Monday, July 15, 2013

Can an actual infinite exist in the real world?

I confess that I'm still intrigued by the kalam cosmological argument (KCA), even though I very much prefer the Aristotelian-Thomistic cosmological arguments.  The KCA is easy to formalize:

1. Whatever begins to exist has a cause. (Premise)

2. The universe began to exist. (Premise)

3. Therefore, the universe has a cause. (From 1 and 2)

If this argument is sound, then the cause of the universe (the sum total of all physical space, time, matter and energy) would have to be timeless, changeless, immaterial and very powerful.  I won't comment on premise (1), except to note that it relies on the ex nihilo principle: out of nothing comes nothing.

I'm more interested in premise (2).  William Lane Craig offers two philosophical arguments and two scientific arguments in support of this premise.  I only want to focus on one of his philosophical arguments.

2a. An actual infinite cannot exist in the real world. (Premise)

2b. A universe without a beginning includes an actual infinite in the real world. (Premise)

2c. Therefore, the universe's past must be finite. (From 2a and 2b)

I'll skip (2c), since I assume it's not the controversial premise of the argument.  Why think an actual infinite cannot exist in the real world?  Here's what I've gathered based on my on-and-off study of the KCA:

2i. In set theory, subtraction and division are prohibited when applied to infinite sets. (Premise)

2ii. Nothing in the real world would prevent subtraction or division when applied to any set. (Premise)

2iii. Therefore, there cannot be an actual infinite in the real world. (From 2i and 2ii)

Keep in mind that set theory is logically consistent.  If a mathematician wants to prohibit certain functions in a mathematical theory, that's fine.  The question is whether or not such a theory can be applicable in the real world.

I've struggled with this argument, but I do see a lot of intuitive support for it.  Nevertheless, I'll ultimately leave this issue to those who are experts on the KCA.


  1. For the sake of argument, let us stipulate that there can, indeed, be actual infinites in the pysical world.

    Now, let us try to investigate what that would mean.

    Is 'space' infinite? If so, then there is an infinite spacial distance separating any two "bits of matter" and an infinite volume of space separating any three of more "bits of matter".

    Is 'matter' infinite? If so, then there is an infinite anount of matter in any given spacial volume.

    Are both 'space' and 'matter' infinite? Then both of the above apply -- there is both:
    1) an infinite anount of matter in any given spacial volume;
    2) an infinite spacial distance separating any two "bits of matter" and an infinite volume of space separating any three of more "bits of matter".

    Is 'time' infinite in the past? If so then this particular moment never can arrive, and we can never *be*, much less know that the past is infinite in extent.

    Is 'time' infinite in the future? If so, how could we even know? The year "Infinity AD" will never get here, so we can never know that the future is infinite in extent.

  2. The future would be considered potentially infinite, and not actually infinite, since it will always and indefinitely be possible for another event to occur before arriving at infinity.

    I'm intrigued by the argument that if the past were infinite, then the present never would have arrived. If the universe is infinite in its past, then why did the present arrive today, and not yesterday, or at any other time in the finite past? After all, an identical amount of time (infinity) had already elapsed between each of these moments.

    1. The present did arrive yesterday, but we happen to call it "today".
      This may sound odd, but the reason for this is that any moment in time is defined in relation to some other moment and the distance (or the time interval)between any two points in an infinite set is finite. To put it simply: if we had started counting a day earlier, 'today' would have 'arrived' yesterday. That is, if the time-intreval between t0 and t5(today) is 5, the time interval between t-1 and t4(yesterday) is also 5.
      In fact, despite Craig's claims, there is no proof against actual infinities. They are extremely counter-intuitive, I'll give you that, but they cannot be proved to be impossible.

    2. Actual infinites are logically consistent, as Craig will be the first the say, but they have arbitrary rules that don't apply to the real world. Examples? The prohibition of subtraction and division. At least, that's how the first philosophical argument goes.

      As for the subset of second argument, this isn't a matter of where we start counting. If the universe's past is actually infinite, then there is a one-to-one correspondence between the number of events that have elapsed and any moment in the finite past. This means the same amount of time had elapsed one-hundred years ago that had already elapsed one-thousand years ago. It's paradoxical, but I'll grant to you that it's not contradictory.

    3. Walter, if you're looking for a knockdown (or the closest thing to it) argument against the possibility of an infinite past, check out Alexander Pruss's brief post on the Grim Reaper Paradox:

    4. I know the Grim Reaper paradox, and while it is a very interesting argument, it is not a knockdown against the possibility of an infinite past. But,I'll give you this, It's much better than any of the arguments Craig presents against an infinite past.
      As to your one-to-one correspondence between the number of events that have elapsed, that is true, but actually irrelevant to the discussion.
      And, although I am not a mathematician, Ray is absolutely right: substraction is not prohibited for infinite sets, but the results are indeterminate. If, e.g. we had an infinite set of marbles in the real world, nothing would prevent you from taking one marble away.

    5. As I said, the one-to-one correspondence issue leads to a paradox, but I grant that it's not strictly contradictory. With respect to Ray's point, there's no need to reiterate it, since I'm already taking his word for it. Perhaps an actual mathematician would be needed to settle the issue. We know the mathematician, David Hilbert, believed actual infinites were impossible, but I'd be interested in seeing some kind of consensus on the matter.

  3. Yes, "the future" is *potentially* infinite, but cannot be *actually* infinite. "The future" doesn't exist yet, hasn't occured yet, so is potential, not actual.

    But, "the past" has occurred, and does exist (or, at any rate, *did* exist). That is, the past cannot be potential, but only actual. So, if "the past" were infinite, it could only be actually infinite.

    Now, we can never get to the year "Infinity AD", that year will never come. But, if "the past" is infinite, then we, in this year, at this instant, stand in relation to *any* infinitely past instant in the same relationship as some instant infinitely in "the future" -- that is, this instant can never *actually* arrive, can never become actual.

  4. (2i) is not true. Infinite sets are frequently and often subtracted. Consider N - {x|x in N for x > 3} = {1,2}. What is typically undefined though is the subtraction of transfinite cardinals, as the set subtraction procedure does not yield a determinative process to define a subtraction operation that yields a unique answer simply on the basis of the cardinals. This isn't any different than, for instance, indeterminate results in calculus.

    Consider for instance the limit as x approaches 0 of sin(x)/x. If you attempt to calculate the limit as it is, you would get 0/0, which is an indeterminate form. But of course, if you simply differentiate the top and the bottom of the fraction by applying L'Hopital's rule, you get the limit of x as x approaches 0 of cos(x)/1, which simply is 1. This isn't a matter of paradox, but a matter of using an inappropriate representation. Other representations - here the set itself being a pretty good one! - do permit precisely the operation you are thinking of.

  5. Ray, I'm not a mathematician, so I'll just take your word for it. Like I said, the KCA isn't my area of expertise.