“To Infinity And Beyond!”

What exactly does it mean to be ‘infinite?’ I know we often use the term for rhetorical flourish (usually in hyperbole or embellishment), but what precisely does it mean? The other night a friend at the Dallas Socratic Society discussion group, Brett, read a fantastic paper, Examining Infinite Set Possibilities. Brett’s aim was simply to sketch out some coherent ways we can speak about infinity and infinite sets. Aristotle long ago, Brett reminded us, gave us the traditional view of infinity:

…it is always possible to think of a larger number; for the number of times a magnitude can be bisected is infinite. Hence the infinite is potential,never actual; the number of parts that can be taken always surpasses any assigned number. Physics 207 b8

So if Aristotle is right, an actual infinite set is just impossible because one can always add another member to the set. But if this is true, then the set is always finite. William Craig uses the notion of the impossibility of an actual infinite to bolster the second premise in the Kalam argument. The Kalam argument, recall, makes the following move:

1. Whatever begins to exist has a cause
2. The universe began to exist
3. Therefore, the universe has a cause

Craig supports (2) by (among other things) arguing for the impossibility of an actual infinite series of events in time:

4. The series of events in time is a collection formed by adding one member after another.
5. A collection formed by adding one member after another cannot be actually infinite.
6. Therefore, the series of events in time cannot be actually infinite.

I should say that Carl Sagan may have been a smart cosmologist, but this argument makes his proclaimation about the universe being “all that ever was, is, or will be,” appear quite silly.

Now while I readily accept the notion that an actual infinite set consisting of concrete individuals is just impossible, the Platonist in me is quite willing to accept the notion of an actual infinite set of abstract objects. After voicing this in the discussion, another friend Sloan—who I should say is far smarter than I—roundly criticized me for embracing this view. But I can’t shake it. It seems to me that sets containing members like possible worlds, numbers and other abstracta, are actually infinite! What’s wrong with me? Am I just a Platonist gone wild!?

4 Responses to “To Infinity And Beyond!”

  1. Bryce Taylor says:

    Hey Xavier and Keith,
    This is Bryce from that Socratic Society meeting. I enjoyed your blog and hope I can see you guys at future meetings. I’m considering writing a paper on the theory of time, perhaps with a focus on the B-theory and, particularly, whether it is fatalistic as Mr. Lee asserted. I look forward to reading your future blogs. Thanks.

  2. Xavier says:

    Hey Bryce, thanks for stopping by. Sounds like you have a tough project ahead of you. I hope I get a chance to read it.

  3. Anonymous says:


    To address your issue, I suggest Summa Theologica I:7:1-4, but in particular the reply to the second objection of article 2.

    Regarding Craig and the Kalam argument, per his textbook Craig rejects the modality of being (re: Aristotle & St. Thomas). That’s strange… and of course incorrect. Moreover, one of the presuppositions of Craig’s argument (and which is, incidentally, a fatal flaw in the arguments of ID theorists) is he believes he can use the results of modern empirical science ALONE to reason to the existence of God. That’s wrong and it flows (partially) from his rejection of the modality of being. Modern empirical science can and does provide us with data and descriptions of physical reality (i.e., science addresses questions of “how?). However, it takes natural philosophy to interpret those data and descriptions and to reason to the existence of the First Cause (i.e., natural philosophy address the question of “why?”). IDers indeed are guilty either of (1) using science alone to argue to the existence of an Intelligent Designer or First Cause (which, as just stated, is impossible), or (2) manipulate or change the bounds of modern empirical science to suit their needs. They end up getting it wrong and doing a disservice to science, natural philosophy, and frankly, to God.

  4. Keith says:

    Thank you for stopping by. We appreciate your thoughts. However, the discussion is neither whether we like or dislike what you call modern emperical science, nor to speculate on whether Craig presupposes certain feelings toward it. Further, the post said nothing of ID. Now, we appreciate that you have strong feelings on, well, it appears everything; nevertheless, we’ll appreciate you even more if you could humor us and stick to the topic at hand.
    If you have any thoughts on infinite set theories, we would love to hear them. Otherwise, good day.–>

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