(This is still a work in progress. Please check back later for the finished version)
Many people object to the doctrine of hell, saying that it's unfair to impose an infinite punishment for a finite offense. Now, there are many possible replies here. In fact, this is one of those objections that persists because it's so mistaken on so many points. The numerous ways to be confused here take on a life of their own: when you provide one reply, your opponent often just moves on to another mistake.
But I am fond of one particular answer. It's rather novel, and it requires only some high school math to understand. So I thought I'd take the time to explain it in detail. I will then go over the other classes of answers in a quick, holistic review.
My answer is very simple: an eternal punishment doesn't have to be infinite in magnitude.
Clever students of math will have already seen that this is perfectly obvious, and figured out everything I'm going to say about it. But let us go on to be explicit: it is a well-established mathematical truth that an infinite geometric series can sum to a finite number. So, 1/2 + 1/4 + 1/8 + ... sums to 1, despite the fact you're adding up an infinite number of terms. Likewise,
0.003 + ...
= 0.333..., despite it again involving an infinite number of terms. The only thing required is that each subsequent term decrease by a ratio less than 1.
This idea has many broad applications in more advanced mathematics, and solves philosophical conundrums like Zeno's paradoxes. It's how Achilles can catch up to a tortoise, despite the fact that this can be described as an infinite sequence of durations: the infinite sequence sums up to a finite duration. Likewise, a sinner's punishment in hell may be eternal, yet sum to a finite total value.
This is established mathematics. In this way, the punishment in hell can be any finite value, or infinite - and tailored to the exact amount that the sinner deserves. For the Bible says that hell is eternal, but does not specify whether it's infinite in the total magnitude of its suffering. So at this point, the objection of the infinitely disproportionate punishment in hell is fully answered. But I think it may be worth exploring the consequences of this answer, to get a better understanding of the issues surrounding hell.
For one, limiting the punishment in this way may seem like it would require hell to improve or offer relief in some way, as the terms in a converging series has to go to zero. This is indeed a mathematical requirement of such series, but it may not play out in the way one may expect, if one expects hell to be less hellish over time.
The way I see it, the amount of punishment suffered in a given time interval can be modeled as:
intensity * duration * worth
Here, "intensity" is the subjective experience of the target - basically, their answer to "how much does it hurt, on a scale of 1 to 10?" "Duration" is just how long the experience lasts. "Worth" is the overall value of the target. Humans have relatively high worth, whereas an animal - say, a rat - has relatively less worth. Something like a worm or a cockroach has less worth still.
We then take the product of these factors for each time interval, then sum them over eternity get the net punishment in hell. So the term for each time interval has to eventually go to zero, for the net punishment to converge to a finite value. And the way I think that hell works, is that it diminishes the worth of the entities that are in it. So the reprobate may start out relatively worthy, but hell will erode that over time, so that they'll come to be worth less than a rat, or a worm, or a cockroach. Eventually, at the end of eternity, they'll be worthless, and their suffering will count for nothing.
Thus the sequence of terms goes to zero, and the total punishment remains finite. Note that this means hell doesn't ever have to "get better": the reprobate's answer to "how much does it hurt" can remain maxed out at 10, but as long as their value decreases according to the rules of converging series, the net punishment can still be finite. In fact, the intensity may increase over time, even on to infinity, and the total punishment can still be finite as long as the worth of the reprobate decreases accordingly.
Of course, all this is speculation: the Bible doesn't give nearly enough information to make any specific calculations. But the mere possibility of such calculations is sufficient to answer the objection of "infinitely disproportionate punishment", and without giving up anything about whether hell gets better or worse over time.
But I do think that there is something to the idea that the reprobate in hell become less worthy over time. After all, Christ promises us that in him, we will have life, and life abundantly. How should the reverse not happen to those who are separated from him? How could they retain any value, when they're alienated from Him who is the source of all value? In heaven, our existence is enriched. In hell, it would be correspondingly diminished. As C.S. Lewis notes, hell is a very small place. In the eternal limit, it is of infinitesimal consideration compared to heaven.
There are other, more common possible answers to the objection of "infinitely disproportionate punishment". I'll go over these in less detail.
One such answer is that God doesn't so much send people to hell, as people choose to go there themselves by rejecting God. This is true enough, but I think it merits an elaboration to demonstrate how this answers the objection. After all, it's not as if God's saying "hey dude, you rejected my help when you had your chance, now you get to suffer forever", which could still be considered unjust.
So first, why do we believe that people send themselves to hell? The people in hell got there by "rejecting God" - and we must understand what that means. God is the source of all goodness. Anyone who rejects him for eternity will be left alone with only their sins. That is to say, hell is not some separate, distant, or external consequence of rejecting God. Rather, it is simultaneous and synonymous with the eternal rejection of all goodness - by rejecting him who is, by definition, its very source. Hell IS the eternal rejection of God.
This is what we mean when we say that people send themselves to hell. As an immediate corollary, we may infer that people in hell do not want to leave. It is "locked from the inside". After all, it would be good for a damned soul to desire repentance, forgiveness, and heaven. But alas, those in hell will reject this good, and instead make the wicked choice to remain.
But we shouldn't overemphasis even this "choice". Remember, hell is the eternal rejection of all goodness. Long ago it was postulated that there is no actual fire in hell: fire is a creation of God and therefore good, but we know that nothing good will exist in hell. Likewise, at the end of eternity, the ability to make a choice will not exist in hell. Nor will the ability to stick by that choice, or even the pleasure that comes from defiance. So the reprobate in hell can be imagined as a gibbering mess of resentment, self pity, impotence, and delusion. That is the nature of their choice. And as time passes on to eternity, it will not even matter whether they have a real "choice" or not, as they'll be utterly worthless.
How does all this help with answering the objection of "infinitely disproportionate punishment"? Well, why would an infinite punishment be disproportionate? It would in fact be perfectly fair, if the offense is also infinite. Consider the above description of the reprobate: resentful, delusional, continually making the evil choice to reject everything good even as they harden to the point where it's not a choice anymore. Does that sound like the paragon of virtue to you? Like someone who wouldn't ever sin for all of eternity?
Of course not. It would be strange indeed if someone became morally perfect by going to hell. No; one of the most pernicious effect of sin is that it'll cause you to sin more. In hell, this effect will run its full course to eternity. That is, the reprobate in hell will continue to add to their sin, forever. And just as a criminal can add to his sentence by committing more crimes in jail, the sins committed in hell merit further punishment. And if the total sin is infinite, the perfect, proportionate punishment would also be infinite.
We must also consider the nature of the offense: In this world, there are some who reject God for what they honestly believe are good reasons. In hell, 'honest belief' is not possible. There the reprobate will continue to reject God, even after they have infinite evidence that the God of infinite goodness has expended infinite effort for their salvation. Now, what kind of punishment might that merit?
So, the objection is wrong. It is wrong on both of its premises. As I have demonstrated above, hell does not necessarily impose an infinite punishment, and the sins of the reprobate is not necessarily finite. Biblical Christianity is perfectly flexible on both of these points. Either one of them, in isolation, is enough to fully answer the objection of "infinitely disproportionate punishment", and we have both. So we can say with confidence that the reprobate in hell will get exactly what they deserve - the exact right level of punishment commensurate with their sins.
But there are some who would still find a way to object, who would consider it "unfair" that any kind of hell should exist at all, or that there should be any punishment for any sins, or that anything should even be considered "sin" in the first place. To these I merely point out that words like "justice" and "fairness" have well-understood meanings, and that it is God who ultimately defines and establishes them. You don't get to say "unfair" to just mean "well I don't like it".
But this "I don't like it" objection, combined with the two mistaken premises of assuming infinite punishment and finite offense, is what allows the objection to persist by going from mistake to mistake. As I said at the beginning, addressing any one of these points allows the objector to simply move on to the next mistake, in a spiral reminiscent of hell itself. After all, the reprobate in hell will tell themselves in a spiral that their condition is monstrously "unfair" - that they've done nothing wrong, and deserve no punishment. In other words, they'll be wrong at every level, like how the objection is wrong at every level. And at some point, there is little that can achieved through further reasoning, like how the reprobate are damned forever.
You may have noted that I have not been sympathetic in my depiction of the reprobate. This is intentional. But there is still hope for anyone on this side of eternity.
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