(wink here again. Just so you know.)

Traditionalists typically advance two arguments about suffering against annihilationists: 1) We merit infinite suffering (I'll probably deal with this point in a different post), and 2) People suffer differing levels of punishment in hell.

Point (1) is intended to prove that you can't have annihilationism since you can't suffer an infinitie amount without infinite time (again, I disagree, but that's for a different post). Point (2) is intended to show that annihilationism is false since there is only one result--annihilation without differentiation, and that contradicts the different levels of punishment shown in the Bible. (This point only works on annihilationists who believe that there is no period of suffereing before annihilation. It falls completely flat against those who do.)

Traditionalists run into trouble if they try to hold both (1) and (2). Basically, there are no real levels of punishment in infinite suffering. You might argue that person A's suffering is 5x more intense than person B's suffering. But when you multiply by infinity, it is exactly the same. 2 x infinity = 5 x infinity = 100 x infinity = infinity x infinity. So to argue that there are levels of punishment in infinite suffereing is to not understand infinity very well.

There are two defenses that I can see against the contradiction: 1) Cardinality. There really are some infinities that are larger than others. You could argue that Person A's suffering lasts for duration Aleph naught, while Person B's suffering lasts for Aleph prime. This strikes me as a distinction without a difference, but I suppose it would be technically true. 2) You could argue that suffering is qualitatively different at different levels--a pain that is quantitatively 5x worse is even worse in some uncountable way as well--it is a qualitatively worse pain.

I have never seen either argument made. Has anyone else?

A while ago I wrote a series of six short posts using these two ideas. I think I've answered your objections there.

It's at http://kiwiandanemu.org/?cat=31. (The first post is at the bottom of the page).

Your discussion is interesting. I'm glad you had to have some place for your thoughts to overflow. :)

Ali - it sounds like you are going towards my "qualitative difference" idea, though you don't fully flesh it out. I was hoping you'd deal with the infinite durations and show how it isn't the case that an infinite duration of "medium security" punishment is equal to an infinite duration of "maximum secruity", because math wise, it is equal.

It's the same sum in terms of cardinality. That doesn't mean it's equivalent math-wise. You can put all the items in one list into one-to-one correspondence with all the items in the other list, and the sum is equal. But at any given moment it's not equal, and for any finite time it's not equal. In fact, you will never reach a time when it's been anything other than a finite time, so it will always add up to having been greater with the maximum than it would have been with the medium.

So I would say that there is a quantitative difference, just one that can only be measured over a finite amount of time (but you will never have reached anything more than that). With infinity you can easily get weird results like that.

Wink, I understand the math, but I don't think it applies in this instance.

(I'm conscious of appearing callous by discussing eternal torment without at the same time expressing the horror it would involve. Even though I won't be crying horror every time I mention it, be aware that I do have a horror of eternal punishment. It's unfortunately impractical to continually express it in a discussion.)

Assume that there are different levels of punishment. A person with an everlasting sore toe will certainly suffer less than a person who is experiencing everlasting burning all over their body. In that case, no matter how long the suffering is, there are physical parts of the first person that will never suffer whereas there is no physical part of the second person that doesn't suffer. A difference is maintained regardless how long the suffering lasts. Sure, if you were to measure "level of pain" and multiply it by infinity, there would be no difference, but that's because you are only factoring in pain levels and not considering other possible factors.

But let's assume that both people experience an everlasting sore toe, and that the first person experiences a lower level of pain in his toe than the second i.e. there is a difference in level of punishment, but not type of punishment. If you were to graph it, with the level of pain on the verticle axis and time on the horizontal axis, at any single point of time along that graph the first person would be suffering less than the second, i.e. even though mathematically

as a wholethey will suffer the same amount of pain, their experience will be that one is suffering less than the other. In that situation, if at any time you asked the second person if they wanted to swap with the first, they'd say yes; if you asked the first to swap with the second, they'd say no.So my two points are these:

1. Your mathematical equation does not take into account that eternal punishment may not just be "amount of pain", but "type of punishment".

2. Your mathematical equation does not take into account that at any given moment the lesser punishment will consistently be less than the greater punishment. And that, I believe, is what the Bible teaches.

I suppose that works. I'm too used to thinking about infinities as total sums. Your point that we'd never reach a time when it'd been anything other than a finite time had escaped my notice.

I agree with Ali.

I would also add that 2 x infinity = 5 x infinity = 100 x infinity = infinity x infinity is not valid mathematically. You can't multiply things by infinity. The statements are not evaluable.

One thing though, I'm not sure think that the cardinality argument could have been used against the position advocated in your post. For a set to be countably infinite there must exist a 1-1 map between the set and the Natural Numbers. I think that this does not exist for any set crossed with time since time (I think) is not contained in the rational numbers but is the positive real numbers. Therefore all sets of eternal suffering must be uncountably infinite sets.

As I did research after my post I saw that there are different cardinalities fpr uncountably infinite sets. I forgot about that...that's what I get for going off of what I remembered from a math class 6 years ago. :P I guess I'm going to have to think about this one more.

If time is atomistic, then it's the natural numbers. If it's dense, it's the rationals. If it's continuous, it's the reals. I don't think there's any consensus on which of those is correct, is there?

So far as I know, there is as yet no consensus on time. My guess is that there never will be.

Really??? Time may not be continuous? Or even dense? I'm curious as to how those arguments work because that makes no sense to me.

According to the Internet Encyclopedia of Philosophy entry on time, classical theories assumed continuous time and some newer ones assume quantized/atomistic time. I'm not sure why it couldn't be dense as a third possibility.

I would also add that 2 x infinity = 5 x infinity = 100 x infinity = infinity x infinity is not valid mathematically. You can't multiply things by infinity. The statements are not evaluable.Pedantic. Those "equations" were just shorthand for f(x) = 5x lim x -> infinity and its relatives. Evaluable, but far more annoying to write than 5 x infinity, which pretty much everyone understands even if it is technically unevaluable.

I would also add that 2 x infinity = 5 x infinity = 100 x infinity = infinity x infinity is not valid mathematically.Pedantic. How about f(x) = 5x lim x -> infinity. Evaluable and functionally equivalent. But a lot more annoying to write and a lot harder for people to understand.