Arguments for a First Cause

Abstract

Cosmological models which postulate an infinite past, such as eternal inflation model, Ekyroptic universe, and Penrose’s conformal cyclic cosmology, face various difficulties related to the Generalized Second Law of Thermodynamics (Aron Wall), and arguments against an infinite regress of causes and events: (1) The argument from the impossibility of concrete actual infinities. (2) The argument from the impossibility of traversing an actual infinite. (3) The argument from the viciousness of dependence regress. (4) The argument from the Grim Reaper paradox. Any one of these arguments would be sufficient. Bounce cosmologies which postulate entropy reversal (Sean Carroll) neglect the problem of causal dependence at the interface. I show that Linford’s suggestion that the universes to either side of the interface might be interpreted as the simultaneous causes of each other entails a vicious circularity. Likewise, a closed causal loop (Gott and Li) is viciously circular. Given that an infinite causal regress and a closed loop is not the case, there is a First Cause.

5.1 Introduction

An important issue in philosophy of religion debates concerns whether there is a First Cause, and if so, what is the First Cause. I shall address the first question in this chapter and the second question in the next. I shall begin by considering some of the scientific cosmological models relevant to addressing the first question, before presenting philosophical arguments against an infinite regress of causes.

5.2 Scientific Issues

According to the so-called Standard Version of the Big Bang (also known as the Friedmann–Lemaître–Robertson–Walker [FLRW] model), matter-energy began to exist at the initial cosmological singularity of the Big Bang. As cosmologists Barrow and Tipler explain, at the initial cosmological singularity, ‘space and time came into existence; literally nothing existed before the singularity, so, if the universe originated at such a singularity, we would truly have a creation ex nihilo’ (Barrow and Tipler 1986, p. 442).

Over the years a number of scientists have proposed alternative models of the Big Bang, and these cosmological models may be classified under the following types:

• Type (1): Originates from a finite past ex nihilo: for example, Vilenkin’s (1982) ‘Creation from Nothing’ model

• Type (2): Originates from Closed Timelike Curves (CTCs) where the universe ‘creates itself’, for example, Gott and Li (1998)

• Type (3): Originates from a timeless initial state, for example, the Hartle–Hawking no-boundary proposal (1983)

• Type (4): Originates from an actual infinite regress

• Type (5): Involves a reversal of time

Type (1) has been discussed in Chaps. 2 and 3, while Type (3) will be discussed in Chap. 6. I shall discuss Types (2), (4), and (5) in this chapter, beginning with Type (4).

The following are examples of Type (4) cosmologies:

• Eternal Inflation model (Linde 1994; Aguirre 2007)

• Baum–Frampton (2007) phantom bounce cosmologies

• Veneziano and Gasperini’s (2003) ‘pre-Big Bang theory’ based on analogues of the dualities of string theory

• ‘Ekyroptic universe’ initiated by a collision between pre-existing ‘branes’ in a higher dimensional spacetime (Steinhardt and Turok 2005)

• Conformal cyclic cosmology (CCC) model (Penrose 2010; see below)

• Static Quantum Multiverse model (Nomura 2012)

There are other proposals, such as the Loop Quantum Gravity model (Bojowald et al. 2004) and Poplawski’s (2010) Black Hole model (which proposes that our universe might have originated from a black hole that lies within another universe), which are not committed to whether there is an actual infinite temporal regress (e.g. whether that universe was born from a black hole in another universe, which was born from a black hole in another universe, and so on) or a finite past.

To begin with our evaluation of Type (4) proposals, it should first be noted that none of these proposals are proven given that we do not currently have a well-established theory of quantum gravity, without which these proposals are, in the words of Ellis (2007, section 2.7), ‘strongly speculative, none being based solidly in well-founded and tested physics’. Ellis (2007, section 9.3.2) also argues that it is not possible for science to prove that the universe is past infinite; ‘observations cannot do so, and the physics required to guarantee this would happen … is untestable.’ It has been explained in Chap. 4 that, while (1) there are evidences for inflationary Big Bang cosmology and that multiverse is possible, this does not imply (2) an infinite inflation and infinite multiverse in the past, as illustrated by Ellis’ acceptance of (1) but rejection of (2).

The conformal cyclic cosmology (CCC) model (Penrose) proposes that the universe cycles from one aeon to the next, with each ‘aeon’ involves a big bang followed by an infinite future expansion that eventually results in the big bang of the next aeon. Penrose claims that anomalous regions have been found in the CMB temperature maps which results from the Hawking radiation from supermassive black holes in a cosmic aeon prior to our own (An et al. 2020). Other scientists are unconvinced by this purported evidence, objecting that there is no statistically significant evidence for the presence of such Hawking points in the CMB (Jow and Scott 2020). Moreover, even if there are such points, there could be alternative explanations other than cycles of aeons. In any case, such points do not prove that there are aeons which are infinite in the past or that there are an infinite cycles of prior aeons.

On the other hand, cosmological models which attempt to avoid a beginning face various difficulties related to the Second Law of Thermodynamics, the Borde–Guth–Vilenkin (BGV) theorem, acausal fine-tuning, and/or having an unstable or a metastable state with a finite lifetime (Craig and Sinclair 2009, pp. 179–182; Bussey 2013; Wall 2013a, 2013b), as well as philosophical arguments against possibility of an infinite regress (Ellis 2007, see section 5.3; it is a pity that a number of cosmologists have continued to ignore these objections; see, for example, Susskind 2012). For example, Vilenkin (2015) argues that the BGV theorem contradicts eternal past inflation. One might try to escape the BGV theorem by (1) postulating an earlier quantum era in which the theorem does not apply. However, Wall’s work on the Generalized Second Law of Thermodynamics (GSLT) indicates that the GSLT would still apply on the quantum era and implies a beginning. One might also try to escape the BGV theorem by (2) postulating a reversal to the arrow of time (Aguirre 2007). To elaborate, some forms of bounce cosmologies which postulate that the universe was born from an entropy-reducing phase in a previous universe and the entropy reverses at the boundary condition (Linford 2020) have been proposed to avoid some of these problems. While it has been objected that such models which attempt to avoid a beginning by postulating a reversal of the arrow of time nevertheless have a type of ‘thermodynamic beginning’ which still requires an explanation (Wall 2014), Carroll replies:

A thermodynamic beginning is not a beginning—it happens in the middle. It’s a moment in the history of the universe from which entropy is higher in one direction of time and the other direction of time. There is no room in such a conception for God to have brought the universe into existence at any one moment. (Craig and Carroll 2015)

Nevertheless, there is a deeper problem which Carroll neglected (this neglect may be related to Carroll’s dismissal of the Causal Principle, which I have responded to in Chap. 2), and that is the problem of causal dependence (Fig. 5.1):

Fig. 5.1

Bounce Cosmology with entropy reversal

According to these types of bounce cosmologies, event x₁ is the first event of universe 1 (U1), while event y₁ is the first event of universe 2 (U2). Events x₁, x₂, … and events y₁, y₂, … have a beginning (i.e. these events are finite temporally and have ‘edges’; see Chaps. 2 and 3). Given the Causal Principle that whatever begins to exist has a cause (see Chap. 3), if the beginning of x₂ is causally dependent on the beginning of x₁, what brought about x₁ at the boundary condition? Likewise, given that the beginning of y₂ is causally dependent on the beginning of y₁, what brought about y₁ at the boundary condition?

Linford suggests that the universes to either side of the interface might be interpreted as the simultaneous causes of each other. He writes:

Supposing (as Craig argues) that there are no instants … Instead, there exists an interval of time whose boundary is the interface between the two universes. Consequently … for every existent temporal interval, U1 and U2 co-exist. There is no need to introduce an independent cause for the interface, and we can interpret U1 and U2 as the simultaneous causes of each other. (p. 24)

However, the view that U1 and U2—or more precisely, x₁(the first event of U1) and y₁(the first event of U2)—are simultaneous causes of each other violate the irreflexivity of causation and amounts to a vicious circularity: on this view the beginning of existence of U1 is dependent on the beginning of existence of U2, which depends on the beginning of existence U1 in order to begin to exist (see the critique of causal loop and Gott and Li’s (1998) proposal below).

One might claim that causal dependency stops at the boundary condition (or moment of minimal entropy), and it is this condition that all later events depend upon in either time direction.¹ This implies that the boundary condition is the First Cause that brought about x₁ and y₁. However, the boundary condition is something that is temporally finite and has ‘edges’, and as argued in Chap. 2, such a thing would require a cause, and thus cannot be the first cause. Moreover, as I shall explain in Chap. 6, the independently existing First Cause must be something with (1) the capacity to initiate the first event, and also (2) the capacity to prevent itself from initiating the first event. These two capacities describe libertarian freedom. Therefore, the First Cause has libertarian freedom and hence is a Creator; it cannot be an impersonal boundary condition which Carrol describes. It would be of no use for Carroll to argue that such bounce cosmologies are mathematically possible, for it has already be explained in Chap. 1 that, even if a cosmological model is mathematically possible, it cannot be a correct model of the cosmos if it is metaphysically impossible.

Against Craig and Sinclair (2009, 2012), who have argued the low-entropy interface between universes should be understood as the beginning of two universes and this beginning requires an efficient cause beyond either of the two universes, Linford (2020) attempts to present a dilemma for them.

Linford argues that, on the one hand, if the direction of time is reducible, then efficient causation would most likely reducible as well given the Mentaculus (see Chap. 2), and thus even if the interface should be interpreted as an absolute beginning for two universes, these universes would probably not require a cause (p. 3). I have already argued in Chap. 2 why this horn of the dilemma is false.

In any case, let us consider the second horn of the dilemma. Linford argues that, on the other hand, if the direction of time is not reducible (as Craig in fact argued),² then ‘we are left without reason to think that the direction of time aligns with the entropy gradient … then the direction of time need not point away from the interface in two directions. So, Craig and Sinclair’s interpretation of the interface as an absolute beginning is unjustified’ (pp. 16–17), and ‘there’s no longer reason to suppose that the direction of efficient causation would align with the entropy gradient’ (pp. 16–17). In that case, events in a cosmological epoch of higher entropy could be the causes of events in an epoch of lower entropy. Linford writes:

the fact that geodesics can be extended through the interface provides some reason to think that the interface should be interpreted as a transition from one universe to another and not as an absolute beginning for two universes. Again, we can imagine God—or the fictional agent occupying the view-from-nowhere—watching as metaphysical time passes, the world unfurling through ages of entropy decrease, until the entropy begins to increase once more. (pp. 26–7; cf. Halper 2021, p. 161, who argues that on Loop Quantum Cosmology [LQC], one has ‘two classical space-times joined by a quantum bridge giving an hour-glass structure but with no reversal of the arrow of time’.)

However, if the direction of time does not point away from the interface in different directions but in the same direction, this would imply that the universe was born from an entropy-reducing phase in a previous universe and would violate the Generalized Second Law of Thermodynamics (see Wall 2013a, 2013b).

Linford (2020, pp. 18–19) claims that ‘we already know that the second law of thermodynamics is a statistical regularity that admits of exceptions’ without engaging with Wall’s work on the general second law. In the entry on ‘Singularities and Black Holes’ in The Stanford Encyclopedia of Philosophy, Curiel (2019) notes that the recent important work by Wall (2013a, 2013b) indicates that the Generalized Second Law ‘seems to admit of proof in ways much more mathematically rigorous than does the ordinary Second Law … Indeed, the Generalized Second Law is the only known physical law that unites the fields of general relativity, quantum mechanics, and thermodynamics. As such, it seems currently to be the most promising window we have into the most fundamental structures of the physical world.’

Halper (2021, p. 161) objects by arguing that Wall’s results may not hold in full quantum gravity, concluding that ‘this really underlies the view from cosmology today. Without a well-verified theory of quantum gravity, we cannot meaningfully describe the origin of our expanding universe and so we are in no position to say that cosmology implies a beginning to the universe.’ Halper fails to note that the same problem (i.e. the lack of a well-verified theory of quantum gravity) also besets the cosmological models (which he cited) which affirms a beginningless physical reality. Wall’s work is a good counterargument to those cosmologists who (despite the lack of a well-verified theory of quantum gravity) have argued that a beginningless model of physical reality is physically possible. Moreover, Halper also fails to note Wall’s (2013a) statement that, since the (fine-grained) Generalized Second Law of Horizon Thermodynamics is ‘widely believed to hold as a consequence of the statistical mechanical properties of quantum gravitational degrees of freedom, it is a good candidate for a physical law likely to hold even in a full theory of quantum gravity’ (p. 2; italics mine). Wall also shows that the Generalized Second Law implies a ‘quantum singularity theorem’, which indicates that, even when quantum effects are taken into account, spacetime will still be geodesically incomplete inside black holes and to the past in cosmological models (like the currently most well supported ones, which start with a Big Bang singularity) (Wall 2013a, 2013b).

Moreover, if (as Linford suggests) the direction of time point in the same direction, then given that an infinite regress of events is impossible (as indicated by the philosophical arguments below), which implies that an infinite regress of transitions is impossible, there must still be a first event which requires a First Cause given the Causal Principle defended in Chap. 2. It should be noted that, while on such models time goes towards infinity in both entropy directions, this does not imply that an actual infinity can be obtained in both directions; rather, this can be understood as potential infinite³ which is always finite at any time.

It might be thought that the law of conservation of energy which states that the energy of a closed system must remain constant implies that ‘energy cannot be created’ and that the universe has no beginning and end. However, what ‘energy cannot be created’ means is just that the total energy of an isolated system is constant. The fact that this law does not imply that the universe has no beginning can be seen by considering the Zero Energy universe Theory. This theory postulates that, ‘in the case of a universe that is approximately uniform in space, one can show that the negative gravitational energy exactly cancels the positive energy represented by the matter. So the total energy of the universe is zero’ (Hawking 1988, p. 129).

When this is applied to the beginning of the universe, it indicates the possibility that the Big Bang has no material cause. Hawking states that ‘When the Big Bang produced a massive amount of positive energy, it simultaneously produced the same amount of negative energy. In this way, the positive and the negative add up to zero, always. It’s another law of nature’ (Hawking 2018, p. 32). Cosmologist Sean Carroll suggests that this theory can be used to explain why the Big Bang is consistent with the first law of thermodynamics, which states that the total energy of an isolated system is constant. For if the positive energy and negative energy of our universe balanced up to zero, then there is no violation of the first law of thermodynamics if our universe began to exist from zero energy.⁴ As explained in Chap. 2, while this response implies that our universe has no material cause, it does not imply that our universe has no efficient cause; indeed, it does not imply that there is no requirement for a Creator to make the total energy of zero to be the way they are such that galaxies and gravity are formed while the total energy remain constant at zero. Given the deductive (Modus Tollens) argument for the Causal Principle ‘whatever begins to exist has a cause’, where ‘cause’ is either an efficient cause or a material cause (see Chap. 3), one should still ask what is the efficient cause which made the positive and negative energy to be the way they are (see Chap. 6). One can argue that God created the universe without using pre-existing material (i.e. ex nihilo), and this would not be a violation of the first law of thermodynamics because the total amount of energy of our universe after creation is still zero.⁵ In any case, the determination of the equation of the law of conservation of energy assumes a closed physical system; hence, it does not rule out a supernatural cause creating it supernaturally, unless we beg the question by assuming that a closed physical system is all there is.⁶

Finally, the limitations of science with regard to the realism–anti-realism debate in physics and cosmology should be noted. Moreland and Craig (2003, p. 334) observes that, in the history of science, many theories (e.g. aether theory) have explained phenomena, generated fruitful research and accurate predictions, yet were later abandoned as false. Hawking recognizes this limitation when, in speaking of the reach of science, he affirms anti-realism by stating that science consists only of models, but when he speaks of scientific results, he argues with inappropriate confidence for a self-contained universe, thus contradicting himself (Giberson and Artigas 2007, p. 118).

I do not wish to overstate the limitations of science by claiming that all models should be interpreted in anti-realist terms. Rather, we should decide on a case-by-case basis and do so if there are good philosophical reasons to think so. The importance of philosophical considerations have already been demonstrated in Chap. 1, in particular, it has been demonstrated that metaphysical considerations are more fundamental than mathematical considerations. Thus, ‘if we have good philosophical reasons for believing that the spacetime universe had a beginning a finite time ago, then if a “successful” scientific model runs counter to this belief, it may be best to interpret the model in antirealist terms’ (Moreland and Craig 2003, p. 344). The five arguments against an infinite regress which I discuss below are based on similar metaphysical considerations which are derived from understanding the nature of the world.

5.3 Introducing the Philosophical Arguments Against an Infinite Regress of Causes and Events

There are at least five arguments which have been offered to rule out an actual infinite regress of causes and events:

• 1. The argument from the impossibility of concrete actual infinities

• 2. The argument from the impossibility of traversing an actual infinite

• 3. The argument from the viciousness of dependence regress

• 4. The argument from the Grim Reaper paradox (Pruss 2018; Koons 2014)

• 5. The argument from Methuselah’s diary paradox (Waters 2013)

Each of these arguments is independent of the other, and any one of these arguments would be sufficient to demonstrate that an actual infinite regress is not the case. Therefore, it is not enough for the objector of finite regress to rebut one of these arguments; rather, the objector would need to rebut all five of them (and perhaps others). It is important to emphasize this point, because it has often been wrongly assumed (e.g. by Zarepour 2020, p. 17) that if one rebuts (say) the argument against concrete actual infinities (aka the Hilbert Hotel Argument), then one has rebutted the KCA. (Zarepour fails to consider that, even if a concrete actual infinite [e.g. an actual infinite number of stars] is possible, this does not imply that an actual infinite can be traversed [e.g. finish counting an actual infinite number of stars one after another], nor does it imply that an infinite causal regress is not vicious, etc.) The argument against concrete actual infinities has been subjected to numerous objections in recent literature; I don’t think the objections are compelling and I have replied to many of them elsewhere and developed a new version of the argument (Loke 2012, 2014b, 2016b, 2016c, 2017b, 2021a). However, to reply to the rest of them here would take up too much space, and is in any case unnecessary for the main argument of this book, since (as I have explained earlier) the KCA does not depend on this argument. Hence, I shall reserve my reply for future publications.

Arguments 4 and 5 have been defended at length by others elsewhere (Pruss 2018; Koons 2014; Waters 2013). A modified and easy-to-follow version of argument 4 can be briefly stated as follows⁷:

A piece of paper is passed down from the past to the present, from person to person. If it is blank, someone would write his unique name on it; if it is not blank, it is simply passed on. Suppose the past is infinite. When the paper reaches you, what is written? Something must have been written by then, yet nothing could have been written because any name that might have been written would have been a different name, that is, the name of the person before. In other words,

• 1. If the past is infinite, then the paper being passed down from the past to the present would have a name written on it.

• 2. If the past is infinite, then the paper being passed down from the past to the present could not have any name written on it.

• 3. Anything that entails a contradiction cannot exist.

• 4. Therefore, the past cannot be infinite. (From 1, 2, and 3)

In addition, Pruss (2018) has argued that an actual infinite causal chain results in a number of other paradoxes, and the simplest and most elegant solution is to accept causal finitism, which implies an uncaused First Cause (see also the Gong Peal paradox defended in Luna and Erasmus 2020).

In what follows, I shall briefly discuss the argument from the impossibility of traversing an actual infinite and the argument from the viciousness of dependence regress.

5.4 Argument Against Traversing an Actual Infinite

Think about a series of events (whether microscopic or macroscopic). Suppose event₁ begins at time t₁, event₁ causes event₂ at t_2, event₂ causes event₃ at t_3, and so on. The number of events can increase with time, but there can never be an actual infinite number of events at any time, for no matter how many events there are at any time, the number of events is still finite: If there are 1000 events at t₁₀₀₀, 1000 events is still a finite number; if there are 100,000 events at t_100,000, 100,000 events is still a finite number, and so on. This illustrates that an actual infinite is greater than the number which can be traversed one after another at any time, because finite (one) + finite (another) = finite.

Since it is impossible to traverse an actual infinite number of events from event₁, it is likewise impossible to traverse an actual infinite number of earlier events to event₁, given that the number of events required to be traversed in both cases is the same. Thus, the number of events earlier than event₁(and likewise, the number of earlier causes and durations) cannot be an actual infinite. Therefore, there must be a first event. As cosmologists Ellis et al. observe:

a realized past infinity in time is not considered possible from this standpoint—because it involves an infinite set of completed events or moments. There is no way of constructing such a realized set, or actualising it. (Ellis et al. 2004, p. 927)

To illustrate, the set of earlier years is formed by a one-by-one process, for example, one year (e.g. 2018) followed by one year (2019). ‘One’ year is a finite number. Finite + finite = finite. Finite + finite cannot form an actual infinite. Hence, it is impossible to form or complete an actual infinite number of years. Thus, the number of earlier years cannot be actual infinite.

As an example of a series of events, one can think of a person marking a stroke each year. An argument against an actual infinite series of earlier events can be formulated as follows:

• 1. If the series of earlier (yearly) events is an actual infinite, then a person marking a stroke each year would have experienced⁸ the accumulation of an actual infinite series of strokes by a one-by-one process.

• 2. It is not possible for a person marking a stroke each year to experience the accumulation of an actual infinite series of strokes by a one-by-one process.

• 3. Therefore, it is not possible that the series of earlier (yearly) events is an actual infinite.

For premise 1, consider the series of years BCE. If the series is an actual infinite, then the series of strokes would be actual infinite, and if the series is finite, then the series of strokes would be finite. The phrase ‘experienced the accumulation of an actual infinite series of strokes by a one-by-one process’ means that the person would experience the existence of a series of strokes which is not given together all at once, but added one at a time (i.e. she would have added one stroke at 1 BCE, added one stroke at 2 BCE … etc.) and forming the series as they do so. The series of strokes is supposed to have been made up by each stroke; none of the strokes existed beginninglessly. Each of them was added at some finite point earlier in time, one by one. At each point in time only a finite number is added to the series, which is formed as a result.

Now, if we suppose that there is no beginning to the process, then there is no specific point in time at which the person would have the experience of the accumulation of an actual infinite number of strokes, for in that case at every year (not any specific year) there is supposed to be a totality of actual infinite. For example, the number would be actual infinite at 1 BCE (since she would have added one stroke at 2 BCE, one stroke at 3 BCE … etc.); likewise, it would be actual infinite at 2 BCE (since she would have added one stroke at 3 BCE, one stroke at 4 BCE … etc.). In other words, there would be no point in time at which there is a transition from having a finite number to having an actual infinite number of strokes. Rather, at each year BCE a stroke would have been added to a series of strokes that was already (supposedly) actually infinite. Nevertheless, we still need to ask how is that series of strokes constituted in the first place. Obviously, it is constituted by a one-by-one process, that is, one (finite) stroke being added (e.g. at 3 BCE) followed by one (finite) stroke being added (e.g. at 2 BCE). But can such a series be infinite?

Premise 2 is based on the fact that a one-by-one process cannot constitute an actual infinite series, because finite + finite = finite.

Objectors to the argument against traversing an actual infinite often claim that the argument begs the question by assuming a starting point, for if there is a beginningless series with an actual infinite number of earlier events, then an actually infinite sequence has already been traversed (Morriston 2013, pp. 26–27).

In reply, the argument against traversing an actual infinite is based on the nature of a one-by-one process, that is, finite + finite = finite, it is not based on starting at a point and therefore does not beg the question. To elaborate, without begging the question by assuming a starting point or by presupposing whether the number of events earlier than any time t_p is infinite or not, think of a series of events in the midst of being constituted by a one-by-one process (Fig. 5.2):

Fig. 5.2

A series of events being constituted 

There is one event P produced at time t_p. (Note that I stated ‘there is one event P produced at time t_p’; I did not state or assume that a total of one event has been produced by t_p, which is false if there are events earlier than t_p.) There is event P followed by event Q produced at t_p and t_q and together they constitute two events, there are events P, Q, and R produced at t_p, t_q, and t_r, respectively, and together they constitute three events, and so on. The series of events is constituted by each event. The series is constituted by a finite number (e.g. ‘one’) of event/s and a finite number (‘another’) of event/s, and together they constitute a finite number of events, not-possibly an actual infinite number of events.

The above conclusion is not based on presupposing ‘a particular time as a starting point’ or that the number of earlier events is not actual infinite; thus, it is not question begging. Rather, the conclusion is based on the nature of the one-by-one sequential process (Finite + Finite = Finite), and how any concrete series of events is constituted. For example, P could be (say) marking a stroke on a piece of paper at a certain year and Q could be adding a stroke the next year in the process of forming a series of strokes. Consider the following demonstration by mathematical induction which, contrary to its name, is a deductive proof which shows that the process would result in a natural number and hence finite number of strokes: A set S is an inductive set if for every element x of S, the successor of x is also in S. Let n be a natural number. Since the set of natural numbers is an inductive set, n + 1 is a natural number for all natural n. If n + m is a natural number, then n + (m + 1) = (n + m) + 1 is also natural number since the set of natural numbers is an inductive set. Therefore, n + m is a natural number (which would be a finite number) for all n and m by mathematical induction.⁹

A sceptic might object that the above argument only implies that the series of events that happened between time t_p and any time later than t_p is finite; it does not imply that the series of events prior to t_p is finite, and therefore does not exclude a beginningless series of events with an actual infinite number of earlier events.

This objection however fails to note that the above argument is intended to show how any series of events is constituted in the first place, and that ‘any series of events’ would include the series of events prior to t_p which, following the above argument, would be finite as well. In other words, the above argument is intended to illustrate how any series of events constituted by a finite number (‘one’) of events being added followed by another finite number (‘another’) of events being added would result in only a finite series of events. Now the series of earlier events is such a series of events. Thus, the number of earlier events must be finite.

Concerning the phrase ‘the accumulation of a series of strokes by a one-by-one process’, a sceptic might object by citing Morriston, who notes that, if the series of years is a beginningless series in which every year BCE was preceded by another, then ‘each year BCE would have been ‘added to’ a running total of years that was already infinite’ (Morriston 2021, p. 8n.5). In other words, while it is impossible to start with finitude and constitute an actual infinite, in the case of a beginningless series of events one does not start with finitude; rather, an actual infinite is already constituted at each moment. At each moment a finite number (one) is added to an actual infinite (and not to a finite number). Thus, if the number of earlier years is an actual infinite, then a person marking a stroke each year would experience a finite number adding to an actual infinite number of strokes at each year. Finite + infinite + = infinite

In reply, it should be noted that the point I am making is different from Morriston’s. His point is that, if the series of earlier events is actually infinite, then there already is an infinite series, to which a finite element is added. In other words, Morriston is thinking in terms of adding a finite element to the series of events which already exists. This does not answer the more fundamental question of how the series is constituted by its individual elements in the first place (and one must be careful not to beg the question by assuming that a beginningless series can exist).

Whereas I am thinking of the more fundamental question of how any series of events is constituted by its elements. I am thinking in terms of what is being added (i.e. one finite element followed by one finite element) to constitute the series (that is the meaning of the phrase ‘the accumulation of a series of strokes by a one-by-one process’). Finite + finite = finite thus refers to the (finite) elements that constitute the series; it is more foundational than the series which they metaphysically ground. The series of events is constituted by one (finite) element being added followed by one (finite) element being added. The essential feature of this one-by-one sequential process is that the series of strokes is supposed to have been made up by each stroke; none of the strokes existed beginninglessly. Each of them was added at some finite point earlier in time, one by one. This point is illustrated by Fig. 5.2 and the mathematical induction proof, which is about how each element which are added into the series at each year constitute the whole, without begging the question either way. That is, without assuming whether the total number of elements in the series of earlier events is already infinite or not, the fact remains that the series is constituted by its elements, and each element is a finite quantity (i.e. ‘one’ event is a finite quantity), and together they constitute a finite quantity because finite + finite = finite; therefore, the series of earlier events cannot be actually infinite. This conclusion is arrived at without first assuming that the total number of elements in the series is not infinite, and hence the argument does not beg the question. In other words, the problem with postulating a beginningless series of events is that, even though it supposedly does not start with finitude but already has an infinitude, nevertheless the infinitude is still supposed to have been constituted by a one-by-one sequential process, which, as shown above, is impossible. It is true that, if the number of earlier years is an actual infinite, then a person marking a stroke each year would experience a finite number adding to an actual infinite number of strokes at each year. However, the problem is that the actual infinite number of strokes which supposedly already exist at each year is supposed to have been experienced by the person to have been constituted by a one-by-one process, which as shown above is impossible.

In summary, the objector claims that there is a distinction between constituting a series of later events from a particular event (a beginning) and constituting a series of earlier events to a particular event, and while it is impossible to constitute an actual infinite in the former case, it has not been shown to be impossible in the latter case. Contrary to this claim, I have argued that the number of elements required to be constituted by a one-by-one process is the same for both cases, and this implies the same impossibility for both cases because finite + finite = finite. Thus, the objection fails.

Against the explanation that ‘any finite quantity plus another finite quantity is always a finite quantity’, Malpass (2019) objects that the question is, ‘how long we have been doing it for. The lesson seems to be that if you only count for a finite amount of time, then you cannot construct an actual infinite by successive addition, but if you do it for an actually infinite amount of time, then you can.’ In other words, if George has been counting −1 at t − 1, −2 at t − 2 … he would have counted an actual infinite and there would be no longer any more number of a negative infinite series to count.

However, such an objection ignores how the series of counting is constituted in the first place. It is constituted by one (finite) element being added followed by one (finite) element being added, which (as explained above) cannot constitute an infinite number. The set (an abstraction) of earlier events is grounded in the existence of its concrete members (each event), which constitute the set. Since the series of earlier events is constituted by individual (i.e. a finite number of) event being added followed by individual(finite) event being added, the series must be finite. Moreover, the supposition that there is already an actual infinite time in the past entails the possibility that finite + finite = infinite, but as explained previously the consequent is impossible; hence, the antecedent is impossible.

A sceptic might object that, if the process of one (finite) element being added followed by one (finite) element being added has been happening from an actual infinite past (i.e. if one repeats this process an infinite number of times), then an actual infinite number of elements would have been added to the series.

My reply is that an actual infinite number of elements is not supposed to have been added all at once, but one after another. To repeat a process actual infinite number of times, one needs to first proceed one time after another, but the problem is that the result of that process is always finite at any time, because ‘one time’ (‘finite’) after ‘another’ (‘finite’) implies finite + finite which is equal finite. Thus, the result of that process is always finite, because the number of each stroke is a finite number, and finite+ finite cannot be infinite. Therefore, one cannot have been adding from an actual infinite past since this entails the impossible consequence that finite + finite can be infinite. This reply does not beg the question because there is an independent proof that natural (finite) number + natural (finite) number = natural (finite) number using mathematical induction (see above).

In their defence of the argument against traversing an actual infinite, Craig and Sinclair (2009, p. 124) state that this argument would not work on a static theory of time because a series that was formed sequentially, one event occurring after another such that the collection grows with time, presupposes a dynamic theory of time. Whereas on the static theory of time, a series of events is not formed by addition of later events (which come-to-be) to earlier events; rather, the collection of events is a collection whose members all tenselessly coexist and are equally real. It would be like an infinitely long ruler with an infinite number of markings, with different features at each marking.

Craig himself has responded to this objection by providing a number of arguments against the static theory of time and defending the dynamic theory of time (Craig 2000a, 2000b). In what follows, I shall argue that, even if the static theory of time is true, it remains the case that a series is experienced to be forming over time by successive addition, and if the number of earlier events is infinite, the formation of an infinite series by successive addition would be experienced by someone living as long as time exists, but, as argued above, this is not possible.

A sceptic would object that on the static theory of time our experience of a series forming over time by successive addition is illusory, just as our experiences of the flow of time and the ‘time arrow’ is illusory; on a static theory there is no ‘progression’ and any specification of temporal points is arbitrary.¹⁰

Nevertheless, even if the static theory of time is true, it remains the case that our experience of ‘a series forming over time by successive addition’ exists, even if this experience is illusory. For example, we can obviously mark a stroke on a piece of paper at time tp, add another stroke at tq and so on, hence forming a series of strokes. In other words, we can still experience a series forming over time by the addition of a finite number of element(s) (e.g. ‘one event’) followed by the addition of a finite number of element(s) (e.g. ‘another event’). This is different from the infinite ruler case which does not involve the experiencing of adding one element after another over time. Likewise, we have experiences of progress and of specifying each year, for example, marking each year with (say) a stroke and adding another stroke the following year (finite + finite). In other words, even if there is no progress (given the static theory), yet it would seem to the person that there is progressive addition of strokes in the sense explained above. These experiences exist even if they are illusory. Illusory just means that our experience of the world is not true of the world; nevertheless, it remains true of the world that we have the experience. For example, it seems to me that I was in Hawaii last night, but in fact I wasn’t at Hawaii; it was a dream. Nevertheless, it is true of the world that I had a dream in which it seems to me that I was at Hawaii. Now my seemings cannot involve a logical or mathematical impossibility. For example, it is impossible that I dream about a shapeless square or 2 + 2 = 5. Such impossibilities cannot exist anywhere, not even in our illusions or dreams. Hence, anything that entails the possibility of a person experience (whether as an illusion or not) such an impossibility must be false. Now if there is a beginningless series of events in which the number of earlier years is an actual infinite, this entails that a person experiencing the marking of each year with (say) a stroke and adding another stroke the following year (finite + finite) would experience the series constituted by what is added, and what is added is not added all at once, but one by one; that is, she would experience¹¹ finite + finite = infinite.¹² (This point remains valid regardless of whether the static theory of time is true.) The consequent is mathematically impossible. Hence, the antecedent is impossible.

Against the argument for the impossibility of an infinite regress, Russell (1969, p. 453) objects that there could be an actual infinite series of negative integers ending with minus one and having no first term. Likewise, Graham Oppy asks us to consider the series …, −n, …, −3, −2, −1. He writes: ‘In this series, each member is obtained from the preceding member by the addition of a unit’ (Oppy 2006a, p. 117).

In response, it should be noted that a negative or positive cardinal number series is a case of abstract actual infinite which exists timelessly rather than constituted by a one-after-another temporal process. Thus, it does not provide a counterexample to the claim that an actual infinite cannot be constituted by a one-after-another process in the concrete world. While each member of the abstract negative cardinal number series …, −n, …, −3, −2, −1 is ‘obtained’ from the preceding member by the addition of a unit, this ‘obtaining’ is in the form of timeless mathematical ordering relation. It is not the case that the abstract number –2 (say) is brought about in time by the addition of a unit to –3. Rather, the abstract numbers –2 and –3 have always existed timelessly, and this is unlike a causal series of concrete entities existing in time (note that static theory of time is not timeless, see Chap. 2). One can have an abstract actual infinite number of negative numbers each of which is timelessly separated from zero by a finite number of negative numbers. The existence of each of the number in the series is not causally dependent on any previous number, nor is it dependent on the actual infinite number which exists outside of the series. However, to constitute a series by a one-after-another causal process is a different matter. In contrast with a series of timeless numbers, a temporal series of causes and effects is constituted over time by adding one element after another, and each effect in time is causally dependent on a prior cause. The process proceeds one after another, constituting a finite number at any time. While the number of (say) cardinal numbers is an (abstract) actual infinite, the number of future events that has been constituted is clearly not an actual infinite because it is impossible to traverse an actual infinite as argued above, and I have argued that likewise the number of earlier events cannot be actual infinite as well.

In summary, a number series is a set the elements of which can be ordered one after another; it is not a series which is constituted over time by adding one element after another. Even if there could have been an actual infinite number of elements (e.g. abstract numbers) each of which is a finite quantity, this does not imply that there could have been an actual infinite series formed by successive addition of elements each of which is a finite quantity. On the contrary, a successive addition of elements each of which is a finite quantity cannot constitute an infinite series, because finite + finite = finite. That is why no infinite series can be formed by successive addition.

The point of the argument is that an actual infinite number of events cannot be constituted at any time. The objector suggests the hypothesis of a series of events that is infinitely long. However, in order to constitute an actual infinite number of events in the first place, the process has to proceed one event after another, and the problem is that the result of that process is always finite at any time. One does not constitute an actual infinite at any time, not at t₁₀₀₀, t₁₀₀₀₀₀, or t_1000000. As noted earlier, actual infinite stands outside of the series, timelessly and abstractly. As Copan and Craig (2017, p. 309) observe: ‘Necessarily, given any finite number n, n+1 equals a finite number. Hence, aleph₀ has no immediate predecessor; it is not the terminus of the natural number series but stands, as it were, outside it and is the number of all the numbers in the series.’ But here we are talking about what happens in a series of events in the concrete world, not timelessly and abstractly. The proponent of KCA is referring to what happens in the concrete world when he/she argues that an actual infinite number of events cannot have been constituted at any time t; therefore, the number of events earlier than t is finite and hence has a first member (i.e. a first event) which (given the Causal Principle) has a First Cause. If one wants to talk about the timelessly abstract which has no causal powers, that would be irrelevant as an objection to the KCA and does not block the conclusion of the argument that there is a First Cause.

Morriston (2013, pp. 26–27) claims: ‘From the fact that we cannot—beginning now—complete the task of enumerating all the events in a beginningless series, it does not follow that the present event cannot arrive or that a beginningless series of events that have already arrived is impossible. To suppose otherwise would be to confuse the items to be enumerated with the enumerating of them—it would be like arguing that there must be finitely many natural numbers because we can’t finish counting them.’

In reply, constituting a series to the present from a beginningless past would require the number of events constituted to be actual infinite, but an actual infinite is too large to be constituted by a one-after-another process. The problem is not due to our ability to enumerate; rather, it is due to the nature of an actual infinite which is too large to be constituted by a one-after-another process. While there can be an infinite number of natural numbers in the abstract, to constitute an actual infinite in the concrete is a separate issue and the real issue here.

One might object that there could be an actual infinite number of points between (say) time t₀ and t₁(just as there is an actual infinite of real numbers between 0 and 1) which is traversed in a manner similar to Zeno’s supposed paradox of motion. In a similar vein, one might object that there could be an actual infinite number of events between t₀ and t₁. Where our universe is concerned, there might not be a first point of time at t₀(Pitts 2008).

Three points can be made in response.

First, having a beginning does not require having a beginning point (Craig and Sinclair 2012, p. 99). It has been explained in Chap. 2 that something has a beginning if it has a temporal extension, the extension is finite, and it does not have a static closed loop or a changeless phase that avoids a boundary. Suppose t₀–t₁ is the first temporal interval. Even if there were an actual infinite number of points/events between t₀ and t₁ and there is no first point of time at t₀, the extension of t₀–t₁ is still finite because the points/events sum up to a duration that is finite in magnitude. Therefore (if it does not have a static closed loop or a changeless phase that avoids a boundary; see Sect. 5.6 and Chap. 6), the series of points/events would still have a beginning, and hence (given the argument in Chap. 3) would still require a cause. (To elaborate, suppose an increasing in strength of electric field occurred at time interval t₈–t₉. Even if there were an actual infinite number of events between t₈ and t₉ and there is no first point of time at t₈, the extension of t₀–t₁ is still finite and has temporal boundaries. We know from experience that such an increasing in strength of electric field is caused when [say] I switched on an electric field generator. Now, the Modus Tollens argument defended in Chap. 3 implies that, if our universe begins uncaused at time interval t₀–t₁, there would be no difference between that event and the increase in strength of electric field at t₈–t₉ where beginning to exist uncaused is concerned, and thus the latter would also begin uncaused, which is not the case; hence, the antecedent is not the case.)

Second, the argument for the impossibility of traversing an actual infinite has a crucial disanalogy with Zeno’s paradox of motion. In the case of the argument, the events in a temporal series are actual. By contrast, in the case of Zeno’s paradox, the interval traversed can be regarded as being potentially infinitely divisible and not actually infinitely divided. In other words, one can keep on dividing the interval by half without ever ending up with an actual infinite number of divisions. The claim that Achilles must pass through an infinite number of halfway points in order to cross the stadium begs the question by assuming that the whole interval is a composition of an infinite number of points which have been traversed (Craig and Sinclair 2009, p. 119). (One might object, ‘but how could one distinguish between those that are actual events and those that are only potential events?’¹³ In reply, this is a question about epistemology [How we distinguish?]. Whereas the arguments in this section concerns ontology: they demonstrate that, ontologically, there cannot be a traversal of an actual infinite in the concrete world. The epistemological question [How we distinguish?] is irrelevant as an objection to the argument which concerns ontology, and therefore fails to rebut the conclusion that there cannot be a traversal of an actual infinite in the concrete world. In other words, even if we are not able to distinguish between those that are actual events and those that are only potential events, it remains the case that—ontologically—there are only a finite number of events that are traversed.)

Third, I shall explain below that it is fallacious to think of time as a continuum of points.

Against this, one might object that, at the level of fundamental physics, events may be analogous to points. In other words, events may not be discrete entities in a causal chain; rather, causality could be continuous in nature.¹⁴

In reply, on the one hand, it has not been proven that spacetime is a continuum made up of actual infinite number of points. Cosmologist George Ellis notes that ‘there is no experiment that can prove there is a physical continuum in time or space; all we can do is test spacetime structure on smaller and smaller scales, but we cannot approach the Planck scale.’ A distinction should be made between mathematical models of the physical world and the physical world itself. Craig and Sinclair (2012, p. 100) explain that it has not been proven that space and time really are composed of an actual infinity of points rather than simply being modelled as such in general relativity. While infinities are useful mathematically, that does not imply that concrete infinities exist, just as the fact that imaginary numbers (e.g. √−1) are mathematically useful does not imply that that they correspond to concrete entities (they obviously don’t!). Infinities, like imaginary numbers, can be regarded as useful abstract tools. Imaginary numbers work as a shorthand for mathematical operations involving real numbers. Likewise, infinities may work as approximations or generalizations. For example, the idea of infinities can be understood as approximations which have proven to be useful in addressing problems concerning pendulums, chemical decay, coagulation kinetics, diffusion, convection, economic equilibrium, and fluid and air flow. Tegmark explains:

Consider, for example, the air in front of you. Keeping track of the positions and speeds of octillions of atoms would be hopelessly complicated. But if you ignore the fact that air is made of atoms and instead approximate it as a continuum—a smooth substance that has a density, pressure, and velocity at each point—you’ll find that this idealized air obeys a beautifully simple equation explaining almost everything we care about: how to build airplanes, how we hear them with sound waves, how to make weather forecasts, and so forth. Yet despite all that convenience, air of course isn’t truly continuous. I think it’s the same way for space, time, and all the other building blocks of our physical world. (Tegmark 2015)

In some cases infinities can be understood abstractly as a limit concept, or as generalizations as in a polygon circle. Regarding infinitesimal calculus, Oxford philosopher A.W. Moore points out that mathematicians using calculus can uphold claims ostensibly about infinitesimals or about infinite additions, knowing that they are only making disguised generalizations about what are in fact finite quantities (Moore 2001, p. 73). Various scientists and philosophers have also argued that time and space could be a set of discrete entities of extended simples (i.e. a spatiotemporal entity that has no proper parts and does not have the shape and size of a point) (Van Bendegem 2011; Hagar 2014). Craig notes that ‘Since the advent of quantum theory, philosophers, and physicists as well, have exhibited much greater openness to taking time and space to be discrete rather than dense. In fact, many think that the continuity of spacetime in general relativity is what needs to go if we are to have a unified physical theory of the world’ (citing Butterfield and Isham 1999, section 3.2; Huggett and Wüthrich 2013). On the other hand, one might defend an alternative view that time might be continuous yet divide into a finite number of smallest parts of finite durations rather than divide into instants/points (see my reply to Puryear in Loke 2016a and Loke 2017a, chapter 2).

It has been explained above using the example from quadratic equation that, while what is mathematically impossible is metaphysically impossible, what is mathematically possible is not always metaphysically possible. Thus, the mathematical possibility of actual infinity does not imply that there are metaphysically possible in the concrete world. Thus, for example, while one can always decompose a real function in terms of infinitely many sinusoidal functions (Fourier series) with countably infinitely many coefficients (Pitts 2008), this can be regarded as true only for mathematical modelling (Chan 2019; for other examples, see Loke 2017a, chapter 2). Likewise, the infinite number of points within the intervals (0, 1) and (0, 2) and the one-to-one correspondences between them merely refer to abstractions; it does not imply that they can be realized as concrete entities. Likewise, while there is an abstract actual infinite number of points between (say) t₀ and t₁ just as there are an actual infinite number of real numbers between 0 and 1, this does not mean that they can exist in the concrete, nor does it imply that such a series can be constituted in the concrete.

While one might claim that it is possible to have a concrete point in between any two points,¹⁵ it is logically invalid to infer from this to the conclusion that there is an actual infinite number of concrete points which can be traversed. This would be guilty of a fallacious modal operator shift, inferring from the true claim,Verse

Verse 1. Possibly, there is some point at which x is divided. To the disputed claim, 2. There is some point at which x is possibly divided. (Craig and Sinclair 2009, p. 114)

It would be like arguing ‘because a leaf could be any colour, therefore it can be every colour’. A leaf obviously cannot be of every colour at the same time because of metaphysical constraints. Likewise, there might be metaphysical constraints such as those which I have explained using the Christmas Present Scenario (Loke 2017a, Chapter 2) which prevent all the points from existing together concretely, even though it is possible that each of the points exists concretely.

On the other hand, Craig observes that the idea of spacetime being a continuum made up of actual infinite number of points results in the ancient Greek paradoxes of motion: Suppose time t is the last point in time at which an object O is at rest, it would be impossible for O to begin to move. To reply that t′ is the first point in time at which O is in motion will not do, for t is supposed to be the last point in time at which O is at rest, and one can always think of t* (where t < t* < t′) for any t or t′ (Craig and Sinclair 2012, p. 100; thus, for example, suppose time 0.00 sec is the last point in time at which an object O is at rest; to say that time 0.01 sec is the first point in time at which O is in motion will not do, for 0.00 sec is supposed to be the last point in time at which O is at rest, and therefore O would have already been in motion at (say) 0.005 sec, that is, before 0.01 sec. The solution to this paradox is to reject the theory that time is composed of an infinite number of points and to adopt the theory that time is composed of durations. In that case, one can suppose time t is the last duration in time at which an object O is at rest, and that t′ is the first duration at which O is in motion, and that there is no duration t* in between t and t′). Additionally, recent defences of the Grim Reaper Paradox provide convincing reason to think that spatiotemporal intervals are not composed of a dense infinity of points or instants (Craig 2018, p. 398). Moreover, I have shown above that there cannot be a traversal of an actual infinite of events because finite + finite = finite; therefore, this is a proof that there isn’t an actual infinite number of point-events between (say) time t₀ and t₁ which is traversed.

Sorabji (2006, pp. 221–222) objects that a beginningless sequence does not face the same difficulties as an endless sequence because traversing the former would involve only one terminus (e.g. the present moment), whereas traversing the latter would involve two termini (e.g. the present moment and some future moment). ‘And [having two termini] is what prevents the future series of traversed years from being more than finite’ (Sorabji 2006, p. 222).

In reply, the cause of the impossibility of traversing an actual infinite in my explanation above is not due to the number of termini, but the fact that an actual infinite has greater number than the number which can be traversed one after another in time, because finite + finite = finite. Sorabji does not provide any solution to this difficulty.

One might object that the reason actual infinite is too large to be traversed by a one-after-another process is because one cannot arrive at the endpoint of that which has no end by beginning from a point (Leon 2011). Since there is no endpoint, one can always increase, and every point that is arrived at is always smaller than an actual infinite, but this is not the case if one does not begin from a point by arriving at the present from an actual infinite number of earlier events. However, whether there is an end or no end should not affect the number that can be traversed by a process. The reason is because, regardless of whether there is an end or not, the number traversed by a one-after-another process is finite, and this is due to the nature of the process in which finite + finite= finite.¹⁶

Pruss (2018, p. 152) objects that ‘imagine someone who, according to our external time, now exists and will always exist. But she lives her life backwards. This person, thus, at this point can be said to have lived an infinite life. And if this is her moment of death, then she has completed that infinite life.’ This objection begs the question by assuming that it is possible that someone now exists and will always exist and that the future is concrete actual infinity. It begs the question against the view that someone will exist forever only if the future is potential infinite in dynamic time. On the other hand, the argument against traversing an actual infinite which I explained above would rule out such a scenario, regardless of whether a person travels forwards or backwards in time.

One might object to KCA by suggesting that events should be understood as ‘becoming’.¹⁷ In reply, becoming (with no end) assumes a potential infinite, but events that are causally prior have already happened and therefore cannot be a potential infinite (Loke 2017a, chapter 2). Likewise, recursive function (a function that calls itself during its execution) involving an infinite loop is a potential infinite in its actual execution (perpetually increasing towards abstract actual infinity as a limit but never actually arriving at an actual infinite in time), and thus is inapplicable as well. (Note that a potential infinite is a series which increases towards actual infinity but does not arrive at actual infinity. The possibility of such a series is not the possibility of completing the counting an actual [infinite] series of past events. Rather, the possibility is the possibility of an abstract actual infinite as a limit concept. It is important to distinguish between abstract actual infinite and concrete actual infinite. The argument against traversing an actual infinite concerns the concrete; that is, a concrete actual infinite cannot be traversed. This is consistent with the existence of an abstract actual infinite as a limit concept.)

It has been objected by Dretske (1965) that, if starting from a point someone (e.g. George) does not stop counting, then George will count to infinity ‘in the sense that he will count each and every one of the finite numbers’. Oppy (2006a, p. 61) likewise argues that ‘one counts to infinity just in case, for each finite number N, one counts past N. But unless one stops counting, one will eventually reach any given finite N’ (see also Malpass 2021).

However, the question that is relevant to the Kalām is not whether George will count an actual infinite (since ‘will count’ concerns future events rather than past events). Rather, the question is whether George can be at any particular time t and counts an infinite number successively by that time, and the answer is no: there is no time at which he could have counted an actual infinite number of elements by counting one element after another. Even if it is the case that George counts as long as time exists, actual infinity will always be greater than the numbers to which George has counted by time t. Thus, the fact is that there is no time at which an actual infinite has been counted (Loke 2014a).

Whereas to have counted as long as time exist if time is beginningless (e.g. George counts 0 at the year 2020, -1 last year [2019], -2 the year before that [2018] … etc.) would have required an actual infinite to have been counted at a particular time (say, the year 2020), which as explained previously is impossible. In other words, Dretske, Oppy, and Malpass are guilty of redefining ‘traversing an actual infinite’ without solving the problem in the context of debating the cosmological argument. (Malpass 2021’s ‘fills the future’ argument is not relevant to the discussion concerning the past: if the number of durations earlier than [say] the year 2020 is an actual infinite, then we could indeed look back [say, in the year 2021] and say that the counting was completed in the year 2020, and the consequent [an infinite set of completed events] is what proponents of the Kalām are arguing against [i.e. via Modus Tollens] in the context of discussing whether the universe began to exist.)

The same problem besets Almeida’s (2018, p. 52) objection that ‘Mathematical induction is valid. If 1 has the property of being a number, and if for each finite n, if n is a number, then n + 1 is a number, then all of the positive integers are numbers. Likewise, if n is traversed, and if for each finite n, if n is traversed, then n + 1 is traversed, then all of the positive integers are traversed. Yes, the whole thing! No fallacy of composition is involved. The mistake is in believing that we would have to traverse something other than finite numbers in order to traverse an infinite series.’ Almeida (2018, p. 53) acknowledges: ‘it is also true that the clock never reaches the infinite time t_ℵ0. But this is because no such time as t_ℵ0 exists. If the clock ticks off the finite times in each Sn—as we have proven it does by mathematical induction – then the clock ticks off infinitely many times.’

In response, the fact remains that an actual infinite cannot be constituted (as Almeida acknowledges, ‘the clock never reaches the infinite time t_ℵ0’), whereas the constitution of a beginningless series of earlier events to the year 2020 would have required an actual infinite to be constituted, which as explained previously is impossible. In other words, Almeida is guilty of redefining ‘traversing an actual infinite’ as ‘ticks off the finite times in each Sn’ instead of ‘reaches the infinite time t_ℵ0’, without solving the problem in the context of debating the cosmological argument. Now I do agree that mathematical induction is valid, but Almeida has misapplied it by wrongly defining the problem.

It might be asked whether an omnipotent God could traverse an actual infinite.¹⁸ Many philosophers and theologians throughout history (e.g. Augustine, Anselm, Aquinas, Hoffman and Rosenkrantz) have explained that divine omnipotence does not require God to do what is metaphysically impossible (e.g. God cannot make a shapeless square because there cannot be any such thing for God to make; see further, Loke 2010, p. 526). Since it has been shown above that traversing an actual infinite is metaphysically impossible, there is no violation of divine omnipotence to say that God cannot traverse an actual infinite because there cannot be any such traversing for God to do.

5.5 The Argument from the Viciousness of Dependence Regress

The argument against a dependence regress had a long history. While Leibniz had argued against infinite regress by claiming that grounds can never be found in this way (see Cameron 2008), Hume (1779/1993) objected that if every item in a collection was causally explained by a preceding item, the whole collection would be explained (Hume–Edwards Principle). A Humean might therefore argue that, if there is an infinite chain of events with each event being grounded by the prior event that caused it, and the chain itself is grounded by another entity which is grounded by another, and so on, then there is no problem with an infinite regress (Cameron 2008, p. 11; Cameron eventually appeal to intuition and explanatory utility to argue against such a regress). One might reply that, if all we want is an account of why each thing exists, then an infinite regress is benign because each thing is explained by its cause; but if we want an account of why there are things at all, it is vicious because we have not explained where existence comes from (Bliss 2013, p. 414; Cameron 2018). However, the objector might say that, in an infinite chain, there is always existence; thus, it is meaningless to ask where existence comes from. I shall reply to this objection in what follows by arguing that there is a problem of dependence and a need for the capacity to begin to exist which is not met by an infinite regress.

My argument can be formulated as follows:

• 1. A dependence regress (i.e. a regress in which each item depends on the prior one) is a vicious regress.

• 2. A causal regress is a dependence regress.

• 3. Therefore, a causal regress is a vicious regress.

While arguments based on dependence have often been used to argue against an infinite causal regress in the case of an essentially ordered series in the Thomistic Cosmological Argument, I have argued in Loke (2017a, chapter 3) that they can be used for an accidentally ordered temporal series as well. The Modus Tollens argument defended in Chap. 3 implies that whatever begins to exist would depend on causally necessary condition(s) (this also implies the transitivity of causal dependence understood as dependence on causally necessary condition(s)¹⁹). Moreover, since dependence is a kind of contingency (defined as a dependence on circumstances), my argument can also be understood as a kind of contingency argument which demonstrates that there is a necessary being which (as explained below and in Chap. 6) is an uncaused First Cause which is beginningless, initially changeless, and has libertarian freedom.

To illustrate the viciousness of a causal dependence regress, think about a series of train wagons in which each train wagon requires a preceding one to pull it if it is to begin to move. Before the last train wagon begins to move, the one before it has to begin to move, and before that train wagon begins to move, the one before it has to begin to move, and so on. No matter how many such train wagon there are, none of them would begin to move, because no prior wagon escapes from the problem of depending on a prior dependent member in order to begin movement (vicious regress). What is required is an engine, a First Puller which does not depend on another train wagon to pull it, and which has the independent capacity to bring about the beginning of movement.

Likewise, before I begin to exist, my parents has to begin to exist, and before they begin to exist, their parents have to begin to exist, and so on. No matter how many prior dependent causes there are, none of them would begin to exist, because no prior dependent cause escapes from the problem of depending on a prior dependent member in order to begin to exist (vicious regress). What is required is a First Cause which exists independently (i.e. not dependent on a prior entity). Since whatever begins to exist has a cause (as established in Chap. 3), this First Cause would be beginningless.

One might object by claiming that the above argument only works within a deterministic and reductionistic worldview, which neglects the fact that different branches of the natural sciences talk about their own causal explanations using different scientific theories (rather than uniformly as in the case of a series of wagons).²⁰

In reply, first, it is not true that indeterminism has been proven; defensible deterministic interpretation of quantum physics exists (see Bricmont 2017).

Second and more importantly, in any case, the basic idea of causally necessary condition is compatible with both determinism and indeterminism, and it is compatible with different descriptions by different scientific theories in different branches of sciences. For example, quantum particles do not begin to exist from non-being. Rather, they emerge from the quantum vacuum with quantum field and which possesses ‘zero-point energy (the energy remaining in a substance at the absolute zero of temperature (0 K), which gives rise to vacuum fluctuations)’ (Daintith 2009). The fluctuation is therefore dependent on the quantum field and the energy in the quantum vacuum, which is a causally necessary condition. Likewise, hydrogen is a causally necessary condition for the formation of water.

Even though the emergence of quantum particles may be indeterministic while the formation of water is deterministic, and even though these two events are explained by different theories, nevertheless both theories (as well as other scientific theories) involve effects which are dependent on necessary conditions (which is what I mean by a cause). Even though the kinds of necessary conditions and the kinds of dependence might be different for each theory, they all involve necessary conditions and dependence nonetheless.

The train car analogy is only an analogy; the point of the analogy is that a regress of causally necessary conditions is a vicious dependence regress, and therefore requires an independently existing First Cause which does not require causally necessary condition. This point does not require the assumption that each effect is deterministically brought about by the causes, nor does it require the causes are describable by the same scientific theory. Even if each effect is not deterministically brought about and are not describable by the same scientific theory, it remains the case that the effects depend on causally necessary conditions and there cannot be an infinite regress of dependence; thus, there must still be an independent First Cause. Hence, the fact that different branches of the natural sciences talk about their own causal explanations does not affect my argument. On the contrary, it has been explained in Chap. 2 (e.g. using the observation that the causal term ‘interaction’ is fundamental to science) that causation itself is fundamental to science, and it has been explained in Chap. 3 that the Modus Tollens argument for the Causal Principle implies that whatever begins to exist would depend on causally necessary condition(s).

To speak of a causal chain is to speak of a chain of causally necessary conditions describable by various theories of science. The above examples show that we are able to say something in empirical/scientific terms about the great chain of causes, that is, the quantum vacuum as a causally necessary condition for quantum fluctuation and hydrogen as a causally necessary condition for the formation of water. The Modus Tollens argument in Chap. 3 shows that the causal principle is true and therefore each entity in the chain would have a causally necessary condition except the beginningless First Cause.

Consider another analogy (which I shall call the Debtors’ Scenario). Suppose I have nothing and the only way for me to begin to have money is to get it from Justin, Suppose Justin has nothing and the only way for him to begin to have money is to get it from Alex. If everyone is like this, no one would ever begin to have money. As Schaffer (2016, p. 95) observes, ‘One cannot be rich merely by having a limitless supply of debtors, each borrowing from the one before. There must actually be a source of money somewhere.’²¹ Money would not simply emerged from the chain.²² What is required is someone who does not need to get money from others and is able to have money, that is, a First Source of money.

This is analogous to the real world in which I have no existence before I begin to exist, and the only way for me to begin existing is to be brought about by prior causes (my parents). However, they also have no existence before they begin to exist, and the only way for them to begin existing is to be brought about by prior causes (my grandparents). If everyone is like this, no one would ever begin to exist just as no one would ever begin to have money in a series of debtors. What is required is something which does not need another thing to bring it about and is able to bring about other things, that is, a First Cause.

To elaborate, suppose entity/event x has a beginning of existence and x causally explains why ybegins to exist. x can causally explain the beginning of y only after xbegins to exist, but the problem is the prior entity x cannot explain why x itself begins to exist given the Causal Principle (established in Chap. 3) that whatever begins to exist has a cause. Thus, x cannot explain why there are entities (x, y) which begin to exist; that is, x has 0 capacity for explaining the beginning of existence of entities (x, y). If there is something w which begins to exist prior to x, then w can explain why (x, y) begin to exist, but the problem is that w also cannot explain why itself begins to exist. Thus, w cannot explain why there are entities (w, x, y) which begin to exist, and relies on there being a prior entity v which also has 0 capacity for explaining the beginning of existence of entities (v, w, x, y) if v itself begins to exist. If every prior entity has a beginning, then no prior entity escapes from the problem of having 0 capacity for explaining why there are entities which begin to exist; indeed, every prior entity has 0 capacity for explaining the beginning of existence of entities; thus, nothing could ever happen. What is required is an independently existing and beginningless First Cause which is not being sustained in existence, which does not need to depend on another thing to bring it about and which has the independent capacity to bring about other things by itself. Thus, given any change in reality whatsoever there must be such an uncaused First Cause. This First Cause does not begin to exist and hence does not face the problem of an infinite regress scenario in which all members of the series have beginnings of existence without anything having the capacity for explaining the beginning of existence of entities. Such a First Cause would answer the question ‘Why is there something rather than nothing?’; that is, there would be no causal explanation for why the First Cause exists, since being beginninglessness and not being sustained in existence implies that it was not brought about.

One might ask whether my argument ‘if every member of a causal series has no capacity to begin to exist without prior cause, then all the members would have no capacity to begin to exist without prior cause’ commits the fallacy of composition. In reply, as Reichenbach (2021) observes, arguments of the part–whole type are not always guilty of this fallacy; it depends on the content of the argument. Sometimes the totality has the same quality as the parts because of the nature of the parts invoked. For example, if every member of a set of entities has 0 mass, then all the members (regardless of the number of members) would have 0 mass, because 0 + 0 + 0 … = 0. Likewise, if every member in a series of debtors has 0 capacity for explaining the beginning of existence of money, then all the members (regardless of the number of members) would have 0 capacity for explaining the beginning of existence of money, because 0 + 0 + 0 … = 0. Similarly, if every member of a causal series has 0 capacity for explaining why there are entities which begin to exist, then all the members (regardless of the number of members) would have 0 capacity for explaining why there are entities which begin to exist, because 0 + 0 + 0 … = 0.

One might object that, in an actual infinite regress of events, the actual infinite series itself does not begin to exist; thus, the whole series itself is exempt from the Causal Principle that beginning of existence has a cause, even though each part of the series (i.e. every event) has a cause.²³ In this case, the whole would have a property which the parts do not have. This is the point of disanalogy with the Debtors’ Scenario: while the whole series of infinite number of debtors would not avoid the problem that the whole series has no money because each part has no money, the whole series of an actual infinite regress of events would have no beginning even though each part has a beginning.

In reply, it is trivially true that if an infinite regress causal series exist it would not have a cause. However, my argument is that such a beginningless series cannot exist in the first place because of the problem of vicious regress; that is, no prior dependent cause escapes from the problem of depending on a prior dependent member in order to begin to exist. Every member in such a causal series suffers from the problem of depending on prior member for beginning of existence, analogous to the Wagon Scenario, where every member suffers from the problem of depending on prior member for beginning of movement, without any source.

In other words, such a beginningless series does not get rid of the problem that all its members have beginnings and are dependent in the similar way that all the debtors are dependent, and this is the point of analogy with the Debtors’ Scenario. It does not get rid of the problem that there is 0 capacity for explaining the beginning of existence of entities/events. Every member in such a causal series suffers from the problem of having 0 capacity for explaining the beginning of events, which is analogous to the Debtors’ Scenario, where every member suffers from the problem of having 0 capacity for explaining the beginning of money in the chain. There needs to be a source somewhere. While on the infinite regress scenario the whole does not have beginning, it remains the case that all the members of the whole has a beginning (y has a beginning, x has a beginning, w has a beginning …), and my argument concerns how to explain the latter. One cannot avoid the latter by appealing to the former; one can only appeal to the former if one postulates a beginningless changeless (eventless) entity enduring through all durations, but such an entity would not bring about the events of our universe which is what requires explanation here. On the other hand, postulating an infinite number of changes which are infinitesimally closely ordered within a duration of time would not work as well (and in any case this postulation has been ruled out by the arguments in Sect. 5.4). The reason is because an infinite number of states each of which has 0 capacity for explaining why there are entities which begin to exist still does not get rid of the problem that there is 0 capacity for explaining the beginning of existence of entities within the series.

It might be objected that, while none of the entities in the series has capacity for causally explaining the beginning of existence of events, the series as a whole has such a capacity.

However, this would not work because every entity in the series has 0 capacity for explaining the beginning of existence of entities; therefore, collectively as a whole, 0 + 0 + 0 + 0 … = 0. (This is analogous to the Debtors’ Scenario, where every member suffers from the problem of having 0 capacity for explaining the beginning of money in the chain; therefore, collectively as a whole there is no such capacity. This does not commit the fallacy of composition; see discussion above) This implies that, collectively, the earlier (or causally prior) entities would not have the capacity for explaining why there are later entities which begin to exist. This means that there is 0 capacity for explaining the beginning of entities within the infinite regress series, which implies that the beginning of my existence would not have occurred if the causal chain leading to my beginning of existence was such an infinite regress. But I had begun to exist. Therefore, there is no such infinite regress but rather there is a First Cause. The above argument is not based on our common-sense intuitions but on the meaning of causal dependence which is required by science itself; for example, water has 0 capacity to begin to exist by itself and depends on the formation of hydrogen at the earlier history of the universe.

It might be objected that the series as a whole has the capacity for causally explaining the beginning of existence of its members because every member would have a cause, and that the beginning of each entity is adequately accounted for by a previously existing entity in an infinite regress.²⁴

However, the problem is that every cause of every member is dependent on prior causes and every prior cause is dependent. The objector might reply that this is not a problem because every prior cause has a cause—which is dependent, yes, but it has a cause. In reply, every member having a prior dependent cause is useless for solving the problem that all the members are dependent, just like every wagon having a prior dependent wagon is useless for solving the problem that all the members are dependent. In order for a cause to bring about the next member the cause has to begin to exist first, just as in order for a wagon to pull along the next member the wagon has to begin to move first, but the problem is that in both cases it is dependent on prior member all of which are dependent. Moreover, as noted earlier, every cause of every member and every previously existing entity in such a causal series suffers from the problem of having 0 capacity for explaining the beginning of entities, which is analogous to the Debtors’ Scenario, where every previous member suffers from the problem of having 0 capacity for explaining the beginning of money in the chain. There needs to be a source somewhere.

Sceptics might object by claiming that this is not a problem, given that in an infinite regress ‘there is always at least one member in existence’ to explain why there are subsequent entities which begin to exist. Sceptics might argue that this is the point of disanalogy with the Debtors’ Scenario: in an infinite regress of debtors, there is no one who has money, but in an infinite regress of causes, there is always at least one member in existence.

In reply, it is trivially true that, if an infinite regress exists, then there is always one member existing. However, such a series cannot exist because the problem of vicious regress remains: how does any of its member begin to exist in the first place? It is dependent on prior member every one of which is dependent. In other words, the postulation that there is always at least one member in existence relies on the assumption that that existent member(s) begins to exist (if none of the members begin to exist, then it is not the case that there is always at least one member in existence), and I have explained that the problem with this assumption is that none of the members prior to that existent member has the capacity for explaining the beginning of entities. This is the point of analogy to the problem that no one has the capacity for explaining the beginning of money in a series of debtors. In order for there to be money in the series, someone must begin to have money, but no one has the capacity for explaining the beginning of money in the series. Likewise, in a series whereby every member has a beginning of existence, in order for there to be something in existence already, something must begin to exist, but the problem is that no one has the capacity for explaining the beginning of entities in the series.

One might object that the things we observe did not begin to exist; rather, they are merely rearrangements of pre-existent matter-energy. For example, the amino acids existed before they became a part of my body.

In response, the fact that my body involves a rearrangement of pre-existing matter-energy does not deny the fact that various events such as new arrangements of the pre-existent matter-energy did begin to exist (e.g. the event of fertilization has a beginning), and that each event (an event is something with a beginning) depends on prior causes (as shown in Chap. 3). My argument does not require beginning to exist from nothing; it only requires that an infinite regress of dependant events/changes is not the case, and therefore there is a first change brought about by an initially changeless and independently existing First Cause (which can bring about the first change by having libertarian freedom, as explained in Chap. 6).

One might object that according to the static theory of time it is always the case that I began to exist (say) in 1975 and I always existed in 1975.²⁵ However, even if the tenseless theory of time is true, it is still the case that later events (say my existence in 1975) is dependent on earlier events (say, the existence of my parents prior to 1975, because if they did not exist prior to 1975 I would not have begun to exist in 1975). It should be noted that there are two different senses of ‘always exist’: (1) existing forever without beginning and therefore doesn’t require a cause (see the view of the Oxford School in Chap. 6); (2) having a beginning but a tenseless fact at a particular duration. To appeal to a static theory of time would be to refer to the second sense, but according to the second sense, later events are still dependent on earlier events. Hence, this does not remove the problem with a series of dependent events all of which suffer the same problem of dependence. The solution to this problem requires something that has no beginning and therefore can be the uncaused First Cause.

One might object by suggesting that perhaps matter do not begin to exist and atoms are in continuous motion bringing about water, myself, and so on, in which case ‘beginning of existence’ is just our way of conceptualizing this motion. According to the conservation of momentum (momentum= mass × velocity), to keep something moving (i.e. changing position), no external cause is needed; thus, one might think that this implies it is false that every change/event has a cause. Carroll writes:

Aristotle’s argument for an unmoved mover rests on his idea that motions require causes. Once we know about conservation of momentum, that idea loses its steam … the new physics of Galileo and his friends implied an entirely new ontology … ‘Causes’ didn’t have the central role that they once did. The universe doesn’t need a push; it can just keep going. (Carroll 2016, p. 28)

The conservation of momentum implies that there is no need for external cause; nevertheless, the momentum is itself the cause for the continuing movement. Suppose a moving object M is at location x₁ at time t₁ and x₂ at t₂. If M has zero momentum at x₁ it would not move to x₂, but if M has a certain momentum it would move to x₂; thus, the momentum is the cause.²⁶

Moreover, the beginning of the state at t₂ is dependent on the beginning of the state at t₁ having momentum, which is dependent on the beginning of a prior state having momentum. Given the viciousness of an infinite regress of dependent states as explained earlier, there must still be a beginningless and initially changeless First Cause with libertarian freedom. (If there is a changeless beginningless entity, that would not require a cause, but that entity would be irrelevant for accounting for the events we observe, unless [as explained in Chap. 6] that entity has libertarian freedom to bring about the first event leading to the series of events we observe; this implies that it would be a First Cause Creator who is initially changeless)

Finally, it should be noted that the argument from the viciousness of dependence regress is compatible with time being a continuum of points (which nevertheless has been demonstrated above to be false). Even if there is an infinite number of points between t₁ and t₂, they sum up together into an interval t₁–t₂ which is still finite in earlier than extension. Therefore, an entity lasting this interval has a beginning, and hence (according to the causal principle established in Chap. 3) would depend on a cause. As argued above, there cannot be an actual infinite regress of dependent entities with beginnings; therefore, there cannot be a beginningless infinite regress of entities/events each of which has a beginning. Rather, there must be a First Cause which is beginningless and not composed of an infinite series of events/changes, but rather is beginningless and initially eventless/changeless.

In conclusion, the above argument can be summarized as follows:

• 1. An infinite regress of dependent entities is a vicious regress.

• 2. An event/change is an entity that has a beginning at the state of having gained or lost a property. (Definition)

• 3. Whatever begins to exist depends on a cause. (Chap. 3)

• 4. An event/change is a dependent entity. (From 2 and 3)

• 5. An infinite regress of events/changes is an infinite regress of dependent entities. (From 4)

• 6. An infinite regress of events/changes is a vicious regress. (From 1 and 5)

What is required is a First Cause which is beginningless and (initially) eventless/changeless and therefore can exist independently. (It will be shown in Chap. 6 that this beginningless entity cannot be an impersonal universe, since it must have libertarian freedom in order to cause the first event from an initially eventless/changeless state. Thus, the impersonal universe itself would have a beginning, which was freely brought about by a beginningless personal First Cause.)

5.6 Can a First Cause Be Avoided by a Causal Loop?

Gott and Li (1998) proposed that a First Cause may be avoided by the suggestion that the temporal series of events at the beginning of the universe is a small time loop, thus allowing it to create itself similar to the way a time traveller could travel to the past and become his/her own mother.

However, such a closed causal loop is contrary to the Generalized Second Law of Thermodynamics (Wall 2013a, 2013b) and faces the following problems.

For a causal loop in dynamic time, the members of a series of events come to be one after another cycles after cycles. Given that an actual infinite regress is not the case (as argued in previous sections), the number of earlier cycles would be finite, and thus there would still be a first cycle with a first event and a First Cause.

A causal loop in static time—in which A requires B to bring about its beginning, B requires C to bring about its beginning, and C requires A to bring about its beginning—is viciously circular. It would be similar to a scenario in which railway wagon A requires wagon B to bring about its motion (i.e. by pulling it), wagon B requires wagon C to bring about its motion, and wagon C requires wagon A to bring about its motion. It is evident that such a viciously circular setup—in which the state of each of the entities in a causal loop is supposed to be dependent on another entity within the loop—would not work. Likewise, in a loop that is supposed to avoid a first cause, the beginning of our universe is required to provide causally necessary conditions for the beginning of existence of other entities within a closed loop, while the beginning of our universe itself requires the existence of these other entities. Such a vicious circular setup would not work as well.²⁷ It faces a similar problem with that of a vicious dependence regress explained previously; that is, since every member of such a causal series has 0 capacity for explaining why there are entities which begin to exist, all the members would have 0 capacity for explaining why there are entities which begin to exist, because 0 + 0 + 0 … = 0.

Against circles of causes, Oppy (2019a, p. 13) argues: ‘It is a fundamental causal principle that, if one thing is a cause of a second thing, and that second thing is a cause of a third thing, then the first thing is a cause of the third thing. However, if there could be a circle of causes … then it could be that there are things that are causes of themselves.’ But nothing can be a cause of itself, since causes by definition are causally prior to their effects (ibid.; Oppy’s argument depends on the transitivity of causation, which I have defended above in Sect. 5.5).

Another objection has been raised by Rasmussen (2018) with reference to the possibility of a scenario which a static closed loop gives rise to. This scenario involves a time traveller whose older self was able to go back in time to teach his/her younger self how to build a time machine and his/her older self knew how because his/her younger self had been told (Lewis 1976, pp. 148–149) Rasmussen (2018) remarks:

Consider that there is no knowledge of how to build a time machine in our world. That’s because no one figured it out (and we can assume for sake of illustration that it could be figured out). Yet, the same is so in the above scenarios: no one figured out how to build a time machine. Thus, no causally relevant difference explains how such knowledge exists in the loop and infinite regress scenarios but not ours. (For replies to objections, for example, by David Lewis and others, see Loke 2017a, chapter 4).

5.7 Conclusion

While the ‘Standard’ Big Bang model affirms that matter-energy began to exist at the initial cosmological singularity of the Big Bang, over the years, alternative models of the Big Bang have been developed, and some of them have tried to avoid postulating a first event. However, such models face scientific objections related to the Generalized Second Law of Thermodynamics, acausal fine-tuning, or having an unstable or a metastable state with a finite lifetime (Craig and Sinclair 2009, pp. 179–182; Bussey 2013; Wall 2013a, 2013b). Bounce cosmologies which postulate that the universe was born from an entropy-reducing phase in a previous universe and the entropy reverses at the boundary condition (Linford 2020) have been proposed to avoid some of these problems. However, these cosmologies neglected the problem of causal dependence at the interface. While it has been suggested that the universes to either side of the interface might be interpreted as the simultaneous causes of each other (Linford 2020, p. 24), this violates the irreflexivity of causation and amounts to a vicious circularity.

Moreover, there are at least five philosophical arguments against an infinite regress of causes and events which have been proposed in the literature, and any one of these would suffice to establish the conclusion:

• 1. The argument from the impossibility of concrete actual infinities

• 2. The argument from the impossibility of traversing an actual infinite

• 3. The argument from the viciousness of dependence regress

• 4. The argument from the Grim Reaper paradox

• 5. The argument from Methuselah’s diary paradox

I have defended some of the above-mentioned arguments in this chapter, and demonstrated it is not the case that there is a causal loop which avoids a First Cause. Therefore, there is a first event and a First Cause. In the next two chapters, we shall discover what the First Cause is.