The Infinite. Part 10. The Axiom of Regularity, Theological Tag-alongs, Mormonism.

[Here is part 9. Here’s a page with all parts.]

I’m being very brief here and I don’t want anyone to get the impression that I take lightly the work of sincere people who have thought long and hard over important questions. I leave that to Brad.

Christians (along with Muslims and Jews) have a historically mediated anxiety to *prove* that their God exists. Various attempts abound through history, and some still have followings/advocates. Two that play into Mormonism are the Ontological Argument (OA) and the Cosmological Argument (CA). The OA is often associated with St. Anselm and consists in arguing that the “being than which nothing greater can be conceived” must indeed exist (necessarily).[1]

This lodestar is certainly beautiful. However, there are some problems with the argument, not least of which is, if you manage to believe that the argument works, it’s not exactly clear what you have proved. What is clear though is that it should be seen as an augmentation of faith rather than taking its place. But the OA is a side-light for us. It is the CA that focuses on something that impacts Mormonism and questions about the infinite we have already considered. Great pedagogy huh?[2]

Before considering the CA, remember Russell’s paradox? It was back in part 8. Russell’s idea was to consider the notion of a collection that contains itself. It was a strange concept in the sense that collecting all the sets that don’t contain themselves causes a problem. There are a load of paradoxes that appeal to the same scenario, self-reference.

There are a couple of approaches to ruling out such discomfort (i.e., a lack of confidence in the power of reason). One is to simply not allow sets that contain themselves in the language. This is a common approach in the formal theory of sets and it is called the Axiom of Regularity. The Axiom of Regularity is closely connected to the CA. (the Axiom of Regularity excludes sets A of the sort, A ∈ A. Such sets imply an infinite regression and succession . . . A ∈ A ∈ A ∈ A ∈ A ∈ A ∈ . . . )


The CA can be traced to among other things, St. Thomas’s “five ways.” The five ways (and especially the first three) center around the discomfort we seem to share with “infinite regress.” Aquinas:

1. All things are in motion: why? How did it start? Motion requires a cause and each cause requires a prior cause. Unless there is an infinite regress of causes, there must be a first cause and this is God.

2. Effects in general, not just motions have the same trace back and the same result.

3. The world is full of contingent beings, that is, they are not here by logical necessity. In contrast, Aquinas argues, there is a necessary being[1]. The fact that contingent beings exist requires another being (like child and parent say). Again we are involved in a regression.

4. The fourth way considers human values like love or truth or goodness. Where do these come from? What causes them? There must be something which is in itself, good, true, beautiful, etc. That is God.

5. The fifth way retreats to order in the universe. Order implies design. What or who did the original designing? This is God.

The CA fails on a number of levels (maybe there was a first cause, but does it need to exist now?). But we are interested in the regression idea. In other words, whatever begins to exist, requires a cause. We (or the universe) began and therefore the world has a cause (Mormons might dither on the language here, but it can be adjusted appropriately).

Why can’t there be an infinite sequence of causes? Such would require an actually infinite causal regression sequence.[3] Some people don’t like this and so it can’t happen (really that’s about it, but see [6]). Could there be an actually infinite prior sequence of temporal events for example? An ordered sequence of events always has either a successor (in reverse) or a limit ordinal “cap” if you will. That is not really needed here though. Assuming the past is Archimedean[4] in some metaphysical sense, then infinite regression of causes is perfectly acceptable. If the past is not Archimedean, then the past would not be just infinite in the usual linear sense, but perhaps some kind of “long” infinite past.[5]

There are versions of the CA that don’t require a rejection of infinite regression. Here’s one:

1. Every being is either contingent or self-existent (uncreated).

2. Not every being can be a dependent being.

3. Therefore there exists a self-existent being.

I won’t discuss this argument but if you think about it a bit, you’ll see that it is not really coercive. Besides, you’re already thinking that Mormonism may negate such an argument in several ways. But one can ask again, why is it necessary that an explanation for a causal sequence is even needed?[6]

First, one strand of Mormon thought beginning with Joseph Smith himself asserts that each human is really founded in its own unique self-existent consciousness.

Joseph: You have always existed. You are self-existent. Any questions?

Another strand of Mormonism asserts that while individuals are not necessary beings, they are founded in an infinite regression of “parents.” This latter idea was the position of a number of early Utah thinkers and a main behind-the-scenes 20th century advocate was James Talmage (he called it the parent-derived theory and it clearly relates to the idea that God had a “father”). In the latter case we have an infinite regression of contingent beings (the sequence includes God as the first step back from us). This is seen as unsatisfactory by many as not answering the question of why there exist contingent beings. Mormons who espouse such a view probably see this as a shoulder-shrug thing and might answer that the sequence itself has necessary “being” — I won’t engage that now (again, see [6]).

Next time I’ll look at some interesting consequences of “Mormon” infinity and the infinite universe.

[Part 11 is here.]
[1] By necessarily, I mean “logical necessity.” In other words, denying it would entail a contradiction. Reductio Ad Absurdum again.

[2] I’m not dissing the OA here, many of the issues about the OA link with Mormon ideas. Just avoiding that a bit for now.

[3] It has been argued that if a causal sequence involves the nonzero time between causes then an infinite causal sequence cannot exist. This is certainly false.

[4] Archimedean means that whatever moment in the past one considers, there is a fundamental time unit further back. (For the illuminati, this just means time is being modeled by the real line.)

[5] I’m ignoring questions of what “past” means. Remember ordinals? There is a first uncountable ordinal (remember, they are well-ordered), we call it ω1. Now take a unit interval of time, say an hour if you want. Stack up ω1 such intervals (we can easily give a formal construction here). This creates an infinite past which is a “long” past. You can go infinitely far back, say ω hours and still have much more to go. There are lots and lots of limit ordinals back there. There are ω1 past limit ordinal hours. That’s a very long time ago.

[6] St. Thomas would offer that some events require a sustained cause. These are called “hierarchical” regressions. Aquinas would put our existence squarely in this category and implicitly assumes as a kind of axiom, the Principle of Sufficient Reason (PSR). The PSR is really a way of saying that there has to be a first cause in a hierarchical sequence of causes. In other words, there is a bit of circularity. To be fair, Mormonism has had some appeal to hierarchical sequence. Passages like, “in whom we live and move and have our being” (Acts 17:28) or Benjamin’s Mosiah 2:21 suggest this, and there is a strand in Mormon discursion to the same effect. God has to be around, otherwise the physical world would cease (perhaps we should say, disorganize). Trust me, if it doesn’t ring a bell, there is a fair amount of this. Even Joseph Smith offers that God holds the world in its orbit, and everything else depends on God for its present stability. But it’s clear that historically at least, many Latter-day Saints don’t subscribe to the PSR. For example, just saying that spirits are uncreate, doesn’t seem to pass the PSR smell test. On the other hand the PSR apparently lived and breathed in the theology of Joseph F. Smith. One more thing that some CA proponents point to is the “big bang.” Can you guess why?


  1. I am enjoying these WVS.

  2. Love this stuff.

  3. I accept your endorsements gladly.

  4. P9 ∈ G Were G is the set of great posts. And P9 this post.

  5. I’ve actually long liked the ontological argument. When I first understood it I had a terrible time refuting it. It’s a very, very subtle argument. I don’t think the argument ends up telling us much about God. And ultimately the argument does fail for a variety of reasons. But it’s an argument I think people need to read charitably and really grapple with. Most people when they think of the argument transform the argument into something much weaker and sillier.

  6. Clark, it is full of nuance. That makes the OA interesting, and vulnerable.

  7. There is an interesting alternative to the sequence of causes, namely the quasi-continuum “something from nothing, but slowly”. Presumably, to ancient Greeks, Gods were too complex to just spring into existence, to always have existed, or to have been created. Instead, there was a steady progression from the static Protogenos (air, earth, fire, water), which begot the kinematic Titans (time and history), which begot the dynamic Gods (war, beauty, wisdom). Through slow increase of complexity, there was no one implausibly miraculous step function.

    Except they were wrong. The mathematical thorn in this side is the need for infinite support of a non-trivial analytic function. You can’t just start from nothing and jump start the process. In other words, Greek mythology is no more plausible for having progressed. They might have saved us the cosmic evolution and gone with the deus ex nihilo approach.

    How does Mormon theology explain the Fall? Why then, and not 20 years earlier, or 5 million years later? If there is an infinite progression (or even stasis), what is the motivator for a singular change of status? Beings that have eternity pre- and post-mortally to act would have an awfully hard time deciding what null set of that infinite existence to mortally inhabit.

  8. “How does Mormon theology explain the Fall? Why then, and not 20 years earlier, or 5 million years later? If there is an infinite progression (or even stasis), what is the motivator for a singular change of status? Beings that have eternity pre- and post-mortally to act would have an awfully hard time deciding what null set of that infinite existence to mortally inhabit.”

    I don’t think we explain any of this well. Timing. It has to be set by an expert. Next installment I’m going to run through some of this. I think there is a tendency to apply PSR here, but maybe it’s stochastic. (frown)

  9. How about: “Created by a random act of quantum mechanics?” This is a riff on an infinite number of monkeys with an infinite amount of time… Anything can be created by this method, even God, even the universe.

  10. RW, your hypothesis why this happened at least once does not explain why it has happened only once. The latter is the harder challenge.

  11. You are correct that the axiom of regularity rules out sets that contain themselves but it is not the case that the axiom of regularity prevents Russell’s paradox as you seem to imply.

  12. em, I only mentioned Russell because of the concept of self reference. Axiomatically, Russell is avoided by something else.

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