Mormonism and the Actual Infinite: How an Actually Infinite Collection of Gods is Metaphysically Problematic
Southeastern Baptist Theological Seminary
April 18, 2014
LDS Theology and an Actually Infinite Set of Gods:
1. Question: Is the LDS conception of gods such that the set of them is actually infinite?
2. Problem: How can an actually infinite collection, formed successively, exist in reality?
A. Craig’s Library Argument. A set is actually infinite if it “has a proper subset equivalent to itself.” Suppose there is a library with an actually infinite collection of books, and two colors represented: red and black. There are an equal number of red and black books. But this means the number of the red books is equal to the number of the red books plus the number of the black books, and this is absurd!
B. No LDS Collection of Gods can be actually infinite. 1. The LDS series of gods is a collection formed by successive addition. 2. A collection formed by successive addition cannot be an actual infinite. 3. Therefore, the LDS series of gods cannot be an actual infinite. This collection also relies on a temporal regress. See al-Ghazali’s orbit argument, as well as the Tristam Shandy paradox. Before any LDS god arrives on the scene, the one prior to him would have.
The Oppy Objections:
First, Graham Oppy claims that small actual infinites, like divisions in spacetime points, can exist in reality, while granting that in most cases it does not apply. Second, Craig’s Library and other such examples only show, at most, some types of actual infinites cannot be instantiated, not all of them. Third, the bouncing ball objection, which completes a “supertask” and so completes an actual infinite by successive addition.
Responses to the Oppy Objections:
1. Confuses Divisions in Spacetime with Spacetime Itself. Oppy takes the points in spacetime to be real, but Zeno’s Paradoxes suggest spacetime exists independently of any divisions made in it.
2. Does Not Account for the Nature of the Actual Infinite. Actual infinites are sets which whole has a subset that is equivalent to itself. The insight of Craig’s Library, Hilbert’s Hotel, et al., is that these absurdities can result from just any actual infinite collection.
3. Supertasks Assume an “Infinitieth” Task. The very nature of forming a collection by successive addition will always result in a finite number. If this were not so, then supertasks would have to complete an “infinitieth” task, whereby the final task in the series counts as an actual, rather than potential, infinite. For any task with a beginning and final point, this seems just to be a regular finite set. For points with a beginning but no end, this is just a potentially infinite set. For this point, there is no “infinitieth” god in LDS theology.
The LDS Objections:
1. A Multiverse Model. This model postulates that an actually infinite collection of gods is instantiated over the range of multiple universes, one god per universe, as argued by Kirk Hagen.
2. An Infinite Past is Not Formed by Successive Addition. LDS philosopher Blake Ostler objects that an infinite past cannot be formed, for at any moment in a postulated infinite past, by definition, the past will already be an actual infinite. Thus, no moment can ever be added to this infinite past.
Responses to LDS Objections:
1. The Problem Remains, Even Accounting for Other “Times.” This multiverse scenario still results in other LDS gods promoting other gods in a succession, so that whatever timeframe of reference they are in, the same arguments against forming this actually infinite collection apply.
2. The “Infinite Past” Objection Highlights the Problem. Some might think Ostler has simply highlighted the problem by showing that any collection formed by successive addition cannot be an actual infinite.
3. The Infinite Past still Requires Each Event to Come into Being. These events do not simply exist as a block, but rather as a growing collection. For any past event, no matter how far back, this event was not true of the world, became true of the world in the present, and then belonged to the past event category.
If actual infinites are not possibly instantiated in reality, or if they can be but cannot be formed by successive addition, then certain important facets of LDS theology are falsified. It appears as though there are good reasons for thinking the LDS set of gods, extending back into the infinite past, is either not actually infinite, or does not exist at all.
Mormonism and the Actual Infinite: How an Actually Infinite Collection of Gods is Metaphysically Problematic
Southeastern Baptist Theological Seminary
April 18, 2014
Christian formulations of God and what he is like often vary, but what tends not to vary among traditional Christian theists is that there is only one God (cf. Isaiah 44:6). Yet popular theology of the Church of Jesus Christ of Latter-Day Saints (LDS), otherwise known as Mormons, suggests a multiplicity of gods. Not only many, but an infinite number of them. It is referred to as the Doctrine of Eternal Progression (DEP). Contemporary LDS leaders now characterize DEP as more of a process of self-improvement, but it has far reaching implications.
LDS apostle Bruce R. McConkie explains that God gained his “attributes of godliness . . . in their fullness” at some time in the past. In fact, God used to be a man, though he is really an “exalted, glorified, and perfected Man.” In the same way that Elohim (name of the LDS god) was once a man and became god, so too can man become a god. In Doctrine and the Covenants, an inspired Mormon text, Joseph Smith wrote of obedient Mormons, “Then shall they be gods . . . therefore shall they be from everlasting to everlasting.” Richard Abanes has noted that this has been understood in LDS theology as meaning that there is a god for every moment in time, and that time itself is everlasting infinitely into the past. From this, it follows only naturally that there are an actually infinite number of gods, extending infinitely into the past and potentially infinitely into the future. The question is this: how can an actually infinite collection, formed successively by addition, exist in reality? This paper shall consider arguments against an actual infinite instantiated in reality, then move to consider objections to these arguments, from LDS thinkers and otherwise. Finally, responses shall be provided. If actual infinites are not possibly instantiated in reality, or if they can be but cannot be formed by successive addition, then certain important facets of “revealed” LDS theology are falsified (namely, that there are gods extending infinitely into the past, as well as an actually infinite set of gods).
Craig’s Library Argument
The first argument to be considered is referred to as Craig’s Library Argument (CLA). CLA is a parallel argument to a famous argument known as Hilbert’s Hotel. William Lane Craig developed CLA in his book, The Kalam Cosmological Argument. Craig asks the reader to imagine a library containing an actually infinite amount of books. In order to understand CLA, one must know how to distinguish between an actual infinite and a potential infinite. A potential infinite is a collection or series of things/events that could keep on being extended toward infinity as a conceptual limit, even though it never realizes this goal. Craig writes, “When Aristotle speaks of the potential infinite, what he refers to is a magnitude that has the potency of being indefinitely divided or extended. Technically speaking, then, the potential infinite at any particular point is always finite.” Georg Cantor was able to elucidate a set theory concerning the actual infinite, where the actual infinite is the unified whole of whatever set is being considered. A curious fact about actual infinite set theory is that it is a set that has a proper subset equivalent to itself. This means that no set, taken as a whole, is numerically greater than an individual part of it (otherwise one would be dealing with a potentially infinite set—which is really just an indefinite set within set theory, not infinite in any way).
Going back to CLA, recall the library with the actually infinite amount of books. There are two colors of books: red and black. Further, there is an equal number of red books to black books.
We must now remember the axiom of the actual infinite: any actually infinite set contains a proper subset that is equal to the set itself. So consider the number of the red books is equal to the number of the red books plus the number of the black books. Given all of the information in CLA, this sounds absurd. The thought experiment can be drawn out further. Assume there is a third color, blue, and a fourth, yellow, (and even to an infinite number of colors) and the absurdities increase. This is because the number of the red books is now equivalent to the number of the red books, the black books, the blue books, and the yellow books! Indeed, Craig remarks, “Would we believe anyone who told us that for each of the infinite colors there is an infinite collection of books and that all these infinities taken together do not increase the total number of books by a single volume over the number contained in the collection of books of one color?” For Craig, these types of thought experiments are supposed to show that one cannot have an actual infinite instantiated in reality, only as a conception or mathematical framework.
There is also the issue of Hilbert’s Hotel. This famous example comes from mathematician David Hilbert. There are several ways to discuss the thought experiment, but one way will be considered. Suppose that there is a hotel with an infinite amount of rooms (this way they can host an infinite number of guests). The infinite number of guests arrives and checks in, and, in theory, there are no more rooms available. A new guest shows up and asks for a room; he is told there are no available rooms. However, the night manager decides he can work something out. He politely requests that each guest shift over one room (the guest in Room 1 to Room 2, the guest in Room 2 to Room 3, and so on, infinitely), and so one room (namely, Room 1) becomes available for the new guest. Craig stretches the Hotel out even further, explaining that if an infinite number of guests were to check in at once, the hotel manager could again accommodate them. This is because he can take the guests in all of the rooms and move them to the doubled number (1 to 2, 2 to 4, 3 to 6, and so on). All of the odd-numbered rooms are now available for the infinite number of guests to check in!
Moreland and Craig then proceed to show that mathematical operations of the actual infinite, performed in reality, result in absurdities as well. If the guest in Room 1 checks out, then even though, in reality, there is one guest who was there who is not now there, reducing the number by one, mathematically, there number of guests remains the same. “In fact,” they note, “we could have every other guest check out of the hotel and repeat this process infinitely many times, and yet there would never be any fewer people in the hotel . . . Can anyone believe that such a hotel could exist in reality?” It is worth pointing out that philosophers who press these arguments are not stating there is a conceptual problem with the actual infinite. Rather, they are stating there is a problem with an actual infinite being instantiated in reality.
Francis Beckwith relates the problem specifically to LDS theology. The number of gods, on a traditional conception of LDS theology, is supposed to be actually infinite. If an actually infinite number of gods exist, then removing ten gods from the set would leave an infinite number of gods remaining. Further, even removing some huge number—say, ten trillion—of gods still will result in there remaining an identical, infinite number of gods. For Beckwith, “Mormon metaphysics . . . is absurd.”
The style of arguments that matches CLA strongly suggests that an actually infinite collection cannot be instantiated in reality. If it cannot be instantiated in reality, then, when LDS theology teaches one can become a god and exist everlasting throughout time (and indeed, has existed everlastingly into the past), then LDS theology is in trouble. Next, this paper shall examine the various arguments against the collection of gods in LDS theology being actually infinite (which is a separate set or family of arguments than what has been previously considered).
No LDS Collection of Gods can be Actually Infinite
Despite the idea that actual infinites cannot exist in reality, this next set of arguments does not necessarily assume this is the case. That is to say, these arguments will work against LDS conceptions of an actually infinite collection of gods independently of the success of the preceding arguments. The argument against the LDS’ actually infinite collection of gods can be formulated as follows: 1. The LDS series of gods is a collection formed by successive addition. 2. A collection formed by successive addition cannot be an actual infinite. 3. Therefore, the LDS series of gods cannot be an actual infinite. There are actually several sub-arguments to support the two premises.
Premise (1) should be defended initially. It should seem obvious that the actually infinite collection of LDS gods is formed successively (it does not matter, for the argument’s purposes, if one god followed another, or more than one at a time. It only matters that the gods were successively added, and not fully instantiated all at once without any time where it lacked one of its members) by addition. In discussing the reasoning behind this assertion, Beckwith points out that the events behind one’s bachelor party, wedding, and funeral do not happen all at once. Similarly, the event of one god promoting the next finite human to godhood is not simultaneous with every other god attaining unto godhood (this is explicated in the D&C, for example, and the Law of Eternal Progression). If that is true, then it seems that the collection of gods is added to, and thus (1) is true by definition.
Premise (2) states that a collection formed by successive addition cannot be an actual infinite. This will be the crucial claim. Craig explains why it is impossible to form an actual infinite by successive addition: “The reason is that for every element one adds, one can always add one more. Therefore, one can never arrive at infinity [in an actual sense].” However, it is rightly noted by most that this assumes an initial beginning point. This is correct. Actual infinites cannot be formed by beginning at some point and then continuing to add, never-endingly, members to that set.
The real challenge is to see if an actual infinite can be traversed by being a beginning-less set that ends at some point (it must be this way, for any infinite set that does not have an ending point is potentially infinite only, by definitive distinction between the two types of infinites). This is why Craig reasons about this example of actual infinite, “Rather at every point the series already is actually infinite, although allegedly successively formed.”
However, several significant problems attend this idea. First, if the idea of counting to infinity seems confused, then it is similarly so for counting down from infinity. The idea would be that we could encounter a man who has finally finished counting down all the negative numbers, finally finishing with -2, -1, and 0. Assuming a beginning-less sequence to counting all the natural negative numbers, and assuming the man actually had an infinite amount of time, why should one expect that the man will actually complete the series? How is it that he can actually traverse the infinite set and complete the counting? Any negative number one chooses in the set will have another negative number that precedes it, so that even if any particular small subset could be counted, and thus traversed, it remains that the actually infinite whole cannot be.
The next argument that supports (2) is from the Muslim philosopher al-Ghazali. He was one of the original formulators of the kalam cosmological argument. Within his argument, he discussed several paradoxes concerning the idea that the actual infinite can really be such when formed by successive addition. His famous example involves the planets Jupiter and Saturn. Suppose for every orbit Jupiter completes around the sun, Saturn completes 2.5 (this is not supposed to be a scientific reflection of what actually happens). If the two planets orbit each other for an actually infinite amount of time, then something interesting happens.
On the one hand, one can see that as the orbits increase, the distance between the two planets’ number of orbits increases. After Jupiter has completed ten orbits, for example, Saturn will have completed 25. After 100 Jupiter orbits, Saturn would have finished 250. The distance between the two continues to grow as the number of orbits is successively added throughout a supposedly actually infinite amount of time. On the other hand, over an actually infinite amount of time, the number of orbits of Jupiter and Saturn are actually equal. How can this be? This is perfectly legitimate within the conceptual constraints of transfinite arithmetic, but in actual reality it is absurd.
The third argument is the famous Tristam Shandy paradox. There is a man, Tristam Shandy, who has been writing down days in his life from eternity past. He writes at the rate of one day per year. If one sees Shandy completing writing of a day now, the most recent day he could possibly be writing about is one year ago. Now imagine that Shandy wants to be a good autobiographer, and write about all of the days of his life in consecutive manner. But suppose he wanted to write about every day of his life? He could not very well have written about a consecutive series of every day of his life without going back another year (to write about two consecutive days two years prior). But his life has consisted of far more than two days, so he has to go back and write three years about the three previous days prior to the time he started writing. In other words, “the longer he has written the further behind he has fallen.” On the thought experiment, Shandy has been writing from eternity past! This means that he is infinitely far behind in writing his autobiography. This shows that traversing an infinite amount of time from a beginningless past to some ending point is not possible. Thus, the idea of traversing an actually infinite amount of time, as must be done to get the LDS theology this paper is examining, is equally problematic.
The next argument for (2) is relatively simple. The idea is that in order for the present moment to have arrived, the one moment before this present moment needs to have arrived. So, let t1 represent the present moment, t0 the moment just prior to t1, and t-1 for two moments prior to t1, and so on. The argument is simply claiming that in order for t1 to have arrived, t0 must have arrived, and in order for t0 to have arrived, t-1 must have arrived, and so on, ad infinitum. In fact, Craig explains, “So one gets driven back and back into the infinite past, making it impossible for any event to occur.” Of course, however, events have occurred. From these facts it follows that the series of past moments or events is not beginningless.
There is definitely an easy application to LDS theology. First, in order for any god—say, Elohim—to arrive on the scene of his godhood, the god who “promoted” him must be in place (that is, he must exist and be a god). Call him Zeus. And before Zeus, there must have been another god, Thor, and so on and so forth, ad infinitum. But then it follows that no god could ever have arrived, for any god in the actually infinite set one chooses will have an actually infinite number of gods who needed to arrive before him. On LDS theology, Elohim has arrived. Therefore, the set of gods is not actually infinite.
Beckwith makes a final relevant argument when he argues that, even assuming one can traverse an actual infinite, all of Eternal Progression should be used up by now. He argues, “But if the past series of events in time is infinite, we should have all reached our inevitable fate by now.” If the set of past events were actually infinite, and there were an actually infinite number of gods, then those who would achieve their Eternal Progression would all have done so. Since no one now living on Earth has achieved godhood (Eternal Progression), it follows that the actual infinite does not apply to the series of past events. If that is the case, then there are not an actually infinite number of gods. These arguments all support strongly the idea that a collection formed by successive addition cannot be an actual infinite. Next, objections and some responses shall be considered.
The Oppy Objections
Philosopher Graham Oppy, no friend to Mormonism, has offered several criticisms of arguments against the actual infinite being instantiated in reality. He offers three decent objections to this idea that shall be considered. First, Oppy claims that “small” actual infinites, like divisions in spacetime points, can exist in reality, even if “larger” actual infinites cannot—and defenders of no actual infinites cannot dismiss this idea. He claims, “If the Cantorian theory of the transfinite numbers is intelligible, then we can suppose that some parts of it find application ‘in the real world,’ while nonetheless granting that most of it does not.” In other words, the absurdities that may result in some of these large-scale cases can be noted and appreciated, even while insisting this still allows for successful instantiations of the actual infinite, so long as Cantor’s theory is even intelligible. This might strike one as a bit of an odd claim, and that will be investigated in a response section.
The second objection is that CLA and other examples (such as Hilbert’s Hotel) do not show nearly as much as Craig, et al., would have one think. In fact, at most, they suggest merely that some types of actual infinites cannot be instantiated, not that all of them cannot be. There is some intuitive force to this type of objection. After all, showing that one particular type of something cannot be done does nothing to the other types of that same thing. Oppy proclaims, “[They] show no such thing . . . the key point to note is that these puzzle cases simply have no bearing on, for example, the question of whether the world is spatially infinite, or the question of whether the world has an infinite past . . . one could hardly suppose that these puzzles show that there cannot be actual infinities of any kind.”
The final Oppy objection is that an actual infinite formed by successive addition can be successfully traversed. He gives the example of the bouncing ball careening off an infinite series of slabs. The idea is that the ball hits the first slab at one minute to twelve, the second at ½ minute to twelve, the third at ¼ minute to twelve, and so on, until the bouncing ball has completed the “supertask” of traversing an actually infinite collection formed by successive addition. He anticipates a potential objector stating that actual infinites cannot exist in reality, but goes on to point out that the argument against an actually infinite collection being formed by successive addition does not presume an actual infinite is impossible to be instantiated in reality. Thus, it seems Oppy’s objection remains.
Responses to the Oppy Objections
There are a number of responses to Oppy’s objections. First, in response to the objection that divisions in spacetime can be actually infinite, it can be said that he confuses divisions in spacetime with spacetime itself. The measurement (divisions) is not identical to the thing to be measured. Zeno’s Paradoxes suggest spacetime exists independently of any divisions made in it. These paradoxes of motion were founded on the idea that before Achilles could cross the stadium, he had to cross ½ of it, and before he could cross ½, he had to cross ¼ of it, and so on, so that he could never cross any distance (or else, the argument might go, he is always crossing an actual infinite). Craig and Sinclair point out the disanalogy between Zeno’s paradoxes and kalam reasoning: “Zeno’s opponents, like Aristotle, take the line as a whole to be conceptually prior to any divisions we might make in it.” However, Oppy’s objection (or at least, the example of his objection) only works if the divisions in spacetime are actual, as opposed to merely potential or conceptual.
Second, in response to the objection that CLA only shows that some types of actual infinites cannot be instantiated, it must be noted that Oppy does not account for the nature of the actual infinite. In short, he does not seem to understand why CLA shows the actual infinite cannot be instantiated. One must recall that actual infinites are sets whose wholes have subsets (or, a singular subset for each set) that are equivalent to themselves. The insight of CLA and others is that these absurdities can result from just any and every actually infinite collection/set.
Finally, in response to the bouncing ball objection, this paper concedes this is the best objection of Oppy’s bunch. First, it seems that his bouncing ball thought experiment is not as clear as it would seem. This is because, despite the ball bouncing at an ever-increasing rate, it seems it could only bounce off of a potentially infinite number of slabs. This conflation of the potential and actual infinite seems to be rampant throughout Oppy’s critique. Second, the very nature of forming a collection by successive addition will always result in a finite number. If this were not so, then supertasks would have to complete an “infinitieth” task, whereby the final task in the series counts as the completion of an actual infinite. This supertask, in order to make the jump from potential to actual infinite, needs the infinitieth task, so that when the clock strikes twelve, the ball has indeed bounced off an actually infinite number of slabs.
This can be illustrated by asking oneself if the number of bounces is odd or even. It does not make sense to provide an answer, since one more or fewer bounce could always be added or have been subtracted. This highlights the conceptual differences between a potential and actual infinite. Next, this paper shall consider LDS objections and responses.
The LDS Objections
Unfortunately, not much has been done in this area by LDS thinkers, but there are at least two major objections that can be lodged against the idea that an actually infinite collection of LDS gods cannot exist in reality. First, there is the multiverse model. Kirk Hagen, an LDS thinker, postulates that perhaps the multiverse provides a way to reconcile the idea of an actually infinite collection of gods and the idea that it seems only one God is around this universe. He writes, “For Mormonism, a compelling reason to consider a multiverse cosmology is to attempt a reconciliation of modern cosmological ideas and the central tenet of Mormon doctrine.” For Hagen, the idea is that these other universes avoid a beginning completely (a dubious proposition, but not one to be explored here), and even have different times in which creatures inhabit, consistent with D&C 130:4. Thus, with different time reference-frames and a multiverse, there can be an infinite collection of gods.
The second objection comes from LDS philosopher Blake Ostler. In an online review of a book done by Craig and Paul Copan, Ostler objects that an infinite past cannot be formed, for at any moment in a postulated infinite past, by definition, the past will already be an actual infinite. Thus, no moment can ever be added to this infinite past. He writes,
While there is never enough time to add up finite numbers to an infinite in a finite amount of time, the number of times is always already infinite if there is no beginning term – and thus the events constitute members of an actually infinite series but there is no time at which the temporal series of events became or was formed as an infinite collection.
Responses to the LDS Objections
In fairness, it should be noted that Hagen is not responding directly to those who are arguing against an actual infinite. He is interested in reconciling what he sees as a fundamental Mormon doctrine (the existence of an infinite amount of gods) with scientific discoveries. However, his response’s conclusion finds relevance with this debate, and so it is considered. This multiverse scenario still results in other LDS gods promoting other gods in a succession, so that whatever timeframe of reference they are in, the same arguments against forming this actually infinite collection apply. One can still think of all of the relevant thought experiments (including, crucially, that everyone who was to experience Eternal Progression would have done so by now).
As to Ostler, his infinite past objection highlights the problem more than anything else. Some might think Ostler has simply highlighted the problem by showing that any collection formed by successive addition cannot be an actual infinite. But that is simply affirming premise (2) of the argument! He may be trying to wield (2) in an attempt to deny (1); that LDS theology does not entail a collection by successive addition, after all. However, these events do not simply exist as a block, but rather as a growing collection. Ostler almost seems to be conflating the mental exercise of accounting backwards the set of all of an actually infinite series with the formation of any actually infinite series itself. One need not have a beginning point in order to form this actual infinite.
Craig notes the past does seem to be formed sequentially. Past events were not always in existence as one whole, but rather were not, then came into being, then moved into the past. It seems Ostler might be missing this point, and either presuming a B-theory of time without argument, or else not recognizing that if an actual infinite cannot be formed through the past, then all that follows is that the past is not actually infinite. It would then follow that the LDS theology that says the number of gods is infinite, or that the time of these gods is everlasting into the past (D&C 132:19-20), is false, for it requires a beginning. Beginnings of this sort would require the God of Perfect Being Theology, or the true God of Christianity.
In reviewing the material for this, Craig Blomberg notes that whatever the Mormon conception of God is, he is something fundamentally different. This is because, on evangelical, orthodox Christianity, God is the Supreme Being—not just of this universe, but of any possible universe there could have been. Blomberg also argues that, despite the fact that Mormons distance themselves, in some cases, from Lorenzo Snow’s pronouncement that “as man is, God once was; as God is, man may become,” as official doctrine, it nonetheless enjoys a kind of fundamental status (see Hagen’s quote about it being the central tenet of LDS theology).
This paper sought to challenge the LDS conception of an actually infinite collection of gods being possible. Several arguments were offered suggesting that this LDS conception was not possibly instantiated in reality. Some objections were considered, and responses given. The conclusion to be drawn is that if actual infinites are not possibly instantiated in reality, or if they can be but cannot be formed by successive addition, then certain important facets of LDS theology are falsified. It appears as though there are good reasons for thinking the LDS set of gods, extending back into the infinite past, is not possible after all.
Abanes, Richard. One Nation Under Gods. New York: Four Walls Eight Windows, 2002.
Beckwith, Francis J. and Stephen E. Parish. The Mormon Concept of God: A Philosophical Analysis. Lewiston, NY: Edwin Mellen, 1991.
Blomberg, Craig L. and Stephen E. Robinson. How Wide the Divide? Downers Grove, IL: IVP Academic, 1997.
Craig, William Lane and James D. Sinclair. “The Kalam Cosmological Argument,” in The Blackwell Companion to Natural Theology, William Lane Craig and J.P. Moreland, eds. Malden, MA: Blackwell Publishing, 2012, 101-201.
Craig, William Lane. The Kalam Cosmological Argument. Eugene, OR: Wipf and Stock, 2000.
____________. Reasonable Faith, 3rd ed. Wheaton, IL: Crossway, 2008.
Hagen, Kirk D. “Eternal Progression in a Multiverse,” in Dialogue: A Journal of Mormon Thought. Vol. 39, No. 2. (2006:), 1-45.
Hartman, Dayton. Joseph Smith’s Tritheism. Eugene, OR: Wipf and Stock, 2014.
McConkie, Bruce R. Mormon Doctrine, 2nd ed. Salt Lake City, UT: Bookcraft, 1979.
Moreland, J.P. and William Lane Craig. Philosophical Foundations for a Christian Worldview. Downers Grove, IL: IVP Academic, 2003.
Oppy, Graham. Arguing About Gods. New York: Cambridge, 2006.
Ostler, Blake T. “Do Kalam Infinity Arguments Apply to the Infinite Past?”, published at http://www.fairmormon.org/reviews_of_the_new-mormon-challenge/do-kalam-infinity-arguments-apply-to-the-infinite-past, accessed April 18, 2014.
Smith, Joseph. Doctrine and Covenants.
 Bruce R. McConkie, Mormon Doctrine, 2nd ed. (Salt Lake City, UT: Bookcraft, 1979), 239. One should understand that while McConkie’s writings here are not considered LDS Scripture, he is considered to be an apostle, and as such, his interpretation of LDS theology is very representative.
 Ibid., 751.
 Joseph Smith, Doctrine and Covenants 132:20.
 Richard Abanes, One Nation Under Gods (New York: Four Walls Eight Windows, 2002), 286-87.
 William Lane Craig, The Kalam Cosmological Argument (Eugene, OR: Wipf and Stock, 2000), 66.
 Ibid., 68-69.
 Ibid., 83.
 Dayton Hartman, Joseph Smith’s Tritheism (Eugene, OR: Wipf and Stock, 2014), 125.
 Craig, 84-85.
 J.P. Moreland and William Lane Craig, Philosophical Foundations for a Christian Worldview (Downers Grove, IL: IVP Academic, 2003), 472.
 Francis J. Beckwith and Stephen E. Parish, The Mormon Conception of God: A Philosophical Analysis (Lewiston: The Edwin Mellen Press, 1991), 66.
 Ibid., 54.
 Craig, The Kalam Cosmological Argument, 104.
 William Lane Craig and James D. Sinclair, “The Kalam Cosmological Argument,” in The Blackwell Companion to Natural Theology, William Lane Craig and J.P. Moreland, eds. (Malden, MA: Blackwell Publishing, 2012), 118.
 William Lane Craig, Reasonable Faith, 3rd ed. (Wheaton, IL: Crossway, 2008), 96-97.
 Moreland and Craig, Philosophical Foundations, 476.
 Craig, Reasonable Faith, 122.
 Hartman, 126.
 Beckwith, 60.
 Graham Oppy, Arguing About Gods (New York: Cambridge, 2006), 140.
 Ibid., 143-44.
 Craig and Sinclair, Blackwell’s Companion to Natural Theology, 119.
 Craig, The Kalam Cosmological Argument, 184.
 Kirk Hagen, “Eternal Progression and the Multiverse,” in Dialogue: A Journal of Mormon Thought. No. 39, 2 (2006:), 28.
 Ibid., 20.
 Blake Ostler, “Do Kalam Infinity Arguments Apply to the Infinite Past?”, < http://www.fairmormon.org/reviews_of_the_new-mormon-challenge/do-kalam-infinity-arguments-apply-to-the-infinite-past>, accessed April 18, 2014.
 Craig and Sinclair, The Blackwell Companion to Natural Theology, 124.
 Craig L. Blomberg and Stephen E. Robinson, How Wide the Divide? (Downers Grove, IL: IVP Academic, 1997), 105.
 Ibid., 106.