Of Countable Infinities and Limited Atonement

For some reason many people struggle to understand and accept the traditional doctrine of limited atonement.  Today I would like to show that people struggle with limited atonement because they have not studied mathematics.  Why they fail to accept it is beyond my powers of psychoanalysis (or rather, lack thereof).  So let’s instead focus on the mathematical illustration.

One of several common objections to limited atonement runs something like this:  If the atonement is limited, that is, Christ died only for the elect, then the efficacy of Jesus’ sacrifice on the cross is narrowed.  By virtue of limiting the scope of the atonement, we cheapen the cross and limit the power of Christ’s sacrifice.  The problem with this objection is that it simply isn’t true.  How can I say that?  Mathematics!

People who have not studied math usually aren’t very good at working with quantitative concepts of the infinite.  This idea that somehow the power of Christ’s sacrifice is limited if we limit the scope of the atonement is a quantitative idea regarding infinite entities.  Under the system of doctrine with which we usually hold limited atonement discussions (largely reformed, calvinistic circles, with the five point calvinists arguing for limited atonement and the four pointers against), it is understood that each sinner owes God an infinite debt as punishment for his sins offending the holy deity.  There are, of course, a finite number of sinners in history.  That number is the number of humans that live in the universe during the period starting at creation and ending at the dissolution of the universe.  Let’s call this number X.

Now let’s do something fun.  Let’s show that there are just as many even integers as there are integers.  The integers are just whole numbers, positive or negative.  Even integers are integers that, when divided by 2, yield an integer quotient.  Let’s define a function!  That function will of course be called f.  Let’s say that f takes an integer and gives us an even integer.  The rule that it follows is that f takes any integer to 2 times that integer.  For example, f(17) = 34, or f(-3) = -6, etc.  Clearly, for any even integer (call it y), we can find an integer (call it x) so that f(x) = y.  Similarly, for every integer, we have just one even integer associated with it.  At this point we get to write a list!  We can list every single integer and every single even integer next to each other like this:

  1. 0 \leftrightarrow 0
  2. 1 \leftrightarrow 2
  3. -1 \leftrightarrow -2
  4. 2 \leftrightarrow 4
  5. -2 \leftrightarrow -4
  6. 3 \leftrightarrow 6
  7. -3 \leftrightarrow -6

Proceeding in this manner, we see that there are just as many integers as there are even integers.  QED

Stop and think about that for a moment!  The set of all even integers is infinite, and the set of all integers is a set that contains the set of all even integers, but they are the same size!  They’re both the same type of infinite (countably infinite, in case you were wondering).  In fact, by associating each integer with the number of its list item in my list above, we see that the set of integers and the set of positive integers have the same size.  This is helpful for explaining limited atonement.  The property of evenness is infinite in its scope, but it doesn’t cover every single integer.  This doesn’t limit the infiniteness of evenness though, as we’ve just seen that there aren’t any more integers than there are even integers.  Similarly, the fact that Jesus pays all the infinite penalties of sin only for the elect doesn’t limit the infinitude of His sacrifice.  It’s perfectly acceptable, by the type of function mapping arguments I’ve used above, to argue that Jesus’ infinite sacrifice could pay for the sins of more than just the elect if God wanted it to; an infinite sacrifice would be sufficient, and since there are only finitely many human beings, we’re not changing orders of infinity by adding more human beings to the list of those for whom Christ should die (I’d use Cartesian products if I were trying to show this by the way).  The objection just doesn’t hold mathematical water.

What’s the point of all this though?  Why am I writing this on a theology blog?  My point is this: there are going to be things in Christian theology that we don’t understand.  There are many things that seem contradictory, that are unintuitive, or that we can’t comprehend at all.  That’s the nature of studying an infinite God that also created your brain; there shouldn’t be a way to completely understand Him.  That has practical implications for how you defend your theological beliefs.  You can’t defend your theological beliefs using your intuitive conceptions of the infinite, what you think a just God would do, how the world should work in your mind.  Instead, you have to use Biblical arguments.  The only way to come to a right understanding of God is by looking at what He says about Himself.  The only way to understand His salvation is in His written Word.  While following the evidence in Scripture wherever it leads, we must be willing to lay aside whatever intuitions, prejudices, and biases we may have and take God at His Word, even if it doesn’t seem to make sense at first.


If you want Biblical arguments for limited atonement (there are many), you can check out John Murray’s Redemption Accomplished and Applied and Mike Riccardi’s summary of one of Murray’s arguments here.   There are many other sources, but I leave you with these two for now.


This post is the first of several I want to write about random examples from the secular world that illustrate divine truths. The examples I’ll write about aren’t authoritative or inspired or somehow magically enlightening; they just give us another way to understand theology through illustration. As such, then, they shouldn’t be taken too seriously, nor should too many objections to weaknesses in the illustrations be raised. I hope you’ll find them enjoyable and helpful.

Of Countable Infinities and Limited Atonement

2 thoughts on “Of Countable Infinities and Limited Atonement

  1. Bruce Quimby says:

    O Bearded one, I like your thinking (what I understood of it, for particularly in the area of Mathematics I tend to be a bear of very little brain). My question is, for those who would say that the theology of limited atonement is limiting to Jesus’ power to atone, if you extrapolate from their perspective, wouldn’t Jesus’ power of atonement be limited even if EVERY human were atoned, since there are a finite number of humans?


    1. thebeardedone says:

      Jesus’ power of atonement would be limited if it extended to every human under the assumption that it only extended to every human. So yes, you are right. I wrote because the objection then often gets rephrased as, “to every human who could exist, and every sin who could commit.” As time goes on, that list could increase without bound, but is at most countably infinite. That’s where math comes in and says, “Well, that’s the same amount of atonement as if there were indeed a finite number people” (using my last paragraph with math). It’s not a very solid objection philosophically, but I understand the heart behind it, so I wanted to write a little about it!


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