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Post Index: A Critique of Craig's Philosophical Arguments Against a Beginningless Past

Craig's Case Against the Existence of Actual Infinites, Part Six

Craig’s Diagnostic Argument: "The Deeper Problem"

(The rest of the posts in this series can be found here.)

According to Craig, the supposedly counter-intuitive implications we've discussed arise in virtue of the following set of claims:

1. A set is larger than any of its proper subsets. (a version of Euclid’s Maxim)
2. If two sets can be put into a one-to-one correspondence with one another, then neither set is larger than the other -- i.e., each set is just as large as the other. (The Principle of Correspondence)
3. Actually infinite sets are possible.

Craig points out that (1)-(3) jointly entail a contradiction.[1] Which proposition should go? Craig thinks that it should be (3). But why should we follow him in thinking this? Well, it's not clear that we should. For yet again, Craig equivocates on a key term.[2] For there are at least two interpretations of the notion of a set A being "larger than" one of its subsets B. On one interpretation, A is lar…

Craig's Case Against the Existence of Actual Infinites, Part Four

Fourth Version: Subtracting the Same Amount Yields Contradictory Results

(The rest of the posts in this series can be found here.)

Hilbert's Hotel Illustration:
Imagine a hotel with an infinite number of rooms and an infinite number of guests. Such a hotel would have some surprising features. So, for example, suppose all the guests in the even-numbered rooms check out. Then the hotel would still have infinitely many occupied rooms, viz., the odd-numbered rooms. On the other hand, suppose all the guests in rooms 4, 5, 6, ... check out of the hotel. Then the hotel will no longer have infinitely many occupied rooms -- it will have three. But how can this be? Just as many guests checked out in the first scenario as in the second scenario. This is because (i) the infinite set of even-numbered rooms (2, 4, 6, …) can be put into a one-to-one correspondence with the set of rooms numbered 4 and higher (4, 5, 6, …), and (ii) any two sets that can be put into a one-to-one correspondence with …

Craig's Case Against the Existence of Actual Infinites, Part Three

Third Version: Making Room in a Fully-Occupied Region Without Removing Occupants; Filling Up an Unoccupied Region Without Adding Occupants

(The rest of the posts in this series can be found here.)

Hilbert's Hotel Illustration:
Imagine a hotel with an actually infinite number of rooms. If such a hotel were possible, then it would have some remarkable features. For example, suppose every one of its rooms were full. Then one could create vacancies in some of the rooms even if none of the guests left. So, for example, the hotel manager could have the guests in room 1 move to room 2, the guests in room 2 move to room 3, and so on for all the rest of the rooms. In this way, room 1 would could become vacant for a new guest, even though not a single guest left the hotel.

In fact, one could create infinitely many vacant rooms without removing a single guest. For one could move the guest in room 1 to room 2, the guest in room 2 to room 4, the guest in room 3 to room 6, and in general, for ea…

Craig's Case Against the Existence of Actual Infinites, Part Two

Second Version: Adding Without Making Larger

(The rest of the posts in this series can be found here.)

Library Illustration: Imagine a library containing an actually infinite number of books. In such a library one could add one, two, three, or indeed denumerably infinitely many more books to the shelves without making the library any larger in terms of how many volumes it contains. This is because (i) the original infinite set of books can be put into a one-to-one correspondence with any set of books that could result from such additions, and (ii) any two sets that can be put into a one-to-one correspondence with one another are the same “size” – each is just as large as the other. But this is absurd; the original set of books is a proper subset, or proper part, of each set of books that would result from such additions, and no proper part or subset of a set could be just as large as the set of which it is a part. Therefore, a library containing an actually infinite number of books is i…

Craig's Case Against the Existence of Actual Infinites, Part One*

(The rest of the posts in this series can be found here.)

William Lane Craig argues that the universe had a beginning as part of an argument that it had a divine "Beginner" or divine temporal First Cause of the universe. He offers four arguments for a finite past: two a priori and two a posteriori. The first is that an infinite past is impossible because infinites are impossible in general; the second is that even if concrete actual infinites are possible, they can't be traversed by successive addition; the third argues that the evidence for the Big Bang indicates an absolute beginning to the universe; and the fourth argues that the 2nd Law of Thermodynamics indicates that the universe is running down, in which case it must've been "wound up" with an initial input of matter-energy, and the latter was the beginning of the universe. In the current series of posts, I will consider Craig's versions of the first argument.

First Version: Parts as Large as the …

Jeremy Gwiazda on Swinburne's Case for Theism

Via Prosblogion, I just learned that Jeremy Gwiazda (a recent PhD from CUNY) wrote a critique of Richard Swinburne's case for theism for his dissertation. The title: Probability, Simplicity, and Infinity: A Critique of Richard Swinburne’s Argument for Theism. Thus, for those seeking to understand and evaluate Swinburne's case, you might well want to order a copy of his dissertation (via the link above). Bonus: Here's a free preview to whet your appetite.

Also: a number of his papers can be found via a search on Phil Papers.

UPDATE: John D over at Philosophical Disquisitions has since explicated a number of Gwiazda's criticisms of Swinburne's work. Here is the link.

Maitzen's Argument Against the Existence of a Sensus Divinitatis

I'm currently re-reading Stephen Maitzen's excellent paper, "Divine Hiddenness and the Demographics of Theism" (Religious Studies 42 (2006), pp. 177-191). As the title suggests, Maitzen appeals to the data of the demographics of theism to defend the argument from divine hiddenness. However, in the same paper, he also offers a powerful argument against the existence of a sensus divinitatis (i.e., a natural capacity to form properly basic belief in God in a wide variety of circumstances or "triggering-conditions", such as (e.g.) looking at the starry heavens, reading the Bible, etc.) postulated by reformed epistemologists of the likes of Alvin Plantinga and others. Below is a summary of that portion of Maitzen's article.[1]

The argument summarized: The sensus divinitatis is supposed to be an innate capacity or faculty in all humans. Now innate human faculties tend to be evenly distributed among humans (e.g., hearing, sight, etc.). But the kind of belief …