In his defense of the KCA, William Lane Craig argues that absurd consequences follow from the existence of an actually infinite number of things. Take the case of a library with an infinite number of books, for example. Assign each book a natural number, {1, 2, 3, …}. Suppose a philosophy student comes and checks out all the odd-numbered books, {1, 3, 5, …}. One absurd consequence is that there would be no fewer books in the library than before, because you can match the first set with the second in a one-to-one correspondence. Despite the fact that the second set is a proper subset of the first, the first is not greater than the second (i.e., it has no more members than the second). But, intuitively, there were twice as many books in the library before the student arrived than when the student left. So, this is absurd.

One objection to this line of thought can be found in Wes Morriston and Paul Draper. The objection goes like this: You don’t understand set theory, Dr. Craig. While you can say the set {1, 2, 3} *is greater than* the set {1, 2} on account of the former having *more* members than the latter (or, the latter having *less* members than the former), you cannot say that the set {1, 2, 3, …} *is greater than* the set {1, 3, 5, …} on account of the former having *more* members than the latter. This is because the notions of “more” and “less” are well-defined with respect to finite sets, but not infinite sets. So we aren’t justified in saying one infinite set “is greater than” another, which is where we’re supposed to see the intuitive absurdity.

I am not aware of a response from Craig if there is one. So, I’ll offer my own.

First, transfinite set theory is permeated with comparisons between higher orders of infinities, where the terms “more” and “less” are used without reserve. So if there is a totally different notion of “measure” at work in comparing infinite quantities, we will have to designate placeholder names like “schmore” and “schless” (and “is schmeater than”) to do the same work as the standard notions. But now can we simply restate Craig’s argument, using the placeholder notions? The objection may just sweep the problem under the rug, or kick it upstairs into the attic. Either way, the problem remains either under our feet or over our heads.

Second, note that the conclusion is not that we *can’t* apply the standard, well-defined notions of “more” and “less” to infinite sets (consequently, “greater than”), but just that weren’t automatically *justified* in doing so. In other words, it’s an open question whether we can, but one would need an argument for or against doing so. So, here’s two arguments for doing so:

First, it is not clear whether set theory can given a coherent interpretation without the well-defined notions of “more” and “less.” How, for example, will one paraphrase the claim that “there are more real numbers than natural numbers” *salva veritate*? Second, the standard notions of “more” and “less” cannot fail to be meaningful when applied to substances of the kind Craig has in mind. It is a modal fact entailed by the natures of concrete substances that, necessarily, the standard, well-defined notions of “more” and “less” apply when performing inverse operations with them. So, if you had an infinite collection of concrete substances, the well-defined notions apply. However, it very well may be that when you have infinite collections of non-concrete objects (schmojects?), this is not true.

Hi, Chad. I’m sympathetic to your approach, but here’s a quick objection to your first argument (in the final paragraph).

Perhaps we can paraphrase the claim that “there are more real numbers than natural numbers”

salva veritateby something something like this: there are moreAs thanBs if a bijection relation can’t obtain between theAs andBs. There are moreAs thanBs if they can’t be placed in a one-to-one correspondence.If I recall correctly, Morriston proposes a basic account of what it means to say that one denumerable set is greater (or less) than another denumerable set. What’s interesting (and, I think, problematic) about his proposal is that, seemingly, he doesn’t intend for the greater (or less) than relation to be quantitative in nature. I believe he suggests something along the following lines: a denumerable set

Pis greater (or larger) than another denumerable setQifPcontains all of the elementsQcontains, andPcontains at least one elementQdoesn’t contain. Applying this to the library example above, the original set of books is (in Morriston’s sense) greater or larger than the second set because they satisfy the conditions established in the definition of “greater than.” As mentioned, the worry is that if the greater than relation isn’t a quantitative notion, but ration a qualitative notion, then it becomes difficult to understand what it means to say thatPis greater thanQ, or how this helps avoid Craig’s criticism sincePandQcontain the same quantity of elements (though the former is supposed to be greater or larger than the latter.)Comment by Marc Belcastro — March 18, 2013 @ 1:39 pm |

Sorry about the typos! I was finishing my third cup of coffee.

Comment by Marc Belcastro — March 18, 2013 @ 1:45 pm |