of suffering, and the only good are the instances of joy. Here are some strange facts about w.

1. For each human Hn in w, Hn suffers for a finite amount of time X.

2. For each human Hn in w, Hn enjoys for a finite amount of time (X + Y) > X

3. For each human Hn in w, Hn’s life is on balance very good.

4. Every evil in w is outweighed by a much greater good.

5. *There will always be an infinite amount of suffering in w*.

6. *There will always be a finite amount of joy in w.*

Since each human’s life is on balance very good in w (we could make it superb, if you like), and each evil is outweighed by a much greater good in w, I claim that God could have actualized w. But it is also true that there is no time in w at which there is not *an infinite amount of suffering*. At each moment t in the world w, there is an infinite amount of evil. So God could have actualize an infinitely bad world.

[Revised and Updated 12.21.09]

[Related posts](http://prosblogion.ektopos.com/archives/2009/01/how-to-rational.html)

*Whether this world is infinitely bad depends on the order in which you add things up (which happens with infinite sums).*

Hi Nick,

Of course arithmetical operations are ill-defined for Cantorian infinites. But the argument would work as well for non-standard treatments of infinite amounts on which they’re numbers (not cardinalities). The argument assumes only that the value of a world is equivalent to the on-balance amount of good in the world. This world w is such that, at every time in the existence of w, there is an infinite amount of suffering and a finite amount of good. That is true despite the fact that each person’s life in w is on balance very good. Of course you look at *each life* longitudinally and individually in assessing it’s overall value. But if you look at the value of a world, you take the value of *all lives* longitudinally, and that always yields an infinite disvalue, no matter how far you take it out to do the sum.

OK. I’ll bite.

But first, a handful of worries with the setup.

“I want to argue that, odd as it might sound initially, there are infinitely bad worlds that God could have created.”

So “there are” worlds that are infinitely bad? I deny this. What might be true is that there are worlds which are such that, had God created them, then they “would have been” infinitely bad. I’m assuming Lewisianism Modal Realism is false. I’m assuming that abstract states of affairs, or maximal and consistent sets of propositions (worlds) can’t be infinitely bad. I think these are plausible assumptions.

“We know that each Hn in w will suffer a finite amount of time, since each Hn = t is a finite number of seconds.”

What do you mean by this? ‘Hn’ picks out a particular human, and ‘t’ a time. This is a typo, yes?

Lastly…

“Since each humanâs life is on balance very good in w (we could make it superb, if you like), and each evil is outweighed by a much greater good in w, I claim that God could have actualized w.”

Just because an evil is outweighed by some good doesn’t make it such that God could allow it (or create it). The good must also “require” the evil (or some evil as bad as it). I take it this is uncontroversial common ground, a la Rowe. But then the above conclusion just doesn’t follow. Your case is not one in which the suffering, at any time, is necessary for the existence of some enjoyment at some other time.

*So “there are” worlds that are infinitely bad? I deny this. What might be true is that there are worlds which are such that, had God created them, then they “would have been” infinitely bad. I’m assuming Lewisianism Modal Realism is false. I’m assuming that abstract states of affairs, or maximal and consistent sets of propositions (worlds) can’t be infinitely bad. I think these are plausible assumptions.*

Christian,

Neither Lewis nor Plantinga deny that every possible world exists, despite their ontological differences. Plantinga allows you to quantify over them, and so does Lewis. So, nothing wrong with using the quantifier “there are” in either case. The difference between Lewis and Planinga on this point is that Lewis sees a difference between saying there are other possible worlds and there actually are other possible worlds. Plantinga sees no difference.

*“We know that each Hn in w will suffer a finite amount of time, since each Hn = t is a finite number of seconds.” What do you mean by this? ‘Hn’ picks out a particular human, and ‘t’ a time. This is a typo, yes?*

Yes, that’s a typo. It should read that for each Hn, n = t, where t is the finite amount time of suffering assigned to Hn.

*Just because an evil is outweighed by some good doesn’t make it such that God could allow it (or create it). The good must also “require” the evil (or some evil as bad as it). I take it this is uncontroversial common ground, a la Rowe. But then the above conclusion just doesn’t follow. Your case is not one in which the suffering, at any time, is necessary for the existence of some enjoyment at some other time.*

Technicalities… 🙂 Two points. First, I didn’t want to labor this point since it is not what people will balk at initially. So, just make the assumption that there are no gratuitous evils in this world despite the fact that we happen not to know why the evils are necessary to the good. We still wind up with the main point of the post that we have an infinitely bad world that God could create. That’s weird. Second, the point about what constitutes gratitous evil and whether God might allow some is in serious dispute, so I don’t think we could in any case take it as dogma.

*I’m assuming that abstract states of affairs, or maximal and consistent sets of propositions (worlds) can’t be infinitely bad. I think these are plausible assumptions*

This is confused, I think. You might as well say that Socrates is not possibly a non-philosopher since in order to be so he’d have to be a state of affairs included in a maximal state of affairs, and he’s necessary not a state of affairs.

Plantingan metaphysics of course allows us to say truthfully that Socrates is possibly a non-philosopher and also that there exist worlds that are infinitely bad. Of course the infinitely bad world does not obtain, but it might have. To say that there exists a world that is infinitely bad is just to say that there exists a maximal state of affairs such that had it obtained, it would be infinitely bad. To say that Socrates is possibly a non-philospher is just to say that there is a maximal state of affairs such that had it obtained Socrates would be a non-philosopher.

But none of this paraphrasing affects the modal claims. It’s true that there are infinitely bad worlds and it is true that Socrates is possibly a non-philosopher.

“The argument assumes only that the value of a world is equivalent to the on-balance amount of good in the world.”

Okay. Suppose I give you this sequence S=(1, -1/2, 1/3, -1/4, …). In that order, the sum of the sequence is ln(2). By the Riemann Series Theorem, rearranging the sequence allows the sum to take any value in the extended real number system.

Now suppose I just give you the set of all the numbers in S and ask you to add them up. I.e., I give you the set S’ = {1, -1/2, 1/3, -1/4, …} and ask you for the value of the sum of the members of S, not specying an order. This is undefined.

Likewise, suppose we have your world with an infinite number of people and an infinite number of times. Some of these time-individual pairs have a positive value and others have a negative value. Now, when we ask for the “on-balance amount of good in the world”, I assume we’re asking for the sum of the values associated with all the time-individual pairs. What I’ve pointed out to you is that this sum approaches positive infinity if you add the numbers associated with time-individual pairs in one way (add up for each individual, then add up the values of each individual), but approaches negative infinity if you add up the sum in another way (add up for each time, then add up the sums associated with each time). For this reason, if the “on-balance amount of good in the world” is just the sum of the values of the time-individual pairs, this is like asking for the value of the sum of the members of S (specifying no order). It is undefined.

Now, you seem to be suggesting that, in some sense, this world really is infinitely bad. What I’m suggesting to you is that this world is, at best, infinitely bad with respect to a particular order of taking the sum. But you could equally well claim that the world is infinitely good. I couldn’t tell if we were in agreement, but I thought I’d make sure.

Mike,

I understand Lewis and Plantinga’s view. I’m making what is, I suppose, a nit-picky claim about the statement of your thesis. Merely possible worlds, on the view that they are abstract, don’t “have” value or disvalue. Abstract things don’t have value. Sets of propositions don’t have value. Fusions of uninstantiated states of affairs don’t have disvalue. So, even if there are possible worlds, they are not “infinitely bad”. Rather, they would be infinitely bad were they actual(ized). I think you are denying this here:

“To say that there exists a world that is infinitely bad is just to say that there exists a maximal state of affairs such that had it obtained, it would be infinitely bad.”

This confuses a categorical claim (one about the way things are) with a hypothetical claim (one about how things would be). If you deny this, we should drop it. It’s besides the point.

“We still wind up with the main point of the post that we have an infinitely bad world that God could create. That’s weird.”

Yeah, this seems right if by infinitely bad world you don’t mean infinitely bad “overall and across time”. But even so, it is pretty weird! At any time, the value of the world at that time is infinitely bad, even though every person can truly say “my life went great!” when looking back.

*What I’ve pointed out to you is that this sum approaches positive infinity if you add the numbers associated with time-individual pairs in one way (add up for each individual, then add up the values of each individual), but approaches negative infinity if you add up the sum in another way (add up for each time, then add up the sums associated with each time).*

Nick,

The argument that I’m making could be made as well with non-standard analyses. So we needn’t fuss all of the odd arithmetic with Cantorian infinities.

Maybe I’ve still not been very clear. I’ve assumed that the only things that matter to the overall value of the world are units of joy and units of sufferings in individual lives: call them pings and dings. Let every time in the age of the world w be finite (recall that that every time in the age of w is mapped to some natural number). So you cannot get to infinity (what that might mean) and be in w. You can determine the value of each life in w by subtracting the dings from the pings. Those are all positive. You determine the value of a world by taking the value of *all lives* in w. As you move outward in time in that world there exists no time at which the value of all lives is anything but infinitely bad. As you move outward, there are always (more and more) on balance good individual lives and infinitely many on balance bad lives. No matter how far you move outward, this is true. As you approach an infinite amount of time, the value of the world get higher and higher. The world itself never gets to infinity, since every time in the age of w is finite.

Is that any better?

*I understand Lewis and Plantinga’s view. I’m making what is, I suppose, a nit-picky claim about the statement of your thesis. Merely possible worlds, on the view that they are abstract, don’t “have” value or disvalue. Abstract things don’t have value. Sets of propositions don’t have value. Fusions of uninstantiated states of affairs don’t have disvalue. So, even if there are possible worlds, they are not “infinitely bad”.*

This saddles Plantinga with a view he does not hold. Here are some truth-conditions.

It is possible that a world is infinitely bad just in case there is some w in which it is true that w is infinitely bad iff. there is some world w which includes the state of affairs that w is infinitely bad.

On your interpretation of Plantinga, there is no such world, since worlds are abstract states of affairs and abstract states of affairs can’t be bad. So, on your interpretation of Plantinga, it is impossible that a world is infinitely bad. But this is clearly mistaken. It follows from this that our world @ could not be any better or worse than it in fact is, since there is no possible world w in which it is true that w is better than the @ or w is worse than @. But that is plainly false, and Plantinga would obviously reject the suggestion that his modal metaphysics commits him to it.

*“To say that there exists a world that is infinitely bad is just to say that there exists a maximal state of affairs such that had it obtained, it would be infinitely bad.”
This confuses a categorical claim (one about the way things are) with a hypothetical claim (one about how things would be). If you deny this, we should drop it.*

Maybe we should, but I’m not sure what the worry is. What I say here is just the Nature of Necessity, p. 49ff. “To say that a state of affairs S

*is*actual in W [this would be your categorical claim] is just to say that S

*would have been*actual had W obtained [this would be your hypothetical claim]”.

*… this seems right if by infinitely bad world you don’t mean infinitely bad “overall and across time”.*

No, it is overall and across time. Keep in mind that the world I’m talking about includes no transfinite times (whatever those might be). Every time in w is finite, and each corresponds to a natural number. So as you move outward, there is to time in w at which w is not overall infiniely bad. Still, each life is on-balance very good.

Mike,

“No, it is overall and across time.”

Then I don’t understand the argument. Every life in w has positive overall value [from assumption 3]. If so, then if we add their overall values, it will also be positive [from the assumption that if n and m are positive, then n + m is positive].

“It follows from this that our world @ could not be any better or worse than it in fact is, since there is no possible world w in which it is true that w is better than the @ or w is worse than @. But that is plainly false, and Plantinga would obviously reject the suggestion that his modal metaphysics commits him to it.”

That would be a plainly false consequence. It’s not a consequence of my interpretation though. To say that our world @ could have been better is just to say that there is some world w accessible from @ that (i) could have been actual (obtained) such that, (ii) had w obtained, it would have been better than @.

In a different way: Merely possible worlds don’t stand in ‘betterness’ relations to the actual world just as I don’t stand in a ‘taller than’ relation to merely possible elves. Merely possible elves don’t have heights, only actual things have heights. I’d be very surprised indeed if Plantinga denied this.

*That would be a plainly false consequence. It’s not a consequence of my interpretation though. To say that our world @ could have been better is just to say that there is some world w accessible from @ that (i) could have been actual (obtained) such that, (ii) had w obtained, it would have been better than @.*

Right, and that is true just in case it is true in that world w *that w is better than @*. You deny this, since, as you say, abstract objects are not evil or good. But then the counterfactual in (ii) comes out false.

*Merely possible worlds don’t stand in ‘betterness’ relations to the actual world just as I don’t stand in a ‘taller than’ relation to merely possible elves*

Really? The car I own is not as good as one I might have owned. Elves are not possible.

*I’d be very surprised indeed if Plantinga denied this.*

Be surprised.

*Then I don’t understand the argument. Every life in w has positive overall value [from assumption 3]. If so, then if we add their overall values, it will also be positive [from the assumption that if n and m are positive, then n + m is positive].*

Each life has a positive overall value, since for each life L there is a finite time t at which L is overall valuable. But w does not have overall positive value since there is no time t in the entire history of w at which every life L has overall positive value. So, just as you can count to each natural number in finitely many steps, but you cannot count all natural numbers in finitely many steps, so each life reaches a natural number (time) t at which it is overall positive in a finite time, but it is not true that all lives in w reach a natural number (time) t at which all lives have overall positive value in a finite time. But w contains only finite times. So w is always overall infinitely bad.

There must be an easier way to put it. It’s this:

It is true for each individual life L in w that there is a finite time t at which L is overall very good, but at every finite time t in w most lives are overall very bad. w contains only finite times, each correponding to a natural number.

Compare: Every natural number is finite. But no matter which nartural number you go to, most of the natural numbers are greater. And the set of natural numbers contains as members only finite numbers.

Here’s a small model of the world W that might be helpful. There are infinitely many lives in W, listed vertically on the left. Times in W are listed horizontally across the top. At the first time-slice of W, t1, each of the infnitely many lives are enduring one decrement of pain. As you move to the second time-slice of W, every life except L1 is enduring one decrement of pain. As you move to the third time-slice of W, t3, every life except L1 and L2 are experiencing one decrement of pain. Of course, at every time-slice of W, there are infinitely many lives enduring one decement of pain, and finitely many lives enjoying one increment of joy.

W_______t1_______t2______t3________t4………tn_____tn+1

L1______-L1______+L1_____+L1*_____+L1…….+L1_____+L1

L2______-L2______-L2_____+L2______+L2…….+L2*____+L2

L3______-L3______-L3_____-L3______+L3…….+L3____+L3

L4______-L4______-L4_____-L4______-L4…….+L4_____+L4

.

Ln _____-Ln______-Ln______-Ln______-Ln…….-Ln____+Ln

.

.

But consider the value of each life in W. The life L1 becomes on balance positive at t3 (see first row, starred). The life L2 becomes on balance positive at t5 (depicted as tn starred, second row). And in general, each life Ln is on balance positive at t(2n+1). So, for each life Ln, there is some finite point in the history of W at which Ln is on balance positive. And of course over time each life becomes on balance very positive.

So clearly each life is well worth living, though there is no stage of W that gets beyond an infinitely negative overall value.

Mike:

*Right, and that is true just in case it is true in that world w that w is better than @. You deny this, since, as you say, abstract objects are not evil or good. But then the counterfactual in (ii) comes out false.*

Right, I deny this. But I think that the counterfactual in (ii) comes out true. So we strongly disagree about the truth conditions for counterfactuals like the one in (ii). OK. Anyway, I’ll say this: Maybe I do in fact have Plantinga wrong. It’s been quite a while since I read his stuff. So I hereby take back any claims that I made about Plantinga! I don’t think that I have (ii) wrong, but we’re not going to resolve that.

*Really? The car I own is not as good as one I might have owned. Elves are not possible.*

Well, the car you own is not as good as some actual car that you could have owned. Here the ‘not as good as’ relation relates actual things. I have no problem with that. On the other hand, if you mean to say that your car is not as good as a car that doesn’t actually exist, but could have existed…then I think that’s crazy. Elves are possible!

To your original puzzle:

*But w does not have overall positive value since there is no time t in the entire history of w at which every life L has overall positive value.*

An answer to a few questions might help me out. Is there such a thing as the overall value of a particular life, say, L1 in w? If so, can we correctly assign to the overall value of L1 a finite positive number?

Your diagram helped me to picture the case, by the way.

*Is there such a thing as the overall value of a particular life, say, L1 in w? If so, can we correctly assign to the overall value of L1 a finite positive number?*

Glad the diagram helped. I’ve been assessing the value of lives in a pretty typical way. I say this:

P. A life Ln is on balance positive in W iff. there will be a time in W such that, from that time onward, the total positive value in the life Ln is greater than the total negative value in Ln.

The fact that there will be a time when Ln is valuable, is not to say of course that it is valuable now or even will be valuable one hundred years from now. So to say that a life is overall positively valuable is to say that its a life worth living. We will be able assign that life an all-in positive value in the future.

On the comparison of possibilities I’m puzzled that you find such common locutions unintelligible. It makes perfect sense to assert that there might have existed someone stronger than any human that has ever existed. It not only makes sense, it’s true. Really, elves are not possible (read your Kripke more carefully!).

Let me assume that your account of “on balance positive welfare” is right. It doesn’t answer my question. If it is meant to: Is that ‘yes’ to my questions or a ‘no’? I can’t tell.

I don’t find the locutions strange at all. I find the account of their truth conditions that claims possible worlds stand in certain relations to the actual world and its inhabitants strange. Possible worlds are a heuristic, they aren’t real.

I think Kripke was wrong about fictional objects.

This is getting like work. These are your questions, I take it.

*Is there such a thing as the overall value of a particular life, say, L1 in w? If so, can we correctly assign to the overall value of L1 a finite positive number?*

Well, let’s see. I think I answered your first question, right? In case you are still not sure, “yes”. So no complaints there. Ok, what about the second question? I think I answered that, too. I said,

*So to say that a life is overall positively valuable is to say that its a life worth living. We will be able assign that life an all-in positive value in the future.*

Doesn’t that say that you will be able to assign it an overall value in the future? Maybe it was confusing because I didn’t say ‘positive’? Ok, add ‘positive’. But does that this mean it is not valuable now? Well, the most you can say now is that Ln will be overall valuable in the future. I don’t know whether you’d call that assigning it a positive value.

Okay, good. So each life has a positive overall value “in the future”. Suppose we are in, or are assigning value from the perspective of the future. We retrospect and correctly assign positive overall value to each life in w. Each has overall positive value in the future. Thus, since the value of w in the future is the sum of these overall values in the future, the overall value of w in the future is positive.

Thus, it’s not true that at any time, the overall value of w is negative. In fact, in the future, it is positive.

*Thus, it’s not true that at any time, the overall value of w is negative. In fact, in the future, it is positive.*

Christian,

This is just badly mistaken, since there is absolutely no point in the future of W (literally none) at which W has overall positive value. The value of a world W is the sum of the value of the world-stages that compose that W. That’s non-controversial. Every world stage S that composes W has an overall value that is infinitely bad, so W is infinitely bad. What you’re having trouble grasping, it seems, is that this is consistent with each life in W having an overall value that is positive. It already feels like a long thread, and I’m not sure there’s much more I can do to help with this. Maybe take another look at the picture above, see if that helps. I’m pretty sure we’re through here.

Whether this world is infinitely bad depends on the order in which you add things up (which happens with infinite sums).

One order: calculate each person’s lifetime net benefit/harm, then add those up. This is, on balance, infinitely good.

Another order: calculate each moment’s net benefit/harm, then add those up. This is, on balance, infinitely bad.

If by “infinitely bad” you mean “on balance infinitely bad”, whether this world is infinitely bad depends on the order in which you do the sum. If by “infinitely bad” you mean that it has an infinite amount of suffering (not net suffering, just total suffering), then this is true regardless of in which order you calculate the total sum.

It is tempting to think that the first way of adding things up better represents the goodness of the world than the first. This way of thinking explains the intuitive judgment that if you reversed things, so that individual n would enjoy himself for n seconds, and then suffer for n^100,000 seconds, things would be worse than the way you described. But this is a moral judgment and not a mathematical one. Mathematically, the sum is undefined until you specify an order.