Exterminous Hypertime

: This paper investigates ‘exterminous hypertime’, a model of time travel in which time travellers can change the past in virtue of there being two dimensions of time. This paper has three parts. Part one discusses the laws which might govern the connection between different ‘hypertimes’, showing that there are no problems with overdetermination. Part two examines a set of laws that mean changes to history take a period of hypertime to propagate through to the present. Those laws are of interest because: (i) at such worlds, a particular problem for non-Ludovician time travel (‘the multiple time travellers’ problem) is avoided; and (ii) they allow us to make sense of certain ﬁctional narratives. Part three discusses how to understand expectations and rational decision making in a world with two dimensions of time. I end with an appendix discussing how the different theories in the metaphysics of time (e.g., tensed/tenseless theories and presentism/eternalism/growing block theory) marry up with exterminous hypertime.

Imagine I time travel from 2021 to 1930 and assassinate Hitler. For this to be possible, a 'non-Ludovician' model of time travel must be true. One such model is 'exterminous hypertime' whereby the changes are possible because time has two dimensions. This paper examines that model, expanding on previous papers of mine on the topic [1] (pp. 73-90) [2]. (Note that this 'hypertemporal theory' is not the same as that discussed by Goddu [3][4][5], van Inwagen [6], and Wasserman [7] (pp. 90-94); for a discussion of the differences, see [1] (pp. 76-90) and note 4).
After an initial exposition of the theory (Section 1), this paper proceeds in three parts. Part one considers the causal links between hypertimes, what I call 'hypercausation' (Section 2). As Section 3 explains, contrary to the likes of Smith [8] (and my earlier self [1] (pp. 83-84)), exterminous hypertime has no problem concerning overdetermination. This paper does not argue that our world is actually one of exterminous hypertimeafter all, for all I know time travel is not even physically possible 1 . Rather, this paper is an exploration only of what is metaphysically possible.
You might wonder what the point of such an exploration is. In Section 4, I examine different possible laws which might govern exterminous hypertemporal worlds. To show why those laws are of interest, Section 5 discusses two applications. Firstly, time travellers in exterminous hypertemporal worlds avoid an obstacle to time travel that threatens certain alternative non-Ludovician theories. Secondly, it allows us to make sense of certain fictional stories that are prima facie metaphysically impossible. That completes the second part of this paper.
In the third and final part, which follows from the discussion about how to understand what agents in fictional stories say and do, I discuss how to understand expectation and rational decision making in a world with two dimensions of time (Section 6). I argue that agents are rational in worrying about the hyperfuture and rational to accordingly change how they act in light of what might hyperlater happen. This paper ends with an appendix on how exterminous hypertime connects with various issues in the metaphysics of time, namely tensed vs. tenseless time, the ontology of time, and the open future.
Imagine I use a time machine to travel from 2021 to 1930 and successfully kill Hitler, preventing World War II. Exterminous hypertime can make sense of this scenario if we assume: PROGRESSION: For any hypertime T j , any regular time t n , and any time traveller, x: if, at T j , x time travels from t n to t m<n then x travels to t m<n -T j+1 2,3 .
Given PROGRESSION, time travellers move progressively forwards in hypertime. They can travel back in regular time, but never back in hypertime. When I kill Hitler, I start at (e.g.,) hypertime T 1 . At T 1 , in 2021, I activate my time machine and travel back to 1930. Given PROGRESSION, I leave T 1 and travel to hypertime T 2 , arriving in the past in 1930. That is: I leave t 2021 -T 1 and travel to t 1930 -T 2 . At T 1 , history is such that Hitler survived 1930 and World War II started in 1939, i.e., Hitler is alive at t 1930 -T 1 and war breaks out at t 1939 -T 1 . At T 2 , history is different because I have assassinated Hitler. Having killed Hitler in 1930, he is dead at t 1930 -T 2 . This further prevents the outbreak of war, ensuring t 1939 -T 2 is peaceful and harmonious. See Figure 1.
This theory is not without its problems. Questions include: What is hypertime exactly? Even though space can have multiple dimensions, is it not mind-boggling to think time can likewise have multiple dimensions? [7] (p. 99) [11] (p. 9) [12] (p. 209). Are the people at other hypertimes just duplicates of people at hyperearlier hypertimes-that is, do I not just end up killing a duplicate of Hitler when I kill the 'Hitler looking person' at t 1930 -T 2 ? [13]. Is it accurate to call my making T 2 different from T 1 a case of 'changing the past'? [14,15] (p. 152) [16] (pp. 365-366) (see also [5]).
These are all good questions. However, I have dealt with them elsewhere [1] (pp. 79-90) and they are not the subject of this paper. Instead, this paper focuses on questions concerning the causal connections between the hypertimes. This theory is not without its problems. Questions include: What is hypertime exactly? Even though space can have multiple dimensions, is it not mind-boggling to think time can likewise have multiple dimensions? [7] (p. 99) [11] (p. 9) [12] (p. 209) Are the people at other hypertimes just duplicates of people at hyperearlier hypertimes-that is, do I not just end up killing a duplicate of Hitler when I kill the 'Hitler looking person' at t1930-T2? [13] Is it accurate to call my making T2 different from T1 a case of 'changing the past'? [14,15] (p. 152) [16] (pp. 365-366) (see also [5]).
These are all good questions. However, I have dealt with them elsewhere [1] (pp. 79-90) and they are not the subject of this paper. Instead, this paper focuses on questions concerning the causal connections between the hypertimes.
STABILITY ensures that the past does not change except when time travel occurs-left unmolested by time travellers, the past is always the same as it was at the immediate hyperearlier hyperinstant. For instance, at a world where no time travel takes place, then

Stability
Not every world with multiple temporal dimensions is a world of exterminous hypertime. Both philosophers ( [17,18] (p. 278) [19][20][21][22] (pp. [30][31][32][33][34] [23] (pp. [46][47][48][49] [24][25][26] (pp. [20][21][22][23][24][25][26][27] [27,28]) and physicists [29][30][31][32][33][34][35] have discussed worlds with multiple temporal dimensions, but such worlds are not anything like those discussed in this paper. More things must therefore be true to characterise worlds of exterminous hypertime. PROGRESSION is one such example. Another example would be: STABILITY: For all n and m, and for some j and k: t n -T m is qualitatively identical to t n -T m−1 except when a time traveller from t j≥n -T m−1 has time travelled to t k≤n -T m . STABILITY ensures that the past does not change except when time travel occurs-left unmolested by time travellers, the past is always the same as it was at the immediate hyperearlier hyperinstant. For instance, at a world where no time travel takes place, then STABILITY entails that t 1930 is the same at every hypertime, as is t 2021 , and, indeed, all times (see Figure 2); whereas, in a world where time travel takes place, time-hypertimes may differ, but only if they are later in time than the arrival of some time traveller from a hyperearlier hypertime. Look back at Figure 1: every instant prior to t 1930 is the same at T 1 and T 2 , and it is only after my arrival at t 1930 -T 2 that the time-hypertimes differ because I have killed Hitler. STABILITY entails that t1930 is the same at every hypertime, as is t2021, and, indeed, all times (see Figure 2); whereas, in a world where time travel takes place, time-hypertimes may differ, but only if they are later in time than the arrival of some time traveller from a hyperearlier hypertime. Look back at Figure 1: every instant prior to t1930 is the same at T1 and T2, and it is only after my arrival at t1930-T2 that the time-hypertimes differ because I have killed Hitler. STABILITY is important. In a one-dimensional temporal world, later times are suitably similar to earlier times. It is only via causal activity that the later times differ from the earlier times. STABILITY is one way of ensuring that a similar claim is true of hypertime. Given STABILITY, hyperlater hypertimes differ from hyperearlier hypertimes only because of the causal activity of time travellers.
Indeed, without STABILITY, it would be hard to recognise moving to hyperfuture hypertimes as being a form of time travel. Imagine STABILITY was false. Given it is false, historical events can play out differently at the different hypertimes, independently of how the world hyperearlier was. Imagine that: at T1, history as we know it plays out; at T2, the Big Bang is instead a Small Splutter and for all eternity the universe is an empty void; at T3, the Big Bang occurs but history plays out differently and, by 2021, squid-headed aliens rule a universe composed of planets and galaxies foreign to those found at T1. At T1, in 2021 I build a time machine. I gather together my friends and, intending to amaze them, I travel back in time twenty seconds. Rather than appearing in the past surrounded by my companions watching me step into my time machine, I appear at T2 in the empty void of space. Confused, I then decide to head back to 1930 to kill Hitler. I appear at T3, surprised to find myself at the mercy of cephalopodic monsters who live in a galaxy that did not even exist when I was younger. Fantastic things are happening when I activate my machine, but travelling back in time does not seem to be one of them. Twenty seconds ago, the past was full of material objects, so how can I have travelled back twenty seconds if the place I arrive at it is devoid of all matter? Ninety one years ago, the Weimar Republic existed, so how can I have arrived in 1930 if the Weimar Republic is instead nowhere to be found? In an unstable world, movement between hypertimes does not seem to give rise to what we would normally think of as 'time travel' [36] (pp. 379-380). Given STABIL-ITY, this problem is avoided. Hence, we should endorse STABILITY. STABILITY is important. In a one-dimensional temporal world, later times are suitably similar to earlier times. It is only via causal activity that the later times differ from the earlier times. STABILITY is one way of ensuring that a similar claim is true of hypertime. Given STABILITY, hyperlater hypertimes differ from hyperearlier hypertimes only because of the causal activity of time travellers.
Indeed, without STABILITY, it would be hard to recognise moving to hyperfuture hypertimes as being a form of time travel. Imagine STABILITY was false. Given it is false, historical events can play out differently at the different hypertimes, independently of how the world hyperearlier was. Imagine that: at T 1 , history as we know it plays out; at T 2 , the Big Bang is instead a Small Splutter and for all eternity the universe is an empty void; at T 3 , the Big Bang occurs but history plays out differently and, by 2021, squid-headed aliens rule a universe composed of planets and galaxies foreign to those found at T 1 . At T 1 , in 2021 I build a time machine. I gather together my friends and, intending to amaze them, I travel back in time twenty seconds. Rather than appearing in the past surrounded by my companions watching me step into my time machine, I appear at T 2 in the empty void of space. Confused, I then decide to head back to 1930 to kill Hitler. I appear at T 3 , surprised to find myself at the mercy of cephalopodic monsters who live in a galaxy that did not even exist when I was younger. Fantastic things are happening when I activate my machine, but travelling back in time does not seem to be one of them. Twenty seconds ago, the past was full of material objects, so how can I have travelled back twenty seconds if the place I arrive at it is devoid of all matter? Ninety one years ago, the Weimar Republic existed, so how can I have arrived in 1930 if the Weimar Republic is instead nowhere to be found? In an unstable world, movement between hypertimes does not seem to give rise to what we would normally think of as 'time travel' [36] (pp. 379-380). Given STABILITY, this problem is avoided. Hence, we should endorse STABILITY.

Explaining Stability
Whilst STABILITY might be inexplicably true, it is less strange to think that certain laws of nature will explain it. The rest of this paper assumes that STABILITY is true because certain hyperearlier time-hypertimes stand in a causal relation to other hyperlater timehypertimes. Dub that causal connection 'hypercausation'. (Note that hypercausation is not a 'metaphysically different' type of causation; the introduction of the term is solely for the purpose of exposition.) Given such hypercausal connections, we can explain STABILITY. When I travel back to t 1930 -T 2 , that time-hypertime is such that Hitler is alive (and the Weimar Republic is running Germany, etc.) because the immediately hyperprior time-hypertime (i.e., t 1930 -T 1 ) hypercausally influences t 1930 -T 2 in a way that makes it almost entirely qualitatively identical to t 1930 -T 1 , differing only with regard to my arrival in a time machine.

Overdetermination
Some people have worried that there being hypercausal connections between timehypertimes gives rise to a problem of systematic overdetermination. Section 3.1 explains the worry. Section 3.2 argues that, even if there were systematic overdetermination, it is not worrying. Section 3.3 argues that, in any case, we need not believe that there is such systematic overdetermination in the first place.

The Threat of Overdetermination
An event may be a 'partial' cause of its effect or a 'whole' cause. For example, if three people lift a table then each person's lifting is a partial cause of the table being lifted. It is only their collective action that wholly causes the lifting.
Given that causation is transitive, effects can have multiple whole causes. If e 1 wholly causes e 2 which in turn wholly causes e 3 , then e 1 and e 2 are both whole causes of e 3 . Such cases are routine and occur all of the time. But some cases of effects having multiple whole causes are weirder. In the routine cases (e.g., of e 1 and e 2 causing e 3 ) we can trace a single chain of causes threading all three events. In the weird cases, an effect has multiple whole causes from separate causal chains. For instance, imagine two assassins independently, yet simultaneously, kill a target, e.g., an assassin's bullet strikes the victim in the heart at exactly the same moment that the second assassin's poison causes a terminal encephalopathic event. This is weird in a way that the chain of e 1 , e 2 , and e 3 is not weird. Call the weird cases 'overdetermination cases'.
Whilst weird, overdetermination cases are possible. Yet, we tend to think they do not regularly take place, i.e., we think systematic overdetermination is not taking place. (As a forewarning, Section 3.2 questions exactly this assumption). If STABILITY is explained hypercausally, then there is at least a prima facie reason to think systematic overdetermination will be a problem-a worry previously noted by both Smith [8] (pp. 691-692) and myself [1] (pp. 83-84).
To see why we might suspect that there is systematic overdetermination, imagine a world at which there is no time travel and consider the 1804 liberation of Haiti. At t 1804 -T 1 , the self-liberated slaves have finally overthrown French rule. Assuming there is no hypertime prior to T 1 , t 1804 -T 1 is the way it is solely because, at earlier (regular!) times at T 1 , the slaves did things to cause the liberation at t 1804 -T 1 . Since t 1804 -T 2 is qualitatively identical to t 1804 -T 1 (since all earlier times at T 2 are qualitatively identical to the earlier times from T 1 ), the liberation of Haiti at t 1804 -T 2 is likewise wholly caused by the actions of slaves from earlier times at T 2 . However, given the hypercausal explanation of STABILITY, the liberation of Haiti at t 1804 -T 1 also wholly causes the liberation of Haiti at t 1804 -T 2 . We have a case of overdetermination! And what goes for the liberation of Haiti goes for every event at every hypertime other than T 1 . So, at such a world, there is systematic overdetermination. Given the assumption that there is no systematic overdetermination, exterminous hypertime would have a problem 4 .

There Is No Problem of Overdetermination
However, this is much too quick. Firstly, as already noted, overdetermination cases are possible, even if they are unlikely. An anonymous referee (for my monograph [1], not this paper) correctly pointed out that these worries about systematic overdetermination only show that we probably do not actually live in an exterminous hypertemporal world. Since this paper is interested only in what is possible-and readily admits that we do not actually Secondly, if the problem of systematic overdetermination is that it is unlikely, then it is not a problem in any case 5 . When we say systematic overdetermination is 'unlikely', the type of likelihood in question is 'objective chance'. Objective chance is intimately tied up with the laws of nature. However, at a world of exterminous hypertime, the systematic overdetermination is mandated by the laws of nature (see Section 4 for examples of such mandating laws). Since it is mandated, the overdetermination is therefore not unlikely. Indeed, since the laws mandate it, it always has a chance of occurring equal to 1-the overdetermination is as likely as it could possibly be. The liberation of Haiti at t 1804 -T 2 may be wholly caused by two distinct events, but it is no coincidence! Thus, even if there is systematic overdetermination, it is not of the problematic kind 6 .
One objection might be that the laws themselves are unlikely, i.e., it is unlikely to find oneself in a world with laws mandating systematic overdetermination. However, whilst debating the objective chance of an event given certain laws is straightforward, discussing the likelihood of the laws themselves is not. We are immediately dragged into the quagmire of issues involved in the fine-tuning debate (see, e.g., [37,38] (pp. 147-154) and [39]). Since this objection quickly becomes stuck on such issues, I set it aside.

Exterminous Hypertemporal Worlds Need Not Be Worlds of Systematic Overdetermination
Section 3.2 argued that if an exterminous hypertemporal world contained systematic overdetermination then we need not worry. This section argues that we do not even need to go that far, because we need not think that exterminous hypertemporal worlds feature systematic overdetermination in the first place.
Introduce another chance function alongside the objective chance function. Call it the 'counterhypercausal chance function'-or 'chance chc ', for short. The chance chc of an event occurring (at some time t) is the objective chance it would have of occurring (at t) if only nothing hypercausally interfered with it. As an example, consider the following case: Case one: Moscovium decay with no time travel. At t 2021 -T 1 a moscovium atom has 0.5 chance of decaying a few millseconds later. And, indeed, it decays. Further, no time travel takes place at this world. At t 2021 -T 2 , consider the same atom. Since no time travel has occurred, it has a chance chc of decaying equal to 0.5. However, given Stability it must decay, so at t 2021 -T 2 it has a chance of decaying equal to 1 7 . This in place, turn back to how a world of exterminous hypertime can avoid systematic overdetermination. The key is to assume that a time-hypertime hypercausally influences another time-hypertime only regarding probabilistic events, i.e., events not determined to happen. In such cases, the hypercausal interaction either ensures that the event happens (if it occurred at the hyperprevious time-hypertime) or ensures that the event does not happen (if it did not occur).
Return to the Haiti example. Imagine that the laws of nature are such that the actions of the self-liberated slaves at t 1803 -T 2 determine most of what goes on at t 1804 -T 2 , i.e., most events at t 1804 -T 2 are non-probabilistic. At t 1803 -T 2 , the remaining events at t 1804 -T 2 have a chance chc of occurring between 0 and 1. It is those remaining events (and only those events) that are influenced by hypercausation. Thus, the earlier events of t 1803 -T 2 are only a partial, not whole, cause of the liberation of Haiti. Similarly, the events of t 1804 -T 1 are only a partial, not whole, cause of the liberation at t 1804 -T 2 . It is only the conjunction of these things that is the whole cause. Therefore, there is no overdetermination (and so no systematic overdetermination) because overdetermination involves distinct whole causes bringing about some effect, not distinct partial causes bringing about some effect. Problem solved.

Hypercausal Laws
This section examines the different hypercausal laws that would allow for a world of exterminous hypertime. Sections 2 and 3 have argued that all such laws must entail STABILITY; however, there are other dimensions along which they may vary.

Local vs. Global
The first possible hypercausal law for consideration is: GLOBAL EFFECT: For any event, e, and all j, k, m and n, if e occurred at t m -T n−1 then whether or not it occurs at hyperlater hypertimes depends upon whether a time traveller has (or has not) time travelled from t j≥m -T n−1 to t k≤m -T n : If a time traveller has not time travelled from t j≥m -T n−1 to t k≤m -T n then e occurs iff it occurred at t m -T n−1 (except where t m -T n−1 does not exist-i.e., where T n is the first hypertime-in which case e has a chance of occurring equal to its chance chc of occurring).
If a time traveller has time travelled from t j≥m -T n−1 to t k≤m -T n then e may or may not have occurred. Its occurrence will be probabilistically governed; the chance (at t l -T n , for all l < m) of e occurring is equal to its chance chc of occurring 8 .
Given GLOBAL EFFECT, time-hypertimes are qualitatively identical to their immediately hyperprevious-yet-simultaneous time-hypertimes unless someone has time travelled to that time or earlier. And if a time traveller has travelled to that time or earlier, then all hypercausal ties are severed-from that point forwards, 'all bets are off' when it comes to how the future can play out, with no hypercausation affecting matters.
To see the upshot of GLOBAL EFFECT, consider four example cases. Start with Case One from above. In Case One, the moscovium atom decayed just after t 2021 -T 1 . Since there is no time travel in Case One, GLOBAL EFFECT entails that it also decays just after t 2021 -T 2 (Which is just as it should be, because this is just what STABILITY requires).
Next, consider: Case Two: Moscovium decay with time travel and interaction. As per Case One, but a time traveller from t 3000 -T 1 has time travelled to T 2 to an instant a few hours earlier than t 2021 . There, the time traveller accelerates the moscovium atom to close to the speed of light, lowering its chance chc of decaying from 0.5 to 0.1.
Given GLOBAL EFFECT, because the time traveller has arrived at a time earlier than t 2021 -T 2 , the chance (at t 2021 -T 2 ) of the atom decaying a few milliseconds after t 2021 -T 2 ends up being independent of the hyperprior hypertime, i.e., the chance is 0.1 (rather than, as in Case One, a chance of 1). So, it may very well not decay, quite unlike with Case One.
Case Three: Moscovium decay with time travel but no interaction. As per Case Two, but the time traveller does not interact with the moscovium atom (for instance, because they arrive on the other side of the world).
Even though the time traveller does not interact with the moscovium atom, given GLOBAL EFFECT, the hyperearlier time-hypertime nevertheless no longer hypercausally affects t 2021 -T 2 . That means that (at t 2021 -T 2 ) the chance of the atom decaying is 0.5; at t 2021 -T 2 the moscovium is as liable to decay as not.
Finally, consider: Given GLOBAL EFFECT, you will almost certainly find history has played out differently. Imagine that an ancestor of Aristotle from 40,999 BC did not die of cancer before reaching child-bearing age at T 1 because of an unlikely stochastic event. At T 2 , your arrival in Andromeda means that the ancestor's survival is again put in doubt. Imagine that what is most likely happens and the ancestor does die. Resultantly, Alexander the Great fails to learn the correct lessons from Aristotle, never defeats the Acaehmenid Empire, and history is radically changed. Even though you spent only a minute or two in the Andromeda galaxy, when you return through the time portal to 2021, you find that the world is now under the control of the NeoPersian Hegemony. Everyone you know has gone; your ancestors of the previous thousands of years never existed; Earth's history is nothing like that which you remember.
So, as Case Three and Case Four make clear, GLOBAL EFFECT means that any time travel interferes with history on a universal scale. Even things far removed from a time traveller can be affected by their act of time travel.
GLOBAL EFFECT is not the only possible law that there might be at a world of exterminous hypertime. Consider: LOCAL EFFECT: For any event e, and all m, n and k, whether an event e occurs at t m -T n depends upon whether someone has or has not time travelled from t j≥m -T n−1 to t k≤m -T n : If a time traveller has not time travelled from t j≥m -T n−1 to t k≤m -T n then e occurs iff it occurred at t m -T n−1 (except where t m -T n−1 does not exist-i.e., where T n is the first hypertime-in which case e has a chance of occurring equal to its chance chc of occurring).
If some time traveller has time travelled from t j≥m -T n−1 to t k≤m -T n then e's occurring or not depends upon whether some time traveller has causally interacted/influenced any factors governing e's occurring or not. If some such factor has been influenced by a time traveller, then e's occurring is now independent of what went on at the hyperearlier timehypertime; e's chance of occurring is equal to its chance chc of occurring. If some such factor has not been influenced by a time traveller (e.g., the time traveller arrived too spatiotemporally distant to influence e's coming about or not) then e occurs iff it occurred at t m -T n−1 .
To see the difference between LOCAL EFFECT and GLOBAL EFFECT, reconsider the cases. Cases One and Two play out exactly as they do given GLOBAL EFFECT. However, in Cases Three and Four, LOCAL EFFECT and GLOBAL EFFECT diverge. Consider Case Three. Arriving in the past, the time traveller does not causally interact with the moscovium atom. Given the second clause of LOCAL EFFECT, e must therefore occur, contrary to what was true given GLOBAL EFFECT. Similar thoughts apply to Case Four. Arriving in Andromeda, the time travellers cannot causally influence events in the Milky Way. Hence, events in Earth's past are still held tight in the grip of hypercausation, playing out exactly as they did at T 1 . Given LOCAL EFFECT, when the time travellers return to the future, they find everything exactly as it was before.
So exterminous hypertime might allow for time travel having a local or global influence. If it is global, then as soon as a time traveller arrives in the past, the entire universe's future is unshackled from slavishly following the course of the previous hypertime. If it is local, only those events that the time traveller interacts with can turn out differently from the previous hypertime.

Propagative Laws
The local/global dimension is not the only dimension which the laws of nature can vary over. Both LOCAL EFFECT and GLOBAL EFFECT have it that when the future is changed, it is changed 'hyperinstantaneously', i.e., when a time traveller intercedes in history, the very next hyperinstant is such that the entire future reflects the changes that have been made. However, we might instead believe that the changes to history 'propagate', taking a period of hypertime to change the future. That is: A time traveller changes the past, but at Philosophies 2021, 6, 85 9 of 22 the next hyperinstant those changes are only reflected at a tiny bit of history; as hypertime moves on, more and more of history changes to reflect the time traveller's actions.
As an initial example, imagine that I go back and kill Hitler. If changes propagate, then the next hyperinstant of hypertime will not reflect that change. Even though, at the next hyperinstant, I killed Hitler in 1930, he is still alive in 1939 and starting World War II. We have to wait until a hyperlater hypertime for history to have changed such that Hitler is now dead in 1939 and World War II does not occur. That is: The changes caused by time travellers propagate over a hypertemporal period.
This subsection details the laws required to make exterminous hypertime 'propagative'. (I will assume throughout that time travel has a global influence, similar to GLOBAL EFFECT, though I see no reason to think that a local version, riffing off of LOCAL EFFECT, could not instead be constructed).
First, we must say that there is a property of 'hyper-resilience'. It is had (or not) by time-hypertimes. Three hypercausal laws govern hyper-resilience: 1.
For all j and k, t j -T k is not hyper-resilient iff something tried to causally influence t j -T k−1 that did not try to causally influence t j -T k−2 ; otherwise, t j -T k is hyper-resilient.

2.
For all j and k, if t j -T k is hyper-resilient then events occur at it iff they occurred at t j -T k−1 . This is the case even if something (e.g., a time traveller coming from the previous hypertime) tries to causally influence them to be otherwise, i.e., hyperresilience 'trumps' all other forms of causal force and activity. 3.
If a time-hypertime is not hyper-resilient then there is no hypercausal influence between its hyperprior time-hypertime and it. What goes on at such time-hypertimes is governed only by what went on at hypersimultaneous earlier times in combination with the causal influence of any time traveller who has arrived from the hyperprevious hypertime.
Call the conjunction of these claims GLOBAL PROPAGATION. If GLOBAL PROPAGATION is a law of nature, then changes to history take a period of hypertime to issue forth through to the present. Consider how killing Hitler would pan out given GLOBAL PROPAGATION. Figure 3 depicts what would happen: time-hypertimes are depicted as the larger boxes; whether or not they are hyper-resilient is represented by a tic/cross in the smaller box, where a tic represents that the time-hypertime is hyper-resilient and a cross indicates that it is not. In accord with law (1), every time-hypertime at T 1 is hyper-resilient. At T 1 I attempt to travel from T 1 to 1930 to kill Hitler. Given law (1), every time-hypertime at T 2 is also hyper-resilient. So, given (2), every time-hypertime at T 2 is the same as it was at T 1 ; since I was not in 1930 at T 1 then, even though I try to travel to t 1930 -T 2 , the hyper-resilience of t 1930 -T 2 trumps my attempt; thus, I do not appear at t 1930 -T 2 . However, no time traveller tried to causally interact with t 1930 -T 1 by trying to arrive there, whilst a time traveller does try and causally interact with t 1930 -T 2 by trying to arrive there. So, given (1), t 1930 -T 3 is not hyper-resilient. Since it is not hyper-resilient then, given (3), events at t 1930 -T 3 can play out differently than t 1930 -T 2 . So, whilst I do not arrive at t 1930 -T 2 from t 1930 -T 1 , I do arrive at t 1930 -T 3 (having arrived there from t 2021 -T 2 ). However, I fail to persist any further at T 3 . This is because the next time-hypertime in T 3 , call it t 1930+δ -T 3 , is hyper-resilient; given (2), it must be qualitatively identical to t 1930+δ -T 2 ; since I did not exist at t 1930+δ -T 2 , then I do not exist at t 1930+δ -T 3 either. However, because something tries to causally affect t 1930+δ -T 3 that did not try to causally affect t 1930+δ -T 2 , t 1930+δ -T 4 is not hyper-resilient. So, I arrive at t 1930 -T 4 and can persist to t 1930+δ -T 4 . I cannot, however, persist to the instant after that. At T 5 , the same reasoning means that I can manage to persist to the instant after, but no further. At T 6 , similar reasoning dictates that I manage to persist one instant more. And so on. Ultimately, we arrive at a hyperinstant, T ω , where my interaction with the past has hyperfinally influenced the future, stopping World War II and bringing about a utopian future 9 .

Applications of Global Propagation
You might wonder why I care about worlds at which GLOBAL PROPAGATION is true. After all, lots of weird things are metaphysically possible, so why be interested in this weird possibility? This section explains two interesting features of propagative worlds.

The Multiple Time Travellers Problem
Some theories that allow for the past to change face the 'Multiple Time Travellers Problem'. Versions of this problem have been discussed by Hudson and Wasserman [9] (p. 42) (see also [7] (pp. 96n73) as well as myself [36,40]. This section details that problem and then explains how GLOBAL PROPAGATION avoids it.
Start by imagining that changes to the past do not propagate through to the future and that GLOBAL EFFECT is true. At t2021-T1 I decide to travel to the past to kill Hitler. However, Malcolm-who is at t2022-T1--also uses a time machine. Malcolm travels to 50,000 BC (at T2), kills Aristotle's ancestor, and brings about the NeoPersian Hegemony. Given PRO-GRESSION, we both arrive at T2 (albeit at different times: me arriving in 1930 and Malcolm arriving in 50,000 BC). At t1930-T2, I arrive and expect to find Hitler and Germany but, because of Malcolm's interference, I instead find that Germany, Hitler, and all of Hitler's ancestors for a thousand generations never existed. As discussed in Section 2.1, if a putative time traveller arrives in the past and does not find such things, that is problematic because they do not appear to really have a time machine. So we have a problem. Call it the 'Multiple Time Travellers Problem': At a world where GLOBAL EFFECT is true, if multiple time travellers go back to the past, only those time travellers arriving at the earliest moment manage to succeed; everyone else arrives somewhere which is not properly seen as being 'their past' and so fails to travel in time.
GLOBAL PROPAGATION avoids this problem 10 . In a world where GLOBAL PROPAGATION is true, neither myself nor Malcolm arrive at T2. However, we do arrive at T3, if only for an instant. Malcolm's actions at t−50,000-T3 do not have any causal effect on events in the (regular) future, so when I arrive at t1930-T3, it is exactly as I expect it to be, i.e., with Germany,

Applications of Global Propagation
You might wonder why I care about worlds at which GLOBAL PROPAGATION is true. After all, lots of weird things are metaphysically possible, so why be interested in this weird possibility? This section explains two interesting features of propagative worlds.

The Multiple Time Travellers Problem
Some theories that allow for the past to change face the 'Multiple Time Travellers Problem'. Versions of this problem have been discussed by Hudson and Wasserman [9] (p. 42) (see also [7] (p. 96 n. 73) as well as myself [36,40]. This section details that problem and then explains how GLOBAL PROPAGATION avoids it.
Start by imagining that changes to the past do not propagate through to the future and that GLOBAL EFFECT is true. At t 2021 -T 1 I decide to travel to the past to kill Hitler. However, Malcolm-who is at t 2022 -T 1 -also uses a time machine. Malcolm travels to 50,000 BC (at T 2 ), kills Aristotle's ancestor, and brings about the NeoPersian Hegemony. Given PROGRESSION, we both arrive at T 2 (albeit at different times: me arriving in 1930 and Malcolm arriving in 50,000 BC). At t 1930 -T 2 , I arrive and expect to find Hitler and Germany but, because of Malcolm's interference, I instead find that Germany, Hitler, and all of Hitler's ancestors for a thousand generations never existed. As discussed in Section 2.1, if a putative time traveller arrives in the past and does not find such things, that is problematic because they do not appear to really have a time machine. So we have a problem. Call it the 'Multiple Time Travellers Problem': At a world where GLOBAL EFFECT is true, if multiple time travellers go back to the past, only those time travellers arriving at the earliest moment manage to succeed; everyone else arrives somewhere which is not properly seen as being 'their past' and so fails to travel in time.
GLOBAL PROPAGATION avoids this problem 10 . In a world where GLOBAL PROPAGATION is true, neither myself nor Malcolm arrive at T 2 . However, we do arrive at T 3 , if only for an instant. Malcolm's actions at t −50,000 -T 3 do not have any causal effect on events in the (regular) future, so when I arrive at t 1930 -T 3 , it is exactly as I expect it to be, i.e., with Germany, the Weimar Republic, and Hitler all in existence. So I do arrive somewhere properly called 'the past'. Thus, I manage to travel in time even though Malcolm also manages to travel in time. As hypertime moves on, more and more of the past will be affected by our changes. For instance, there will be a hypertime, T ω , at which ten years of time apiece have been 'affected'. At T ω : 1940 AD onwards is exactly as it originally was at T 1 , i.e., Hitler is alive, World War II is taking place, etc.
If we hyperwait long enough, then we get to a hyperlater hypertime, T ω , where Malcolm's interaction with the past finally catches up to 1930. At T ω , I arrive in 1930 (having come from t 2021 -T ω −1 ) and am bewildered to find Germany has gone, the Weimar Republic does not exist, Hitler was never born, etc., and all such things have been replaced by the NeoPersian Hegemony. In that case I have not time travelled from T ω −1 to T ω . Nevertheless, GLOBAL PROPAGATION salvages the possibility of me managing to time travel back to 1930 for at least some period of hypertime (namely all of the hypertimes between T 1 and T ω ).
Thus, we have at least one reason to take note of worlds at which GLOBAL PROPAGATION is true: At such worlds, multiple time travellers can go back in time to different destinations.

The Metaphysical Possibility of Time Travel Fictions
Recognising the possibility of propagative worlds also helps with determining which time travel stories are/are not metaphysically possible. This was Lewis's task in his famous contribution to the philosophy of time travel [15]. He demonstrates that various time travel stories are possible (such as stories by Heinlein). Lewis was clear, though, that not any old time travel story was thereby possible and it remains interesting to ask of other stories whether they are possible or not. (Elsewhere [2], I have already discussed the possibility of other fictions, not covered by Lewis's theory, nevertheless being possible-this paper furthers that investigation).
GLOBAL PROPAGATION allows us to capture the metaphysical possibility of it 'taking time' for changes to the past to 'reach' the present. This trope appears in a variety of stories, such as Baxter's Timelike Infinity [41], Mark Millar's Chrononauts [42], Robert J. Sawyer's 'On the Surface' [43], Star Trek: Enterprise's 'Carpenter Street' [44], and Red Dwarf 's 'Timeslides' [45], to name just a few. However, as this section explains, it turns out that only some such fictions turn out to be possible given GLOBAL PROPAGATION.
Consider a narrative that GLOBAL PROPAGATION does not bear out the possibility of: Red Dwarf 's 'Timeslides' [45]. Because of misadventure, the protagonist, Lister, is the last man alive in 3,000,000 AD. Finding a time machine, he returns to when he was a young adult (say, 2174 AD), changing history so he avoids his fate. He then returns to the future. At first, nothing changes. Asking why the past has not altered, Lister is informed that 'it'll take a few seconds for the timelines to sort themselves out'. Sure enough, a few seconds later the changes to history catch up to the present and Lister ceases to be; history is now different, with him having now lived a successful life in the past.
Whilst 'Timeslides' involves changes to the past taking a while to reach the present, its narrative is nevertheless not possible at a propagative world of exterminous hypertime. See Figure 4. At the earliest hypertime, T 1 , the past is such that Lister lives an unsuccessful life and goes back in time to change this. At T ω , the past up until 2175 AD has changed. At T ω , still more of the past has altered; Lister is now rich and famous in 2180 AD, rather than poor and unknown. At T ω" , even more of the past has changed and history is different up until 3000 AD. And so on, until we get to a hypertime, T ω , at which all of history reflects the changes Lister made. However, consider what happens at each hypertime. At T ω , Lister returns to the future and . . . keeps on living. He does not vanish or fade away. Similarly for T ω" and all other hypertimes hyperprior to T ω . Additionally, at T ω there is no Lister going back in the past in the first place because he lived a successful life and never got trapped in deep space. So there is no hypertime at which Lister stands around waiting to be washed away by the changes he made to the past, only to have those changes 'catch up with him'. 'Timeslides' is not the sort of narrative which GLOBAL PROPAGATION bears out 11 .
Tω′, still more of the past has altered; Lister is now rich and famous in 2180 AD, rather than poor and unknown. At Tω″, even more of the past has changed and history is different up until 3000 AD. And so on, until we get to a hypertime, Tω″′, at which all of history reflects the changes Lister made. However, consider what happens at each hypertime. At Tω′, Lister returns to the future and… keeps on living. He does not vanish or fade away. Similarly for Tω″ and all other hypertimes hyperprior to Tω″′. Additionally, at Tω″′ there is no Lister going back in the past in the first place because he lived a successful life and never got trapped in deep space. So there is no hypertime at which Lister stands around waiting to be washed away by the changes he made to the past, only to have those changes 'catch up with him'. 'Timeslides' is not the sort of narrative which GLOBAL PROPAGATION bears out 11 . However, that does not mean that propagative worlds cannot capture any narrative that involves changes to history taking time to ripple forwards. Given a suitably liberal interpretation, some such fictions are possible given GLOBAL PROPAGATION. Consider two examples.
The first example is Baxter's Timelike Infinity [41]. Rebels have headed into the past to change history. One character, Jasoft Parz, wonders how their changes will affect him. He worries that there are no 'estimates of the rate -in subjective terms -at which the disruption was approaching, emerging from the past as if from the depths of some dismal sea.' [41] (pp. 182-83) Whilst we might interpret Parz's worry as reflecting the idea that changes to history are propagating through regular time, there is a legitimate interpretation whereby his worry is well-founded because the disruption is approaching hypertemporally. At the current hypertime, Parz will be unaffected. But Parz knows-and is concerned about-what hyperwill happen to him, and whether the changes to the past will affect him at some hyperlater hypertime (Such an interpretation requires it to be rational for Parz to worry about what will happen to him at hyperlater hypertimes; I pick this issue up in Section 6.).
The second example is Millar and Murphy's Chrononauts [42]. The characters worry that changes in the past are coming towards them, threatening to radically alter how things are. One character asks 'How long have we got?', asking how much longer they have, in the present, before the changes wipe them out.
It is natural to read this as a worry about changes to the past propagating through regular time. However, again, there is a legitimate interpretation whereby the changes are However, that does not mean that propagative worlds cannot capture any narrative that involves changes to history taking time to ripple forwards. Given a suitably liberal interpretation, some such fictions are possible given GLOBAL PROPAGATION. Consider two examples.
The first example is Baxter's Timelike Infinity [41]. Rebels have headed into the past to change history. One character, Jasoft Parz, wonders how their changes will affect him. He worries that there are no 'estimates of the rate-in subjective terms-at which the disruption was approaching, emerging from the past as if from the depths of some dismal sea' [41] (pp. 182-83). Whilst we might interpret Parz's worry as reflecting the idea that changes to history are propagating through regular time, there is a legitimate interpretation whereby his worry is well-founded because the disruption is approaching hypertemporally. At the current hypertime, Parz will be unaffected. But Parz knows-and is concerned about-what hyperwill happen to him, and whether the changes to the past will affect him at some hyperlater hypertime (Such an interpretation requires it to be rational for Parz to worry about what will happen to him at hyperlater hypertimes; I pick this issue up in Section 6).
The second example is Millar and Murphy's Chrononauts [42]. The characters worry that changes in the past are coming towards them, threatening to radically alter how things are. One character asks 'How long have we got?', asking how much longer they have, in the present, before the changes wipe them out.
It is natural to read this as a worry about changes to the past propagating through regular time. However, again, there is a legitimate interpretation whereby the changes are propagating through hypertime instead. Given that interpretation, we read the question 'How long have we got?' as concerning what period of hypertime it takes for the changes to reach the present. Given that interpretation, the comic still makes sense. And-assuming it is rational to worry about disasters in one's hyperfuture-there is little reason not to interpret the strip that way.

Hyperexpectations
Section 5.2 argued that certain fictions are possible given GLOBAL PROPAGATION and assumed that it is rational to worry about what happens to you in the hyperfuture. This section argues for that assumption by discussing connected issues concerning persistence, expectation, and rational decision making in a world with two temporal dimensions.

Personal Hyperfutures
Set aside the question of Jasoft Parz and the Chrononauts for the time being. Consider instead the following case: Case Five. Survival in a propagative world. Global Propagation is true. I am wondering whether to time travel from t 2021 -T 1 to t 1930 -T 2 , i.e., I am wondering whether to time travel to 1930 AD. If I did, then the world would be as depicted in Figure 3. But-as Figure 3 makes clear-when I leave t 2021 -T 1 , I fail to appear at t 1930 -T 2 .
The worry is this: Since I fail to appear at t 1930 -T 2 , am I not effectively killing myself by travelling in time?
I believe this worry to be misplaced. To see why, consider a world with just a single dimension of time where I travel back in time from 2021 to the 1930s, intending to retire and never to return. We might worry that this effectively kills me, since-at 2021-I stop existing in the future. However, there is a well-rehearsed objection to this, relying on the distinction between me having an 'external future' and a 'personal future'-if you are unfamiliar with the distinction between external time and personal time, see [15] and [1] (pp. [42][43][44][45]. In that I do not exist at 2022, 2023, 2024, etc., I do not have an external future. However, I still have a personal future lying in the 1930s. It is my having a personal future that is important to rational decision making, hence why my time travelling retirement plan is rational. Whilst it is true that, in Case Five, when I activate my time machine at t 2021 -T 1 I have neither an external future nor a personal future, this line of thinking nevertheless helps. Given exterminous hypertime introduces two dimensions of external time, we should do the same for personal time: I have two types of personal history, my regular personal history and my personal hyperhistory. Just as my future stages are those stages which I immanently causally influence, my hyperfuture stages are those that I immanently causally influence by how I am now and by what I hyperpresently do. So in Case Five, there is a stage of me in the hyperfuture (e.g., at t 1930 -T 3 , t 1930 -T 3 , t 1930 -T 4 . . . t 1930 -T ω . . . ) that I have immanently causally influenced, thus it is one of my personal hyperfuture stages 12 . So whilst it is true that I may have no more personal future, I nevertheless have a personal hyperfuture. And it is in that personal hyperfuture that I end up having a personal future where I activate the time machine and go on to survive in the 1930s.
An event lying in one's personal hyperfuture can be action guiding; my still having a post-time-machine-activation personal future at some hyperfuture time means that it is rational to travel back in time in Case Five. This is because it is rational to care about one's personal hyperfuture in the same way that one cares about one's personal future-your personal hyperfuture stages are stages of you, after all. Since they are stages you can still causally influence-in this case, hypercausally influence-it is natural to worry about what they hyperwill be like. Compare: In a world of one-dimensional time, it is rational for me to presently make sacrifices in order to benefit my future self. It is, for instance, rational to refrain from enjoying my delicious piña colada in order to flee a nearby erupting volcano, thus ensuring that my future self can drink many more cocktails later on. In a world of exterminous hypertime, the analogue applies and I should be motivated to make sacrifices to benefit my hyperfuture self, ensuring that my hyperfuture self has a personal history that lies in the (regular) past. Thus, if you want to travel back to the 1930s, it is rational to get into use the time machine in Case Five.
Indeed, we might be more motivated to worry about our hyperfutures than our regular futures. My biology is such that I am likely to live in the region of eighty or so years. However, given how hypercausation functions, I could potentially persist in the hypertemporal direction for hypereternity. That said, time travellers might end up in a situation where they have to make hard choices. Imagine it is T 1 and a supervillain plans to go back in time and destroy all life on the planet from 50,000 BC onwards. If that happens at T 2 , you will have existed only hypermomentarily. But imagine the supervillain makes you an offer: In return for not foiling his scheme, the villain promises you an elixir giving you an extended life span, with excellent health throughout, for four hundred years. What do you do? Which is better? A longer 'temporally regular' existence but at only one hypertime? Or a shorter 'regular' existence but at a potentially infinite number of hypertimes?

Hyperexpectations
Having bifurcated personal time, we should similarly bifurcate the notion of expectation. Consider the following case.
Case Six. The Retrorifle Case. I live in a world where GLOBAL EFFECT is true. Rather than time travelling to 1930 in order to kill Hitler I instead fire a bullet from my 'retrorifle'. Such a gun fires a bullet back in time from t 2021 -T 1 to t 1930 -T 2 , shooting Hitler dead. However, I myself do not travel in time; I remain at T 1 .
In Case Six, what should I expect to happen when I fire the gun? At first glance, I should expect nothing to happen when I pull the trigger of the gun since my personal future lies wholly within T 1 and my retrorifle only causes history to be different at hyperlater hypertimes. So, we might think that I should not expect my firing the rifle to change the world around me.
There is something both substantially right about this, but also substantially wrong. Whilst my personal future will not reflect Hitler having being assassinated and World War II not happening, in my personal hyperfuture the world's history is such that Hitler is dead. Having bifurcated personal time, we should do the same for expectation: Whilst I should not expect anything to change when I pull the retrorifle's trigger, I should nevertheless hyperexpect things to change. All that said, return to Jasoft Parz and the rationality of his worries. Given what I have said about expectations and hyperexpectations, Parz's worries are rational because he is correct to hyperexpect that he, and the world around him, will be different in the hyperfuture. Indeed, given the changes made to the past, he might not even exist at all once the changes hyperreach him-his chance at existing for the rest of hypereternity is in jeopardy! So, as the fiction depicts, Parz's worries are reasonable. Similar thinking applies to Chrononauts. As the characters stand chatting, they should not expect any changes to the past to hyperpresently affect them. However, they are nevertheless correct to worry-and take action in light of those worries-because of what they hyperexpect to happen.
So, GLOBAL PROPAGATION can help make sense of the metaphysical possibility of certain fictional narratives which we might previously have thought to be impossible.

Conclusions
It is interesting to see how exterminous hypertime can (i) avoid problems facing certain other models of time travel and (ii) expand the range of time travel fictions that turn out to be metaphysically possible. I stress again, though, that this is where one's interest should likely stop, for there is no reason to think our world is a world of exterminous hypertime. Having said that, were you to ever find yourself in 1930 with a dead Hitler at your feet, then exterminous hypertime should be on your list of candidates as to which metaphysical theory might account for what you have just witnessed.

Appendix: The Metaphysics of Time
A referee asked about exterminous hypertime's commitments to the different theories in the metaphysics of time, e.g., must exterminous hypertime be tensed or tenseless, eternalist or presentist, etc.? This appendix deals with those issues, arguing that exterminous hypertime is compatible with any metaphysical theory of time except those committed to an open future.

Tenseless vs. Tensed Exterminous Hypertime
Consider a tenseless eternalist exterminous hypertime. Given the world is tenseless and eternalist, no time is metaphysically privileged, nor is any hypertime, nor is any timehypertime 13 . All time-hypertimes exist simpliciter, whether they are later, earlier, hyperlater, or hyperearlier. I see no reason to think anything about this tenseless theory is problematic. If anything, it is the most natural understanding of exterminous hypertime.
It gets more interesting when we consider the tensed views. In regular worlds of one-dimensional time, tensed theorists say some time is metaphysically privileged. When we extend that to worlds of exterminous hypertime, it is most natural to think instead that some singular time-hypertime is metaphysically privileged 14 .
The rest of this appendix discusses what else we can say about the relationship between exterminous hypertime and the tensed theories of time.

The Dimensionality of the A-Series
Tensed theorists think that 'privilege forms a series'-namely, the 'A-series'. For instance, at a world of one-dimensional discrete time, consisting of times t 1 , t 2 , t 3 . . . , then first t 1 would be metaphysically privileged, then t 2 , then t 3 , etc. Represent that A-series as: Like the temporal dimension at that world, that A-series is one-dimensional. Tensed exterminous hypertemporal theorists would have a problem if, at worlds of exterminous hypertime, the A-series also had to be one-dimensional. At such worlds, there would be two options as to how such an A-series could be arranged, with each turning out to be a bad option.
Option (i): The series is ordered thus: This option means that the spotlight of metaphysical privilege must work through every time at the first hypertime before it can move onto the second hypertime and start working through the eternity of times at that hypertime, and so on for all hypertimes. This is problematic because such a series is too similar to the series arising at a world of one-dimensional time where, after an eternity has passed, the universe 'resets' to how it was at the start and history starts cycling through again. Imagine that, at such a world, ostensible time travellers who step into their time machines in an earlier circle appear in 'the past' in the next cycle-in a sense, they have not travelled back in time, but have instead travelled an eternity into the future to a point where history has repeated itself. As far as I know, no-one has presented such a theory of time travel, but this short description should suffice in explaining it 15 . Given that theory, the A-series of events would be exactly like that given by option (i). So if the tensed exterminous hypertemporal theorist accepted option (i), I would worry that they were not really distinguishing their theory from this theory of time travel relying only on one-dimensional time.
There is also an apeirophobic problem with the A-series of option (i). Given option (i), metaphysical privilege only moves onto the next hypertime after an eternity has passed. There are Zeno-esque concerns that you will never reach the relevant point where privilege moves to the next hypertime-similarly, it is less clear that you should expect something to happen if its happening is literally an infinite amount of time away. If that were true then we should never expect metaphysical privilege to move to the next hypertime and so should no longer 'hyperexpect' things that happen in our hyperfuture to come about.
All this said, I assume exterminous hypertemporal theorists will avoid option (i). Option (ii): The time-hypertimes form the following series, whereby both time and hypertime are always 'flowing forwards': Given option (ii), some time-hypertimes (e.g., t 1 -T 1 or t 2 -T 2 ) are never privileged, which seems bizarre. Imagine the standard tensed theorist believed that there were events that are past but which somehow have avoided ever having the light of presentness shine down upon them. That would be crazy! Similarly, it would be weird if, at t 2021 -T 2021 , it was true that Hitler was alive in 1930 in virtue of how t 1930 -T 2021 is, even though t 1930 -T 2021 failed to appear in the A-series and failed to have ever been metaphysically privileged.
So, it is problematic to combine a one-dimensional A-series with two temporal dimensions. However, this is not a problem, since the exterminous hypertemporal theorist should simply say that the A-series is two-dimensional! Given a two-dimensional A-series it makes no sense to think that there is a single time-hypertime that comes 'next' in the A-series. Rather, there is a time-hypertime that comes next in one dimension of the A-series and a different time-hypertime that comes next in the other dimension. For example, if t 1 -T 1 is metaphysically privileged then it makes no sense to ask which time-hypertime comes next simpliciter. Rather, we should say that t 2 -T 1 comes next in one dimension of the A-series whilst t 1 -T 2 comes next in another dimension of it. Once we accept that it can be two-dimensional, this problem about ordering the A-series goes away.
Some tensed theorists may struggle with the idea of a two-dimensional A-series. They might, for instance, be wedded to the metaphor of God being sat outside spacetime, in his own (one-dimensional) temporal stream, watching different parts of spacetime 'light up' as the spotlight of metaphysical privilege shines upon it. Such a metaphor leaves no room for a two-dimensional A-series. However, such tensed theorists are surely going to think two-dimensional time is also impossible-to think that time can be two-dimensional, but that the A-series must be one-dimensional, seems particularly strange. So, since this paper has already assumed that two-dimensional time is possible, I will assume that a two-dimensional A-series is equally unproblematic.

Presentist/Moving Spotlight Exterminous Hypertime
Armed with a two-dimensional A-series, exterminous hypertime can be married with different tensed ontologies. Consider a presentist theory whereby only one time-hypertime exists. The non-present non-hyperpresent time-hypertimes do not exist; however-just as other times did and will exist given regular presentism-those time-hypertimes did exist, will exist, hyperdid exist, and/or hyperwill exist. (Additionally, on this view, it makes no sense to ask which time-hypertime will be next simpliciter in the A-series, only whether a time-hypertime will be next in in one-dimension of the A-series or the other).
Similar can be said of the moving spotlight theory. Only one time-hypertime is metaphysically privileged. When you are at a metaphysically privileged time-hypertime, the spotlight will next move to different time-hypertimes depending upon whether we consider the dimension of the A-series corresponding to regular time or the dimension corresponding to hypertime-pick a different dimension, and a different time-hypertime will be 'next'.

Growing Block Theory and the Open Future
Things become more complicated when we consider growing block theory. Firstly, we must pin down the details of what growing block theory amounts to in a world with two temporal dimensions. It is natural for such growing block theorists to say that nothing hyperlater than the metaphysically privileged time-hypertime exists. Equally, it is natural to say that any earlier-and-hyperearlier, as well as any earlier-and-hypersimultaneous, time-hypertimes exist. However, this does not settle the status of all time-hypertimes. Growing block theorists have two options when it comes to hyperearlier-yet-later timehypertimes: Either they exist or they do not. Figure 5a depicts a growing block whereby such time-hypertimes do exist; call it 'Bigger growing block theory'. Figure 5b depicts a growing block theory whereby those time-hypertimes do not exist; call it 'Littler growing block theory' (because its block is smaller than that of Bigger growing block theory). Both have problems when it comes to their compatibility with exterminous hypertime. In each of the diagrams below, the black dot indicates which time-hypertime is metaphysically privileged. The greyshading indicates the size of the growing block; every time-hypertime covered by the grey shading exists.
For all j and k: tj precedes tk iff the times that exist when tj is privileged are a subset of the times that exist when tk is privileged.
Bigger growing block theory can (and, presumably, should!) accept the analogue: For all j, k, m, and n: tm-Tn precedes tj-Tk iff the time-hypertimes that exist when tm-Tn is privileged are a subset of the time-hypertimes that exist when tj-Tk is For all j and k: t j precedes t k iff the times that exist when t j is privileged are a subset of the times that exist when t k is privileged.
Bigger growing block theory can (and, presumably, should!) accept the analogue: For all j, k, m, and n: t m -T n precedes t j -T k iff the time-hypertimes that exist when t m -T n is privileged are a subset of the time-hypertimes that exist when t j -T k is privileged.
Given that analogue principle, there would then be a one-dimensional A-series like the following: However, that is just the ordering from option (i) above! Since that ordering was problematic, so too is Bigger growing block theory.
Littler growing block theory also has problems, although this time they stem from the incompatibility of exterminous hypertimes with an open future. Open future theorists believe that facts about later times are indeterminate. Given exterminous hypertime involves two dimensions of time, open future theory must be redescribed. Clearly, that redescription should say that what is hypersimultaneous-and-later is indeterminate, as is anything hyperlater-and-later (and, presumably, anything at all that is hyperlater). However, there are two options concerning the status of facts about hyperearlier-and-later time-hypertimes: (a) Facts about hyperearlier-and-later time-hypertimes are determinate; or (b) Facts about any later time-hypertime-whether that time-hypertime is hyperearlier, hypersimultaneous, or hyperlater-are indeterminate.
Option (a) leads to the same problem we had with Bigger growing block theory. Given (a), the facts become determinate in a certain order. All facts at one hypertime go from being indeterminate to being determinate (in order of earliest to latest) and only once facts about all time-hypertimes at any given hypertime are settled, do facts at the hypernext hypertime begin to get settled (again, in order of earliest to latest in the regular temporal series). So there would, again, be a one dimensional A-series of the form: t 1 -T 1 ⇒ t 2 -T 1 ⇒ t 3 -T 1 ⇒ . . . ⇒ t 1 -T 2 ⇒ t 2 -T 2 ⇒ t 3 -T 2 ⇒ . . . ⇒ t 1 -T 3 ⇒ t 2 -T 3 ⇒ . . . I have already argued that such a one-dimensional A-series is problematic. So option (a) is a bad option.
Option (b) has its own problem. Consider the following scenario. It is presently and hyperpresently t 1999 -T 1999 . Given option (b), facts about hyperearlier/hypersimultaneous earlier times are fixed, so facts about t 1930 -T 1998 and t 1930 -T 1999 are fixed. Let us say, for example, that it is hyperpresently true that Hitler exists at both those time-hypertimes and at neither does a time traveller turn up to kill him. Given option (b), it is further indeterminate whether, at t 2021 -T 1998 , a time traveller leaves that time-hypertime to go and kill Hitler in 1930. Next, imagine time passes and t 2021 -T 2021 becomes hyperpresent/present. There would then also be determinate facts about t 2021 -T 1998 . Imagine that those facts end up being such that, at t 2021 -t 1998 , a time traveller does go back to 1930 to kill Hitler. Given PROGRESSION, that time traveller must arrive at t 1930 -T 1999 . However, we have assumed for purpose of example that no such time traveller exists at t 1930 -T 1999 . So we would now have a contradiction. So option (b) is problematic. Therefore, since option (a) is problematic as well, we cannot pair exterminous hypertime with an open future. (Is it a problem that exterminous hypertime is incompatible with an open future? I presume not. Very few theories of time travel can make room for the future being open [1] (p. 71 n. 3), so it is not concerning that exterminous hypertime ends up in the same boat).
That conclusion in place, return to Littler growing block theory. Stereotypically, growing block theorists are open future theorists-because the future times do not exist, facts about them are unsettled [46] (p. [27][28] [47] (p. 357) [48,49]. Once we add in exterminous hypertime, the natural extension of that stereotype is to say that if a timehypertime does not exist then facts about it are unsettled. So, if they also indulge in that stereotype, Littler growing block theorists will endorse option (b), which I have argued leads to a contradiction. In short: Unless one is willing to ditch the growing block theorist's traditional commitment to an open future, growing block theory is incompatible with exterminous hypertime. (Similarly, any other tensed theory that allows for an open future will have the same problem-for instance, if you are a presentist or moving spotlight theorist who believes the future is open, then that tensed theory will also be incompatible with exterminous hypertime.)

Summary
Exterminous hypertime is compatible with the tenseless theory of time and-as long as you are willing to accept a two-dimensional A-series-certain tensed theories of time. The tensed theories that are incompatible are any that accept that the future is open.