(Note: This is Part II of a summary of my paper, "Craig, the Kalam Cosmological Argument and the Instantiation of Transfinite Set Thoery." There's a lot of preminary discussion, definition of terms used in Part II and off-topic bits of mental flotsam and jetsam in Part I that's relevant to the stuff in Part II. Of course, since about three people read this blog and I have no idea if any of them are interested in this stuff, I have no reason to believe I'm talking to any one right now. Still, having gotten this far, there's no turning back, and one might as well observe protocol.)

So, to recap, Craig's formulation of the Kalam cosmological argument for the existence of God is:

1. An “actual infinite” is impossible.
2. If the universe did not begin to exist at some point, this would constitute an infinite temporal regress of events.
3. An infinite temporal regress of events would be an “actual infinite.”
4. From 1 and 3: An infinite temporal regress of events would be impossible.
5. From 2 and 4: The universe began to exist at some point.
6. Everything that begins to exist has a cause of its existence.
7. From 5 and 6: The universe has a cause of its existence.

The gist of my paper is to attack this at three points, which are separate but I think feed into each other:

(1) Craig is, or claims to be a presentist. There are, at the risk of another diversion (I promise much shorter than the kind of Homer-on-steroids digressions that featured in Part I), a few different terms worth clarifying here.

*A-time, A-series, and A-theory all refers to time as we experience it ("clock time," maybe), as something moving forward from past to future. This is also referred to as "the tensed theory of time." An A-Theorist is a philosopher who holds that this intuitive picture correctly describes reality. Temporal passage is real.

*"Presentism," like a lot of philosophical terms, is used by various thinkers to mean various things, but usually refers to an extreme form of A-theory that says that only the present exists. An alternative, more moderate form of A-theory (advocated by Quentin Smith, for example, in his book "Language and Time") says that the past and the future exist in some sense, but all times have objectively existing "MacTaggart properties" of pastness, presentness and futurity. That is to say, the American Civil War exists and has the property of pastness, the war in Iraq exists and has the property of presentness and the alien invasion of 2012 exists and has the property of futurity. In six years, the alien invasion (assuming for the sake of argument that alien beings will invade the earth in 2012, which is not universally accepted) will have the property of presetnenss, the war in Iraq will (one hopes) have the property of pastness and the American Civil War will have receded even further into the past. I think Ted Sider calls this theory "moving spot-light eternalism," which sounds cool and gives the sense of the thing, so I'll use that here.

*B-time, B-series, and B-theory all refer to time seen "from the outside," so to speak ("calendar time.") This is also referred to as "the tenseless theory of time." A B-Theorist is a philosopher who holds that past, presentt and future are all equally real, indeed that terms like "past" and "future" have no objective reference except as indexicals (i.e. as pointing to the temporal location of the person speaking). When I describe East Lansing as "here" and some one who lives in Leeds describes Leeds as "here," it's not the case that one of us must be wrong. It would be nonsensical to speak of one privileged location and one alone having the property of "hereness," and "nowness" is the same way. In any objective, observer-independent sense, we can't talk about times being "past" or "present," but only of being related to each other tenselessly by "before" and "after" relations. St. Augustine advocates something very much like B theory on purely religious grounds in the philosophical chapters at the end of his "Confessions," so its certainly been a perspective that's been around for a while, but it really took off in the 20th century because many (not all) philosophers of time regard it as logically following form (or at least being strongly indicated by) Einstein's Special Thoery of Relativity with its destruction of absolute simultaneity.

OK, got all that? Back to my first criticism of Craig:

(1) Craig claims to be a presentist. This is important to the matter at hand because it impacts the sensibility of his claim (fleshed out in Part I) that while "potential infinities" are perfectly OK, "actual infinities" would be absurd. If the future already exists (whether in a B-theoretic sense or in the "moving search-light eternalism" sense), then the actual/potential infinity distinction becomes meaningless. To say that something (say, the set of days that have elapsed since the creation of the world) is a potential infinity means it keeps on getting larger--technically, it doesn't make sense to talk about sets getting larger, but you know what I mean--and there is no reason to believe that it will ever stop getting larger at some point in the future. Now, if the future actually exists, this makes no sense. There would only seem to be two options:

(i) The temporal series will never stop. That means that the set of future days is (drum roll please) an "actual infinity" of the sort that Craig is committed to saying are impossible. So if he were to say (he doesn't) that the future actually exists and it will have an infinite duration, he would have to give up claiming there are no actual infinities.

(ii) The temporal series will stop at some point, and there will be day (say, fifty million years after God's creation of time) at which point there is nothing afterwards. It's the end of the road. If this is the case, then the actual/potential infinity distinction fares no better than it would if (i) is the case. Anything you could possibly call a potential infinite (the number of days to elapse, the number of books that have been written, whatever) is nothing of the kind. After all, if the future already exists and it will have a finite duration, then any collection of things (days, books, whatever) that you'd be tempted to say has a potentially infinite number of members actually has a specific, finite number of members and is incapable of expanding any further. So if Craig were to say (he doesn't) that the future actually exists and it will have an infinite duration, he would have to give up the actual/potential infinity distinction on which he builds his argument.

So far, so good. I'm just agreeing with Craig, by saying that it's good that he rejects ontologies like B-theory and moving-spotlight eternalism, because they are manifestly incompatible with his actual/potential infinity distinction. He's quite right to say that, given his presentism, he's not being inconsistent by allowing a future that might never end. After all, according to presentism, the future doesn't exist yet, so it can't possibly count as an actual infinity.

Here, however is the rub:

If Craig's own presentist ontology is true, then he cannot possible justify his third premise, which was, remember:

3. An infinite temporal regress of events would be an “actual infinite.”

The ontology of presentism cuts both ways. If it saves Craig from havng to consider an infinite future progress of events an actual infinity because the future doesn't exist yet, then it should stop him from saying that an infinite past regress of events is an actual infinity. After all, the past doesn't exist any more. If he wants to water down his presentism to say that the past does exist or semi-exists or whatever in some form, then fair is fair and the future exists or semi-exists or whatver as well. You can't say "only the present exists," and then call an infinite past an actual infinity without saying the same thing about an infinite future.

(This is indeed why both Aristotle and Aquinas, good presentists both, didn't consider a possible infinite past to be an actual infinity.)

So, if we take his presentism seriously, the whole argument collapses at this point. The only way to save the argument for the next round is to accept for the sake of argument that presentism *and* B-theory *an* "moving spotlight eternalism" are all false, and adopt a fringe position held by a few philosophers here and there--notably, C.D. Broad and I tihnk Richard Gale--but never catching on too much. To the best of my knowledge it doesn't have a name of its own, but let's just say "open-future B theory." In other words, the past and the present exist but the future doesn't. (There is something intuitively appealing about this, even if its rare for any one to flesh it out as a formal theory and defend it.)

If this is the correct ontology, then the actual/potential infinity distinction doesn't reduce to meaninglessness (which it does if either B-theory or the moving-spotlight form of A-theory are true), and the past qualifies as an actual infinity if it doesn't have a starting point (which it wouldn't if presentism is true.) So, although its a premise damn few people would agree with (A- and standard B-theories between them holding the allegiance of almost every one whose interested in this sort of thing), for the remainder of the discussion, for the sake of argument, we'll accept this.

...which brings us to the next objection.

(2) Craig has given us no reason to think that actual infinities are impossible, especially considering that (as we discussed in Part I) he certainly doesn't try dispute the *mathematical* validity of transfinite set theory. Sure, he basically says talking about infinite numbers is a useful and interesting mathematical abstraction, but it can have no application in the real world.

Well, why not?

In order to establish this premsie, Craig brings up various classical paradoxes of the actual infinity. The following example, taken from his essay "Infinity and the Past," gives the flavor of all of them (sorry about the mis-spelling of "color", the guy's a Brit):

'...if an actual infinite could exist in reality, then we could have a library with an actually infinite collection of books on its shelves. Remember that we are talking not about a potentially infinite number of books, but about a completed totality of definite and distinct books that actually exist simultaneously in time and space on those library shelves. Suppose further that there were only two colours of books, black and red, and every other book was the same colour. We would probably not balk if we were told that the number of black books and the number of red books is the same. But would we believe someone who told us that the number of red books in the library is the same as the number of red books plus the number of black books? For in the latter collection there are all the red books--just as many as in the former collection, since they are identical--plus an infinite number of black books as well. And if one were to imagine the library to have three different colours of books--can we honestly believe that there are in the total collection of books of all colours no more books than in the collection of a single colour?"

Now, this is very weird to be sure, but what sort of impossibility is being claimed here--physical impossibility, logical impossibility or what?

It can't be logical impossibility, because that would mean disputing the strictly mathematical legitimacy of transfinite set theory which, that being mathematically well-established stuff, Craig wisely says he's not trying to do. Anyway, you can only disprove mathematical claims with actual proofs or counter-examples, not by milking intuitions that there seems to be something wrong with them.

You could say, quite reasonably, that building such a library would be a physical impossibility because libraries are the sort of thing that are constructed at specific moments in time, and you can't get an infinite set from a finite set by a process of successive adittion of finite numbers, for obvious reasons. At any given time in the construction of the infinite library, there would only be a finite number of books on the shelves, however large. This is, however, not the sort of physical impossibility that Craig is claiming, as he's willing to grant for the sake of argument as the set-up for the paradox that its' a completed infinite library. So "physical impossibility" would have to mean incompatibility with some unknown law of nature. What kind? It would seem utterly arbitrary to postulate such a law.

Finally, two points about the intution aspect of things:

(i) Yeah, it seems very counter-intuitive that every member of Set A could be a member of Set B, but that Set B has some extra members that Set A doesn't, but there aren't by virtue of this fact a greater number of members in Set B. This should, however, be expected. Our intuitions are formed, I'd think, by our experiences of dealing with ordinary fintie objects.

Intuitions formed at the level of "mid-sized dry goods" (e.g. humans, cats, cars, trucks, pieces of paper, billiard balls) break down at the quantum level. I find it wildly counter-intuitive that something could be a particle and wave. Surely, as a theist, Craig believes that intuitions about how to understand the nature and intentions of other minds formed at the level of the beings we encounter in our day-to-day experiences--that is to say, beings of only limited power, knowledge and benevolence--break down at the level of a being of unlimited power, knowledge and benevolence. Hence divine incomprehensibility and the via negativa and all that.

Similarly, it should be no surprise that intuitions (about things like size of collections and addition and subtraction) formed at the level of finite things break down on the level of infinite things. In none of these cases (weird quantum result, theological head-trips or transfinite sets) does this seem sufficient for a ruling of ontological inadmissability (i.e. sufficient for saying that the things in question don't really exist.)

(ii) The counter-intuitive heft of these results can be deflated nicely when we're more careful about applying the terms. The number of red and black books in the infinite library is "the same" as the number of black books in a set-theoretic sense of "the same number." All this means is that you can put to the two sets into one-to-one correspondence, matching up each individual in one to one and only one individual in the other, without ever running out of one or the other while unmatched members of the other set are left over. For reasons that probably don't need too much unpacking, by definition you'll never have this problem with infinite sets and their subsets. (There *are* interesting cases of transfinite numbers that can't be put into one-to-one correspondence with the whole number counting series--if you ever get a chance to see a proof of this, it's neat--but that's a separate issue from the simple cases we're dealing with here that can be mapped onto the whole number counting series, just from different strating point.) The sets are not however "the same" in the sense of Liebniz's analysis of identity, whereby saying that A is identical to B means that anything you can accurately say about B you can accurately say about B. Set A has members Set B doesn't, so they aren't "the same" in that sense. Once those two senses of "sameness" have been carefully distinguished, I think it starts to sound a lot less weird and counter-intuitive to say that there is 'the same number' of red and black books as there are of black books.

So, to review, (a) the actual/potential infinite distinction doesn't work on most theories of time, and even on Craig's theory, it doesn't do what he wants it to do, and (b) even if you (by going for the open-future B-theory thing) accept for the sake of argument that the distinction does work, he's given us no good reason to think that actual infinities are really impossible, which is a key premise. Let's accept, however, for the sake of argument that none of these objections are decisive. Pretend that actual infinities really are impossible. This sets up what I have to regard, as Zaphod Beeblebrox what say, as the clever bit:

(3) If Craig is right and actual infinities are impossible, and the rest of his premises are right as well, then the argument is self-referentially incoherent (that is to say, the premises contradict the conclusion.) How's that?

Well, and this really will be the quickest pat of the whole discsusion, to see why this is, ask yourself a very simple question: Has God always existed?

I think there are exactly two choices:
(i) God has existed for a finite amount of time, or
(ii) God has existed for an infinite amount of time.

Now, you can try to dodge this unpleasant choice by rejecting the premise on Augustinian grounds and saying that God is somehow outside of time. I'm not convinced that this is a tenable, internally consistent position, but let's accept for the sake of argument that it is. This would mean that the tenseless, B-theory of time is true. After all, if God is timeless, then from God's perspective, all times exist and are equally real. If even from God's perspective 2007 didn't exist yet (or even existed but had the objective property of futurity) but next year from God's perspective 2007 now existed (or existed and had the property of presentness), then it would sound like God is pretty darned thoroughly *in* time, wouldn't it? So, if you think God's existence is atemporal, then the actual/potential infinity distinction is meaningless and the whole argument grinds to a halt there. Given that, God's existence better be temporal, in which case, again, either:

(i) God has existed for a finite amount of time, or
(ii) God has existed for an infinite amount of time.

Now, if (ii) is true, then (given the rest of Craig's premises) the span of God's existence would constitute an actual infinity. Since that's impossible (indeed, the whole argument rests on this alleged impossibility), we'd better pick (ii), that the duration of God's existence is finite.

This, of course, runs us up against one of Craig's premises:

6. Everything that begins to exist has a cause of its existence.

So, if (i) is true, then God needs a creator of His own. Whatever caused God's existence (let's call it G2), however, must either be timeless (in which case the actual/potential infinite distinction collapses as discussed in God's case) or have existed for an infinite amount of time (in which case the argument collapses into self-refutation, as discussed in God's case), or have existed for a finite amount of time *and thus need a cause.* Not only is this an unwelcome result from the perspective of Judeo-Christian montheism, there is an infinite regress here. That is to say, the set constituted by God, God's cause, God's cause's cause, etc., would be an *actual infinity.*

Thus, the argument as reconstructed above (with the alleged impossibility of transfinite set theory being insantiated in spatio-temporal reality as a premise) is self-refuting.

As such, theists would be well-advised to either remove this argument from their philosophical arsenal, or to reformulate it in a way that does without the premise that there can be no actual infinities.