As discussed, despite temporal events being “not existents” to support the claim ‘you cannot count back from infinity’, they need to be at least theoretically countable. It is obvious, therefore, that although Craig and Durston are both proponents of the claim, Craig and Sinclair’s 2009 version of P1:

“.......a collection formed by successive addition cannot be an actual infinite.”

is at odds with Durston’s statement (unpublished manuscript) that an infinite temporal regression would mean that:

".......the number of seconds in the past is a completed countable infinity.”

Here we see Durston making the common mistake of equating an infinite set with some very large number. Craig and Sinclair are, of course, correct in this instance. An infinite set is never defined as such solely by how many elements it contains. Durston (unpublished manuscript) appears to be confused about the whole matter of infinities. After first defining a ‘completed infinity’ as “an infinity that one actually reaches” and telling us that “a completed infinity is a set” he then contradicts himself by stating that it is “impossible to count to a completed infinity” and "the number of elapsed seconds in the future is a potential infinity”. At this point we shall leave Durston to his obvious confusion.

But even if the A-theory was true, mathematically speaking it would remain the case that an infinite temporal set is not achieved by successive addition. This is because, as we have seen, there is no legitimate mathematical operation that allows counting to (or from) an infinite set by successive addition or subtraction. ‘Counting to infinity’ is in no way analogous to counting to a trillion or a googolplex. Similarly, ‘counting back (or forward) from infinity’ is in no way analogous to counting events forward from tbeginning or backward from tnow. The mistake being made (and as we shall see, is made all too often) is a failure to understand that an infinite set never corresponds to any number. Infinity is not a number. It is a property held by numbers and you cannot perform standard arithmetic inverse operations such as such as ‘counting forward’ or ‘counting back’ on the properties held by numbers. To illustrate:

Let {N} be the infinite set of all natural numbers. If we attempt to subtract all the even natural numbers from {N} we effectively have:

∞ - ∞ = ∞

because the odd numbers that remain also constitute an infinite set. Similarly, we would also have:

∞ + 1 = ∞
∞ * 10 = ∞
∞ * 10^7 = ∞
∞ * 10^7 * 5 = ∞
∞ + ∞ = ∞
∞ * ∞ = ∞

And so on. It matters not whether the members of an infinite set are natural numbers, existent or non-existent moments in time or worldly things such as grains of sand. These are not contradictory results. They are the only possible answers under these dubious mathematical circumstances. Standard inverse arithmetic operations such as addition, subtraction, multiplication and division as formulated in the Peano axioms underlie number theory. But they are not intended for use with infinite sets. The results in the arithmetic operations above only appear absurd if you expect to be able to perform standard arithmetic operations on infinite sets with the expectation that they should produce similar results to adding and multiplying finite integers. ∞ + 1 = ∞, for example, is only absurd if you hold to the mistaken believe that ∞ + 1 is somehow a larger amount than ∞ alone and so can be represented by a larger number. Despite this fact, Craig & Sinclair (2009) argue that if the past constitutes a temporal infinity:

“.......then there have occurred as many odd-numbered events as events. If we mentally take away all the odd-numbered events, there are still an infinite number of events left over; but if we take away all of the events greater than three, there are only four events left, even though in both cases we took away the same number of events.”

In other words, according to Craig and Sinclair, in this particular case:

∞ – ∞ = 4

Craig and Sinclair are no doubt relying on their audience being naïve to set theory and defaulting to their intuitions or, more accurately, their sense of incredulity. But mathematics does not rest on intuition. It rests on a formal logical system. Once again, there is nothing intrinsically problematic about infinite sets; it is the unlawful ways we might manipulate these sets that creates apparent absurdities (Hauser & Woodin, 2014; Morriston, 2013; Woodin, 2011a; 2011b). Attempting to simply subtract the infinite set of numbers > 3 from {N} is a gross mathematical error. Craig and Sinclair are misusing the concept of infinity by treating both an infinite set and its infinite subset as if they were numerical quantities both representing a finite number of elements. But infinity is a property of numbers and you cannot manipulate infinities in the same way you can manipulate finite numbers or finite sets of objects.  It is not simply the case that infinities are, for pragmatic reasons, treated as a specific exception to number theory; infinities are in no way part of number theory. If we allow ourselves to apply the Peano axioms willy-nilly to infinite sets we produce absurdities such as this:

If  ∞ + 1 = ∞
and
∞ + 2 = ∞
we can reasonably deduce that:
∞ + 1 = ∞ + 2
now if remove the ∞ from both sides we get
1 = 2
and if we subtract 1 from each side we get
0 = 1

Although this kind of arithmetical procedure is perfectly coherent when applied to finite numbers, it is obviously illegitimate when dealing with infinite sets. But it being so does not mean that an infinite temporal regression is impossible. It means that the arithmetical operations allowed by the Peano axioms are only legitimate under specific, well-defined mathematical conditions. There is nothing mathematically awry about infinite regressions and there exist no mathematical proofs discounting infinite sets (Woodin, 2011a). This does not mean that an infinite set cannot be split, however. Let:

{A} = the infinite set of all natural numbers;
{B} = the infinite set of all even natural numbers;
{C} = the infinite set of all natural numbers >3;
{D} = the infinite set of all odd natural numbers;
{E} = the finite set of natural numbers < = 3.

Because each element in {B}, {C} and {D} can be placed in one-to-one correspondence with an element of {A} it follows that {B}, {C} and {D} are each a subset of {A}. Also, because each element in {B} is in one-to one-correspondence with each element in {C} then, in terms of cardinality, {B} = {C}. We can now perform the following operations:

{A}\{B} = {D}
Which means: if infinite set {B} is removed (note: not subtracted) from infinite set {A} then infinite set {D} will remain, which is what Craig and Sinclair (2009) have done and claimed to have produced an absurd result, seemingly disregarding the fact that {A} = {D}  because {A} and {D} have the same cardinality.

However, despite {B} = {C}:
{A}\{C} = {E}
Therefore: {A\B} ≠ {A\C}

This result is not absurd for the following reason: we cannot subtract an indeterminate number of elements (such as Craig's set of natural numbers > 3) from an infinite set that is also of indeterminate number (such as all of the natural numbers) but we can remove a finite subset of elements from an infinite set to create a second, finite set if we specify in advance precisely which of the finite number of elements from the infinite set are going to be removed (East, 2013; Hinman, 2005). In this case by removing {C} from {A} rather than {B} from {A} we are left with two sets, the infinite set {C} whose cardinality is identical to {A} and the removed remainder of {A} which is now the finite set {E} or {0, 1, 2, 3}. There is no mathematical contradiction here because the arithmetical operation of subtraction has not occurred. Consider, then, the following three statements:

S1: It is not possible to precisely define ∞ – ∞; but
S2: We should expect to be able to perform the arithmetic operation of subtracting an indeterminate number of elements from an infinite set to achieve a finite set.
S3: We can remove a finite number of elements from an infinite set to achieve a second, finite set if the exact elements being removed are specified in advance.

S1 is true. Any answer would be indeterminate. S2 is plainly false; yet this is what those who claim ‘you cannot count back from infinity’ expect should be the case when they attempt to count back to infinity only to find it produces absurdity. If S2 was not false they should just as legitimately be claiming that an infinite temporal regress is impossible because ‘you can’t count back from the infinite set of even numbers’ or ‘you can’t count back from the infinite set of prime numbers’. S3, of course, is true as we have shown with {A\C} = {E}.

As alluded to, whenever the alleged absurdities regarding infinities are employed in service of some theological goal some appeal to intuition is invariably made. For example, Craig (2008) supports his premise that ‘the temporal series of events is a collection formed by successive addition’ by stating that this is “obvious”. This rests on shaky ground for three obvious reasons. First, when has human intuition ever been a reliable guide to truth? Science alone has surely taught us otherwise. Any number of examples can be marshalled in evidence, such as the strong perception that the ground under our feet is stationary or measurable changes in the mass of an object relative to both its motion and the relative coordinates of the observer. Second, our intuitions regarding mathematics have evolved through experience with finite numbers representing finite sets of objects. We are so committed to quantising finite objects in terms of finite perimeter conditions such as ‘best’, first’, greatest’ and ‘maximum’ that conceptual spaces beyond ‘best’ and first’ do not fit comfortably, even within mathematics and logic (Oppy, 2006). Moreover, our intuitive understanding of mathematics is often inconsistent and contradictory. For example, it is rare to find someone with no mathematical training who is able to accurately define the concept of ‘finite’. Yet some people with no mathematical training can give a reasonable definition of infinity (Suber, 1998). The fact that we cannot enumerate or even conceive of all the distinct elements within an infinite set is an irrelevant truth here; we cannot conceive of all the distinct elements within the vast majority of finite sets either. A chiliogon (a 1,000-sided regular polygon), for example, is a perfectly computable and physically realisable geometric shape but it is impossible for humans to visualise. Third, there is more than a hint of sophistry here. Applying intuitions about finite numbers to infinities because finite numbers accord with our empirical observations disregards the formal logical bases by which infinities become meaningful concepts. Despite the availability of consistent mathematical-logical mechanisms, human intuition regarding infinities is being represented as the more legitimate a-priori metaphysical principle. Cantor (1955) recognised the fallacious nature of this view in his letter to the mathematician and historian Gustav Eneström:

“All so-called proofs of the impossibility of actually infinite numbers are false in that they begin by attributing to the numbers in question all the properties of finite numbers, whereas the infinite numbers, if they are to be thinkable in any form, must constitute quite a new kind of number as opposed to the finite numbers, and the nature of this new kind of number is dependent on the nature of things and is an object of investigation, but not of our arbitrariness or prejudice.”

The difficulties many people encounter when dealing with infinite sets likely reflects the limitations of our cognitive mechanisms. This is especially evident when Cantor’s Principle of Correspondence meets Euclid’s Maxim. Craig appears to be capitalising on mismatches such as this. This is exemplified in a story told by Craig (1979; though quoted verbatim here from Craig, 2015):

“So, Ghazali says, let’s imagine our solar system, and here is Saturn. And let’s imagine that for every one orbit that Saturn completes, around the sun, Jupiter, which is closer in, completes two.......Now, notice that the longer they orbit, the further Saturn falls behind. If Jupiter has done ten-trillion orbits, Saturn has only done five-trillion, and the longer they orbit, the farther and farther Saturn falls behind. If they continue to orbit forever, they will approach a limit at which Saturn is infinitely far behind Jupiter.......Now, let’s turn the story around, says Al Ghazali. Suppose that they have been orbiting the sun, from eternity past, now which one has completed the most orbits? Well, the answer mathematically is that the number of orbits completed is exactly the same.......As I say, this is his argument, in the 12th Century. It’s just amazing to read this stuff.......You can’t get out of this argument by saying that infinity isn’t a number, because it is a number in this case. We’re dealing with an actually infinite number of orbits."

Although a similar paradox did originate with the medieval Islamic philosopher al-Ghazali, Craig is incorrectly attributing this particular version to him, apparently unaware that heliocentrism was unknown at the time of al-Ghazali’s writing in the 12th century. It’s not the only thing he got wrong. There is simply no mathematical rationale for claiming that infinity is “a number in this case.” Infinity is never a number. It is a property of numbers. Nevertheless, Craig goes on:

“Al Ghazali asks, “Is the number of orbits completed odd or even?” And, you know, what the answer is, mathematically – it’s both. It is! It’s both odd and even. So, that again just shows, I think, the absurdity of trying to form an actually infinite number of things by successive addition.”

This assertion is misleading. A typical definition of an even number is n = 2k where k is an integer. The definition of an odd number is therefore n = 2k + 1. It is only integers themselves that can possess the property of being even or odd. Because an infinite set is not a number, never mind an integer, it cannot be considered even or odd. Furthermore, all numbers that are not integers can be neither even nor odd and not only is there an infinite set of them but the cardinality of that infinite set is higher than the cardinality of the infinite set of integers. There is no paradox here at all once it is understood that the infinite orbital counts of the two planets can be mathematically distinguished, not by ordinality but by their differing cardinality (Oppy, 2006). Again, Craig’s claim of absurdity rests on his viewing infinity as a number and considering the defining property for an infinite set to be the number of elements it contains. He is attempting to redefine infinity in terms of finitude. In finite arithmetic we can identify even numbers by starting with any set of natural numbers and subtracting every number that satisfies n = 2k + 1. If we do this, the remaining even numbers will obviously be less numerous than our set of natural numbers. But this does not work with infinite sets because an infinite set is properly defined as a set of elements where the elements of a proper subset (in this case, odd or even numbers) can be put into one-to-one correspondence with the elements of the whole set.  If the entirety of Jupiter’s and Saturn’s orbits were a finite set then, yes, a contradiction would result. But they are not a finite set. To complain that the infinite subset of Jupiter’s orbits is the same size as the infinite subset of Saturn’s orbits you must deny the very definition of an infinite set. Unsurprisingly, Craig (2010a) is not convinced by the logic of set theory:

“These developments in modern mathematics merely show that if you adopt certain axioms and rules, then you can talk about actually infinite collections in a consistent way, without contradicting yourself.”

“Merely” is the word doing all the heavy lifting for Craig’s derision. Remove it and the statement's meaning changes completely. One wonders, then whether Craig would be so keen to include the same word in this slight rewording of his quote:

“These developments in modern theology [merely] show that if you adopt certain presuppositions and logical rules, then you can talk about an actually existing God in a consistent way, without contradicting yourself.”

And what of these mathematical rules for dealing with infinities? There are two choices. The first, set theory, is logically consistent and non-contradictory and the second, finite arithmetical procedures, returns logically inconsistent and contradictory results and, by Craig's own reckoning, absurdities. So which of these two does Craig choose to employ when employing infinities to make his case? The latter. But claiming that a temporal infinite regression cannot be the case because standard arithmetic procedures do not work on infinities is surely no more plausible than stating that a zero quantity of something is impossible because:

0 - 0 = 0
0 * 0 = 0
0/0 = undefined/uncomputable
1/0 = undefined/uncomputable
0/2 = undefined/uncomputable

and equations with a missing numeral, such as:

0 *_ = 0

have no unique solution; they are indeterminate and so have an infinite number of possible solutions. But then, as we have seen, so does this equation:

∞ * _ = ∞

Despite the apparent absurdities resulting from procedures involving zero surely no-one would seriously claim that zero does not exist as a coherent mathematical entity. Quite the opposite. There is much nonsense surrounding zero in the popular literature such as Seife’s (2000) hyperbolic contention that:

“.......within zero there is the power to shatter the framework of logic.”

If the universe does constitute an infinite temporal set then according to the B-theory of time (and set theory) it must have always constituted an infinite set. It could neither have spatially expanded nor aged itself to infinity. It would have come into existence as an infinite set, in which case there would be no individual t or temporal relations between multiple ts that would demonstrate otherwise. As discussed earlier, within any temporal continuum (i.e., a continuous range of points between, but not including, any two ts) there must also exist an infinite temporal set. Imagine then that an angel has been ‘counting forward from infinity’ from the depths of eternity and is now finishing the sequence (for whatever reason).  Let’s say she has been counting at a rate of one natural number per second for infinite time. She now decides to stop counting. It makes no difference whether this terminus is labelled tnow or tnow-1 or tnow- 1trillion or whenever. Regardless of when she finishes an infinite amount of seconds will still have elapsed. It makes no difference whatsoever that tnow is a different number to tnow-1 because all ts hold the property of being equally valid elements within an infinite temporal set. Although our angel would have been (somehow) ‘counting forward from’ (or more correctly for) an infinite number of ts she would (at whatever t) always have infinitely many numbers left to count had she not decided to end the count. And whenever she ended the count she could justifiably label tnow because she would only ever have been counting within an infinite temporal set and never toward an infinite temporal set. The fact that a temporal regression is infinite does not preclude any tnow from occurring or being experienced if the regression extends both backward in time infinitely and continues to progress infinitely into the future. If, at any t, she provided a progress report, our angel could accurately respond that she had counted infinite past ts and had infinite future ts left to count.

Thus, claiming that it is implausible that we have arrived at tnow by traversing an infinite temporal regression (and so there must be a tbeginning) does not negate the possibility that the time elapsed since tbeginning comprises an infinite temporal set. Imagine a straight horizontal line in space that is infinite in length and divided up into millimetre marks (spatial moments, if you like). It would be absurd to sequentially shift one’s attention along the line for eternity, stop at some random point, analogously label it as spatialnow and then claim that it is impossible to have reached this mark. It is absurd because wherever you stopped would be spatialnow and there would always be infinitely many millimetre marks to the right and left. Returning to the temporal domain, Draper (2008) remarks:

“If the temporal regress of events is infinite, then the universe has never had a finite number of past events. Rather, it has always been the case that the collection of past events is infinite. Thus, if the temporal regress of events is infinite, then the temporal series of events is not an infinite collection formed by successively adding to a finite collection. Rather, it is a collection formed by successively adding to an infinite collection. And surely it is not impossible to form an infinite collection by successively adding to an already infinite collection.”

Countering evidence such as this requires a strict finitist view which negates the very notion of the existence of infinite sets of any kind:

P1: An actual infinity cannot exist
P2: An infinite temporal regression would be an actual infinity
C: Therefore: An infinite temporal regression cannot exist

Importantly, arguments such as this are not based on empirical data. As Craig (2011) states:

“The primary argument[s] that I give for the finitude of the past are philosophical arguments, based on the impossibility of the existence of an actually infinite number of things, and then secondly on the impossibility of forming an actual infinite by successive addition. So I see the scientific evidence as merely confirmatory of a conclusion that has already been reached on the basis of philosophical arguments.”

The scientific evidence Craig refers to is, of course, the A-theory of time along with his neo-Lorentzian interpretation of special relativity. Nevertheless, in the absence of empirical evidence, several thought experiments have been fashioned to illustrate the alleged absurdities that might occur if an infinite set existed. The paradox most favoured by ‘you cannot count back to infinity’ claimants is that of Hilbert’s Paradox of the Grand Hotel (Gamow, 1947). There are a number of slightly different formulations of this paradox but all convey the same idea. Hilbert’s Hotel has an infinite amount of rooms, consecutively numbered, all of which accommodate a single person and so an infinite number of guests are present. Someone arrives at the hotel looking for a room and in order to accommodate him the manager shifts everyone into the next room. So the guest in Room 1 moves to Room 2 and the guest in Room 2 moves to Room 3, ad infinitum. The new guest is then able to be accommodated in room 1. In spite of this additional guest the amount of guests in the hotel remains the same. In this case, once again, it appears that:

∞ + 1 = ∞

Suppose that a guest decides to leave and her room becomes vacant. The amount of guests in the hotel is unchanged, because:

∞ - 1 = ∞

Then, if all the guests in the odd numbered rooms decide to check out, the hotel would still have an infinite amount of guests even though an infinite number of guests would have left, because:

∞ - ∞  = ∞

However, according to Craig this situation is absurd because if all guests in the rooms numbered 6 and above were to check out, then exactly five guests would remain. So in this case:

∞ - ∞  = 5,

a tactic, as we have seen, that Craig regularly employs to demonstrate that infinities cannot exist. From Craig (2008):

“It is indisputable that if an actually infinite number of things were to exist, then we should find ourselves landed in an Alice-in-Wonderland world populated with oddities like Hilbert’s Hotel.”

But it is not indisputable. All this is merely a rehash, involving the same mathematical errors explained earlier. It is perfectly acceptable to remove a finite number of guests from the infinite set that is Hilbert’s Hotel (e.g., all guests in rooms numbered 5 and below) because we would be specifying exactly which guests were checking out. It is not acceptable, however, to subtract the undetermined number of guests in the rooms numbered 6 and above from the infinite set that is Hilbert’s Hotel because the actual number of guests checking out cannot be specified (infinity is not a number!) and so arithmetic operations such as subtraction cannot be performed. Again, Craig misrepresents the mathematical operations that are allowable with infinities. He tries to convince us that it is paradoxical that we can have two infinite sets with equal cardinalities and differing yet indeterminate numbers of elements (e.g., every room occupied vs. only the even numbered rooms occupied). He’s not alone in his confusion. This sentence comes from Wince (2009):

“If you use Cantor’s set theory to explore numbers, you get to the uncomfortable result that there are different sizes of infinity.”

But on what grounds would someone who understood mathematics "use Cantor's set theory to explore numbers" in the first place? And whether or not we get an “uncomfortable result” is wholly irrelevant to the soundness of the formal-logical system of set theory. To labour these points to destruction; infinity is not a number, it is a property held by numbers and infinite sets do not come in a ‘one size fits all’ package. They differ in size. Looking at it in terms of primality might be helpful. Some numbers have the property of primality but primality itself is not a number. It is a property held by some numbers and the amount of numbers exhibiting this property is infinite. Imagine, then, Hilbert’s Prime Hotel containing a prime number of rooms in which a prime number of guests checks out. Assuming each room checking out was only ever one of a twin prime (see e.g., Zhang, 2014) in every case the hotel will be left with a prime number of guests. Yet in every case the actual number of guests will not differ before and after check-out; both will be prime. Nothing absurd here.

Ironically, al-Ghazali (unwittingly) semi-dealt with Craig’s objections to Hilbert’s Hotel nearly a millennium ago when he postulated (see McAllister, 2011) that if God created an immortal angel every day and if the universe they lived in had existed for an infinite number of days then the result would be an infinite, co-existing population of such immortal beings. This, of course, is not really true as a growing series can never reach infinity; al-Ghazali's  knowledge of infinity derived from Aristotle. Nevertheless, al-Ghazali did realise that if God had created an infinite set of immortal angels they would be able to arrange themselves into one infinitely long line. Then, on his command, they could rearrange themselves into two or three lines each of infinitely long yet differing length. Or, he could have an infinite amount of angels congregate randomly so only a specific number of angels remained neatly in line, thereby removing a finite set from an infinite set. Al-Ghazali’s domain of an infinite population of immortal angels being shuffled around is analogous to a celestial Hilbert’s Hotel.

There are further problems for those denying outright the possibility of an infinite set. They would first need to specify those properties of an infinite set that would always be unobservable under any circumstances. They would then need to identify the discoverable, observable boundaries that would automatically disqualify any set from being an infinite set. As discussed, simply identifying one boundary condition, such as the natural number 1 or tbeginning does not necessarily preclude an infinite set. As a thought experiment Hilbert’s Hotel meets none of these requirements.

It has been argued that the real strength of Hilbert’s Hotel lies in its physicality, i.e., because it demonstrates that this kind of infinite set is physically impossible then the notion of an infinite temporal universe is defeated. However, a specific example of a physical impossible infinite set does not mean that all physical infinite sets are equally impossible. Hilbert’s Hotel is physically impossible, granted, but Oerter (2010) has formulated an astronomical scenario analogous to Hilbert’s Hotel which although physically highly improbable, is certainly not either physically or logically impossible. He starts by substituting galaxies for guests and employs ordinal numbers as identifiers of these galaxies instead of room numbers. He then asks us to imagine a spatially infinite universe containing an infinite number of galaxies (i.e., an infinite set whose elements are galaxies). From a specific location on Earth a beam, which travels endlessly, is emitted in a straight line into space. Naturally, this beam can intersect an infinite number of galaxies and at any one time we can number those galaxies which are intersected, e.g., the first galaxy the beam intersects is ‘Galaxy 1’, and the next is ‘Galaxy 2’ etc. ad infinitum.

The universe, of course, is not static, so galaxies and Earth are continually in motion. If an infinite number of galaxies are intersected by the beam at tnow it is physically possible that at tnow+1 a new galaxy has shifted position and become intersected by the beam. It is also physically possible that the new galaxy is closer to Earth than ‘Galaxy 1’ was at tnow. Thus at tnow+1 we would need to renumber all the intersected galaxies. The new galaxy becomes ‘Galaxy 1’; the previous ‘Galaxy 1’ is relabelled ‘Galaxy 2’ etc. ad infinitum. How many galaxies are intersected by the beam at tnow+1? An infinite set. Now imagine that, at tnow+2, all the even numbered galaxies are no longer intersected by the beam, but all the odd-numbered galaxies remain intersected. How many galaxies are intersected by the beam at tnow+2? An infinite set. The arrangement of galaxies along the beam at tnow+1 is directly analogous to an additional guest arriving at Hilbert’s Hotel and the arrangement of galaxies at tnow+2 is directly analogous to all the guests in even numbered rooms checking out. Craig’s argument is that because Hilbert’s Hotel is physically impossible it is also metaphysically impossible. Oerter’s (2010) analogy, on the other hand, is physically improbable, but it is not physically impossible; so on what grounds is it metaphysically impossible?

Simply treating infinity as a theoretically countable quantity and offering examples of logical contradictions in highly specific, carefully selected cases such as Hilbert’s Hotel does not provide sufficient evidence that infinite sets are forever impossible in all domains. Hilbert’s Hotel is comprised of physical objects such as rooms and guests yet there is not even a requirement that infinities be physical. Recall that Craig (2001) sometimes finds the need to differentiate between ‘things’ and ‘events’ when arguing against infinities, as did Cantor (1887) who argued for infinities both in concreto and in abstracto. Thus, if a physical object such as a universe is finite it does not preclude that there are elements within that universe, such as time, that are continuous and infinite. Furthermore, if abstract objects such as numbers are considered as things then quite obviously Hilbert’s Hotel fails to show that infinities are impossible or absurd. And the irony may be that this best candidate for an infinite set may necessarily be immune to Craig’s alleged absurdities. Hedrick (2014) proposes (somewhat tongue in cheek) ‘Hilbert’s Platonic Heaven’ in which all numbers reside in their perfect forms. He then goes on:

“Now suppose a new number comes along wanting to be added to the collection. In this case, we simply need to shift all of the other numbers around in such-and-such a way, and as a result we will have accommodated the new number. Or suppose a number decides to leave this Platonic Heaven. Nevertheless, the same number of numbers remains as before.”

Mathematics, however, does not concern itself with metaphysical speculation, nor with physical or abstract objects per se but with the possible relationships between those objects. Abstract objects cannot be moved around, either naturally or intentionally, in the same way as physical objects. So, if the hotel’s guests stayed in their rooms and didn’t bother to check out, no absurdities would arise and the infinite set of Hilbert’s Hotel can exist in metaphysical comfort. Something else that cannot check out or be shuffled around are ts. This makes Craig’s use of Hilbert’s Hotel disingenuous in another important way. Previously, the argument against an infinite series of temporal events or past moments was made on the basis that it would be impossible to reach tnow by sequential count. Astute readers, then, will have noticed that Hilbert’s Hotel does not model an infinite temporal regression and so represents a poor analogy. Hilbert’s Hotel is a theoretical example of an infinite set, existing at tnow in which all the elements (rooms, guests) must exist concurrently. However, under presentism the ts making up a temporal sequence can never exist concurrently, only successively. Even if it were possible to subtract from tnow past ts (that according to presentism do not exist) it is not at all possible (and would be a pointless exercise) to shift ts around like the guests in Hilbert’s Hotel in order to make the point that an infinite set of them cannot exist. Craig, of course, disagrees (cited in Morriston, 2012):

“You can still imagine what it would be like for a person in room one to be in room two, the person in room two can be in room four, and you can generate the same absurdities. You don’t have to go to the trouble of moving the persons physically.”

Once again, Craig is grasping at (metaphysical) straws. Yes, even if it is physically impossible for the guests of Hilbert’s Hotel to shift rooms we can still use our imagination as to what would result if they actually did shift rooms but I’m not sure what point is being made. Plenty of things are possible in our imagination without holding any physical or logical consequence. Escher’s drawings can be perfectly well imagined, for example. We can’t go to the trouble of actually shifting t’s around either, so does anything of consequence really entail if we imagine that we have done so? What the Hilbert’s Hotel paradox really demonstrates is not that an infinite set cannot exist, but that we cannot use the same arithmetical procedures on infinite sets that we can apply to finite numbers and sets.