If the Hotel Can Have an Infinite Number of Rooms, Why Can't the Universe Have an Infinite Number of Days?

In my previous post, wherein I attempted to provide an undercutting defeater of Craig's Hilbert's Hotel argument for the impossibility of an actual infinite number of things, I allowed for the conception of a hotel, the set of whose rooms is infinite, and has a cardinality of aleph-null. And yet, I also provided a small argument at the end that I believe may better perform the same function as Hilbert's Hotel in the greater Kalam. But as I thought about what I had done, a possible reply came to mind: If the Hotel can have an infinite number of rooms, why can't the universe have an infinite number of days? I do not have a polished answer, buy I here journal my initial musings.

In the Hotel conception, I said that God built the hotel by the power of His word, and sustains it by His power. I don't actually believe that Hilbert's Hotel exists in this world, but I can conceive of it pretty good, and I think I have illustrated such a conception in a satisfactory way. At any rate, if one claims that he does not know whether he can conceive of such a hotel or not, I think I have made it more difficult to say that it is impossible to conceive of it, such that an argument for the impossibility of such a hotel, or any actual infinite number of things, is not as easy as pointing to the abnormal phenomena such a hotel might give rise to.

Nevertheless, it might appear incoherent for me to posit such a hotel and at the same time argue for the impossibility of an actual infinite number of days in the universe. After all, why can't God simply just create such a beginningless universe and sustain it by the power of His word? Such a question might be unfair for an atheistic infinite universe argument to utilize at face value, but really it is just a call for internal coherence on my part. Well, I believe that there is a fundamental difference between the Hotel and the number of days in the universe.

In the case of the hotel, the rooms are instantly built by God, such that there is no temporal transgression of an infinite spatio-temporal distance. One minute the Hotel is not, the next minute, God speaks such a structure into existence. The set A of all Hotel rooms is at T1 empty, and then the set B of all Hotel rooms is all at once at T2 infinite, and of the cardinality aleph-null.

But in the case of the days in the universe, they cannot exist all at once. Rather, this moment exists and naught else. Thus, by "days" we mean to denote a segment of temporal transgression made by temporally bound objects. There does not exist, "now", all moments in time. This view, Presentism, is something I hold but do not care to explicate and defend here. I am interested in researching it further, and perhaps writing on it someday. My Philosophy of Religion professor, Thomas Crisp, is also a presentist. I am not sure whether my argument for the beginning of the universe is altered whether or not Presentism is accepted, but I will go forward as if it is.

This difference is all that is necessary to establish why it is that a hotel with an infinite number of rooms is conceivable, but a universe whose temporal past is not. For, as was shown in my argument in the previous post about the Hotel, there is no room with the number -∞ (negative infinity). Nor is there a room with the number 'infinity' on its threshold. For infinity is the state of a set, not a number unto itself. Any given number is finite, while infinite sets of finite numbers may be considered.

Similarly, if 'now' is labeled "0", and yesterday "-1", then it is seen that there cannot be a day with the label "-∞". Any given day in history will have a finite label. But why can't the set of all past days be infinite? Because of the impossibility of transgressing an infinite spatio-temporal distance. Just as the present must transgress the moments in today to get to tomorrow, so it would have had to transgress the past in order to get to today. This is what was illustrated in my thought experiment by the runners who attempted to begin their race infinitely far away. They will begin by crossing room numbers 0 through -10, and then they will transgress rooms -10 through -20, and continue running for eternity, without ever finding a starting line from which to run back to room 0. This is a textbook example of a potential infinite. At any given time, the runners will be in front of a room with a finite number, and the cardinality of the set of rooms they will have transgressed will be ever-increasing. But a potentially infinite set will never become an infinite set. In the same way that even God, if He counts one number at a time, will never reach infinity. He will be counting forever, and the set of numbers He utters will be potentially infinite. For at any given time, He will be uttering a finite number. Not even God can transgress an infinite spatio-temporal distance.

The universe cannot have existed from infinity past because it is impossible to cross an infinite distance, or an infinite amount of time. The Hotel can have an infinite number of rooms, because its construction was instantaneous. This would be like God thinking of an infinite set of numbers all at once - something He is no doubt capable of doing.

An Argument Against Craig's Hilbert's Hotel Illustration and a Proposed Solution

Jon Wright pointed out to me that the prior post did not handle the particular type of transfinite subtraction Craig says cannot be done. Namely, subtraction involving infinite quantities. I spoke about this with my math professor, Matthew Weathers. I proposed my argument, and he helped clear up some concepts for me, and explained the nature of cardinalities in a helpful way. Generally drawing from that experience, while sitting here in a class called "Mission in Political Context", I here provide an undercutting argument against Craig's Hilbert's Hotel argument for the impossibility of the existence of an actually infinite number of things. Afterward, I propose an alterative argument that serves the same purpose in the overall Kalam.

In the broader context of the Kalam, Craig says,

Perhaps the best way to bring home the truth of (2.11) is by means of an illustration. Let me use one of my favorites, Hilbert's Hotel, a product of the mind of the great German mathematician, David Hilbert. Let us imagine a hotel with a finite number of rooms. Suppose, furthermore, that all the rooms are full. When a new guest arrives asking for a room, the proprietor apologizes, "Sorry, all the rooms are full." But now let us imagine a hotel with an infinite number of rooms and suppose once more that all the rooms are full. There is not a single vacant room throughout the entire infinite hotel. Now suppose a new guest shows up, asking for a room. "But of course!" says the proprietor, and he immediately shifts the person in room #1 into room #2, the person in room #2 into room #3, the person in room #3 into room #4 and so on, out to infinity. As a result of these room changes, room #1 now becomes vacant and the new guest gratefully checks in. But remember, before he arrived, all the rooms were full! Equally curious, according to the mathematicians, there are now no more persons in the hotel than there were before: the number is just infinite. But how can this be? The proprietor just added the new guest's name to the register and gave him his keys-how can there not be one more person in the hotel than before? But the situation becomes even stranger. For suppose an infinity of new guests show up the desk, asking for a room. "Of course, of course!" says the proprietor, and he proceeds to shift the person in room #1 into room #2, the person in room #2 into room #4, the person in room #3 into room #6, and so on out to infinity, always putting each former occupant into the room number twice his own. As a result, all the odd numbered rooms become vacant, and the infinity of new guests is easily accommodated. And yet, before they came, all the rooms were full! And again, strangely enough, the number of guests in the hotel is the same after the infinity of new guests check in as before, even though there were as many new guests as old guests. In fact, the proprietor could repeat this process infinitely many times and yet there would never be one single person more in the hotel than before.

But Hilbert's Hotel is even stranger than the German mathematician gave it out to be. For suppose some of the guests start to check out. Suppose the guest in room #1 departs. Is there not now one less person in the hotel? Not according to the mathematicians-but just ask the woman who makes the beds! Suppose the guests in room numbers 1, 3, 5, . . . check out. In this case an infinite number of people have left the hotel, but according to the mathematicians there are no less people in the hotel-but don't talk to that laundry woman! In fact, we could have every other guest check out of the hotel and repeat this process infinitely many times, and yet there would never be any less people in the hotel. But suppose instead the persons in room number 4, 5, 6, . . . checked out. At a single stroke the hotel would be virtually emptied, the guest register reduced to three names, and the infinite converted to finitude. And yet it would remain true that the same number of guests checked out this time as when the guests in room numbers 1, 3, 5, . . . checked out. Can anyone sincerely believe that such a hotel could exist in reality? These sorts of absurdities illustrate the impossibility of the existence of an actually infinite number of things.

-http://www.leaderu.com/truth/3truth11.html

Now, his immediate premises and sub-premises are,

2 The universe began to exist.

2.1 Argument based on the impossibility of an
actual infinite.

2.11 An actual infinite cannot exist.

That the universe began to exist I agree with. That an actual infinite temporal regression of physical members such as days is impossible I agree with. But I believe that there are a potentially infinite number of physical terms, meaning that the set that contains each day of time is a potentially infinite set. Specifically, it has a beginning (creation), but it will not have an end. Thus, the cardinality of the set of days that exist or have existed is finite (it includes however many days have passed since creation), but its cardinality (the number of days in the set) will always be increased as days pass. God has no intention that I know of to stop creating and/or sustaining the passing of days.

But this is to say nothing of an actual infinite. If by denying the possibility of an actual infinite Craig is required to believe that the number line does not contain members that are real objects (and informed by his other writings we can be confident of this), then I disagree. I think that numbers are real, that they are objects, and that the set of all whole numbers is infinite, in fact, there are many infinite sets of numbers.

At any rate, I can give an example of a meaningful, coherent subtraction of an infinite set from another infinite set. Let's subtract all whole negative numbers from all whole numbers:

Set X {...-3, -2, -1, 0, 1, 2, 3...}
- Set Y {...-3, -2, -1}
= Set Z {0, 1, 2, 3..}

So the cardinality of X is aleph-null, and so is that of the sets Y and Z. Here is where I feel that Craig has misunderstood the nature of transfinite arithmetic. For by "the same number of guests" Craig really means 'the same cardinality of the set of guests'. Let me explain. I believe Craig's contrual is something like the following.

Aleph-null
- Aleph-null
= Aleph-null

Thus, the cardinality of the set produced by the difference in the cardinalities of the sets X and Y is the same as the cardinalities of each set. This would not be a problem, but for other instances of transfinite subtraction, and Craig gives a few. Consider another example,

Set X {...-3, -2, -1, 0, 1, 2, 3...}
- Set X {...-3, -2, -1, 0, 1, 2, 3...}
= Set Z' { }

The cardinality of X is aleph-null, and when X is subtracted, the resulting set Z' is empty, and therefore has a cardinality of 0:

Aleph-null
- Aleph-null
= Zero

Furthermore, we can construct all kinds of examples of subtracting infinite sets from other infinite sets such that the difference may be virtually any number. Consider,

Set A {1, 2, 3...}
- Set B {2, 3...}
= Set C {1}

The cardinality of A is alpeh-null, and the cardinality of B is aleph-null, but the cardinality of C is 1. Thus:

Aleph-null
- Aleph-null
= One

Therefore, subtracting an infinite set with a cardinality of aleph-null from another
set with a cardinality of aleph-null is an operation that is not well defined, and it is useless to us.

But is this the whole picture? Hold that thought, and follow me while I retrace a famous logical contradiction drawn from the principle of square roots:

√25 = 5
√25 = -5
∴ 5 = -5
But 5 ≠ -5

The above fallacy lies in the incompleteness of the statement √25 = 5, and the incompleteness of the statement √25 = -5. Rather, the two should be resolved with the statement √25 = ±5.

Similarly, I believe that Craig's fallacy is one of incompleteness. As I stated last time, one has not exhaustively defined a set if one has merely described its cardinality. Thus the operations within the ilk:

Aleph-null
- Aleph-null
= One

are not complete. While the unhelpful statement that 'aleph-null - aleph-null = 0 or 1 or 2 or n or aleph-null', isn't well defined, it is coherent. However, there is no reason to only perform transfinite arithmetic at the level of cardinalities. In fact, I can think of a very good reason not to perform transfinite arithmetic at the level of cardinalities...

In the last post I gave an application of the set theory paradigm that went like this

If the intersection of a set X and a set Y is empty, and the cardinality of X is 3, and the cardinality of Y is 2, then the cardinality of the union of X and Y is 5.

But, what if the intersection of X and Y is not empty? In such cases, even transfinite addition becomes undefined. For example, the intersection of a set X {1, 2, 3} and a set Y {2, 3, 4} is the set Z {2, 3} such that to add the cardinalities of X and Y would look like this:

3
+ 3
= 2

But we know that this is not a true equation. However, if the members of the set with a cardinality of 3 and the second set that also has a cardinality of 3 are defined, then the intersection of the two sets is coherent:

Set X {1, 2, 3}
∩ Set Y {2, 3, 4}
= Set Z {2, 3}

Thus, adding cardinalities is just as "absurd" as subtracting cardinalities. But this is exactly what Craig does when he subtracts 'an infinite number of guests' from 'an infinite number of guests' to get 'an infinite number of guests'. He calls an absurdity what is in reality an undefined expression. He should rephrase his expression to say that 'there is some infinite set D such that a proper infinite subset E of D may be subtracted from D, resulting in a third infinite subset F'.

I believe I have given an example of such an operation:

Set X {...-3, -2, -1, 0, 1, 2, 3...}
- Set Y {...-3, -2, -1}
= Set Z {0, 1, 2, 3..}

While Craig believes that Hilbert's Hotel is absurd, I can imagine such a hotel as existing. Picture this:

You are standing, facing the hotel. You look to the left, and you can see the building extending beyond sight. You look to the right and sure enough, the hotel rooms continue on beyond your visual range. God has constructed this hotel by the power of His word, and He sustains its existence. You walk straight ahead, which leads you to room #0, which is the clerks office. She tells you that all the room #'s are occupied, but she thinks she can squeeze you in. She makes an announcement over the loud speaker that every occupant of a room with a positive number is to move to the room with the number one above his. You walk outside and turn around to face the hotel again. Now you begin to see every occupant in the rooms to your right open their doors, walk farther right, and enter into the next rooms. This leaves room #1 open, and you walk in and place your luggage on the fold-out table you remove from the closet. Hilbert's 'Hotel is a strange place' you think, but you're sure they make their money.

In set theory terms, the cardinality of the set containing all room numbers is aleph-null. Moreover, the cardinality of the set containing all guests is aleph-null, and when you check in, the resulting set has an additional member (you), but it also has the cardinality aleph-null. And yet, just like in the examples I gave above, this phenomenon is coherent (and now even conceivable). But am I failing to handle his argument in its full force?

Let's explore Craig's illustration a little more. One example of absurdity he gives is:

But Hilbert's Hotel is even stranger than the German mathematician gave it out to be... Suppose the guest in room #1 departs. Is there not now one less person in the hotel? Not according to the mathematicians-but just ask the woman who makes the beds!

I judge Craig to be accurate when he says that there is "one less person in the hotel". But he is mistaken when he accuses "the mathematicians" of disagreeing. What I think he means is that mathematicians would maintain (and I think rightly so) that the cardinality of the resulting set is the same as the initial set. But this does not mean that the initial set didn't lose a member! What Craig should say is something like 'in the set H containing all the hotel room numbers {-1, 0, 1, 2, 3...}, the guest in room #1 leaves, represented by the removal of the set I, {1} from H, resulting the set J {-1, 0, 2, 3...}. And yet, the cardinality of H is aleph-null, and the cardinality of the set J is also aleph-null, even though J contains one less member than H (J is a proper subset of H)'. Allow me to express this in a linear fashion:

Aleph-null
- One
= Aleph-null

As I demonstrated earlier, this is not a logical contradiction, for the subtraction of a finite number from aleph-null results in aleph-null. More specifically, the subtraction of a finite number of members from an infinite set with the cardinality of aleph-null results in an infinite set with the cardinality of aleph-null as well, even though the resulting set is a proper subset of the original set. In the example Craig gives:

Set H {...-3, -2, -1, 0, 1, 2, 3...}
- Set I {1}
= Set J {...-3, -2, -1, 0, 2, 3...}

So the guest in room #1 may leave, and this is not absurd. But that was just the subtraction of a finite number of members from an infinite set. Can we also meaningfully subtract an infinite number of members from an infinite set? Craig says

Suppose the guests in room numbers 1, 3, 5, . . . check out. In this case an infinite number of people have left the hotel, but according to the mathematicians there are no less people in the hotel-but don't talk to that laundry woman! In fact, we could have every other guest check out of the hotel and repeat this process infinitely many times, and yet there would never be any less people in the hotel

By this I understand Craig to be expressing a phenomenon that goes something like 'in the set H containing all the hotel room numbers {-1, 0, 1, 2, 3...}, the guests in rooms with odd numbers leave, represented by the removal of the set L {1, 3, 5...} from H, resulting in the set M {-1, 0, 2, 4, 6...}. And yet, the cardinality of H is aleph-null, and the cardinality of the set L is also aleph-null, and the resulting set M has a cardinality of aleph-null even though M contains less members than H (specifically, M is a proper subset of H). Thus, the difference between the infinite set H and the infinite set L is another infinite set'. Allow me to express this in a linear fashion:

Aleph-null
- Aleph-null
= Aleph-null

As I demonstrated earlier, this is not a logical contradiction either, for just as the square root of 25 is 5 or -5, so the subtraction of aleph-null from aleph-null may be 0 or 1 or 2 or n or aleph-null. And although this is coherent but not meaningful, meaning may be added by simply defining the members in each infinite set:

Set H {...-3, -2, -1, 0, 1, 2, 3...}
- Set L {1, 3, 5...}
= Set M {...-3, -2, -1, 0, 2, 4, 6...}

Thus, Craig's argument is demonstrated to be unsuccessful in revealing any absurdity. As far as a clear and persuasive argument for the fact that the universe had a beginning, permit a modest formulation of my own. I have offered such arguments before, but will briefly indulge once again.

The morning after your first night's stay in Hilbert's Hotel, you put on your pajamas and open your door to pick up the copy of the L. A. Times whose thump against your door woke you up. You look up, and with the hotel to your back, you look to your right and see a crowd of runners gathered in front of the clerk's office. You inquire as to the event, and your neighbor in room #2 explains that there is to be a race, whose participants will start the race in front of room #-∞. 'How can they ever get started?' you ask. 'They will never stop walking down the line of negative numbered rooms' you assert. 'And as soon as they decide to turn around and begin the race, the room in front of which they do so will necessarily be finite, not infinite!' you argue, beginning to get concerned. 'There is no room whose number is -∞. Each room has a finite number, but the set of all rooms is infinite! Even if God Himself allows them to run toward the clerks office where the finish line is while He constantly pulls them in the direction of the negatively numbered rooms, their running will be in vain, for whatever progress they make will be more than cancelled out by God's pull - they can neither start nor finish the race'. You try to warn the runners, but they ignore you and take off toward the infinite. You feel sorry for the runners, whose racetrack is the set of rooms with negative numbers {...-3, -2, -1}. And furthermore, to get started, they have to cross that same track while searching for the starting line! How can they cross a distance that consists of a set of rooms with the cardinality aleph-null? How can they cross it twice?

Such a mental exercise demonstrates the absurdity of positing a universe whose temporal past extends infinitely. While it is coherent to posit a universe with a beginning and no end, it is incoherent to say that the universe never got started - never had a beginning. For, with today as the finish line of the race (and indeed, today has come to pass), the cursor of history - the ontological "now" - will have to transgress a set of days whose cardinality is aleph-null. This is logically impossible.

Therefore, while Craig's Hotel argument is susceptible to an undercutting defeater, there is another argument that may take its place, rendering his overall cosmological argument stronger.

An Old Journal of My Introduction to General and Special Relativity Wherein My Skepticism and Dogmatic Newtonianism Was Dissolved

Forward:
Dear Derek, in the presence of many witnesses, in light of our brief conversation earlier, I here submit to you some of my thoughts from a Physical Science class I took in Fall of 2005. One of the assignments involved thorough interaction with a chapter of choice in Hewitt’s, Suchocki’s, and Hewitt’s, Conceptual Physical Science. I selected a chapter we hadn’t gone over in class: Chapter 35: Special and General Relativity. Please keep in mind that I have learned a lot since this writing, that this writing is a direct summary and response to a chapter of a textbook, and that this was a general ed, non-science major class.

I also highly recommend these two resources:
http://www.pbs.org/wgbh/nova/elegant/
Zero: The Biography of a Dangerous Idea

•••

Original Writing:
35.1 Summary: To record the state of something we use three dimensions, which have no robust meaning unless we also state the time that the dimensional descriptions applied to the object in question. This seems to indicate that space and time may be bound. Einstein’s theories support this; that things exist in a space-time continuum.
35.1 Comments: I have seen the PBS special “The Elegant Universe”, so I am going into this chapter with a little bit of background information about the theories of relativity, but not much. I have some speculations of my own, which begin with critiques of contemporary explanations of “space-time” observations. I would therefore like to note my initial hesitance to calling space and time, or matter and energy, the same thing; no matter how intimately they may be connected.

35.2 Summary: Einstein’s Special Theory of Relativity, based on Newtonian principles, states first that “Observers can never detect their uniform motion except relative to other objects”. A restatement of this postulate is that “All laws of nature are the same in all uniformly moving reference frames”. An example is given of being able to pour coffee while flying through the air on a jet plane. This is supported by any experiment that has been carried out. If it has been carried out on earth, the fact that the earth is in motion, and the experiment was carried out as if there was not any acceleration of the location it was carried out in (indeed, there was not any relative motion of the lab relative to the earth and its atmosphere), then it becomes more and more clear that if the entire frame of reference is moving relative to an outside object, then relative motion inside the reference frame is only relative to other objects inside the reference frame – the motion of objects outside the reference frame are not factors in the motion of the objects inside.
The second postulate is that “The speed of light in free space will have the same value to all observers, regardless of the motion of the source or the motion of the observer. The speed of light is a constant.” This postulate is confirmed by our measurements of the speed of light. If both postulates accurately describe reality, then when two observers in relative motion to each other and a light source disagree about the time of a certain event, they may both be correct – the only thing to budge in the equation is time. A note is made that points in space and time are both quantized, i.e. there are fundamental units for them.
35.3 Comments: I like that the wording of the first postulate does not deny objective uniform motion or location. It does not affirm them either. If you start with relative motion, you will get relative motion. In other words, if we measure motion relatively, or call motion “change in position relative to a frame of reference” then of course we will conclude that motion is relative. Rather, the postulate simply states that it is impossible to detect uniform motion (if it is a phenomenon that actually occurs in our universe). I am however, skeptical of even this. What, logically, was Einstein’s reason for believing that humans will never discover a universally objective frame of reference, nor be able to develop the technology to measure motion in relationship to it?
I have also wondered about the famous experiment involving two clocks on two airplanes flying in opposite directions around the world. Their clocks showed a difference in displayed time at the end. But couldn’t that have been due merely to the physical strain exerted upon the inner workings of the clocks during the experiment? I would wonder then, what were the clocks measuring? Is it possible that what we call “time” is actually two different things: ‘age’ and ‘time’. ‘Age’ being the physical change or decay of an object, and ‘time’ being the objective, steady progression of events in this world. Time might be the cursor of history marking where everything is. How else would it be that the pilots of the planes could see each other after the flight? Wouldn’t they be living in two different planes of reality – different “times”? One would be a few split seconds ahead of the other! Or perhaps at the end of the day they both wound up in the same time plane, but traveled through less “time” to get there. I wonder then, what the nature of the “time” is that they traveled through; could they have seen each other during their journeys? How much of the Special Theory has to do with mere perception, and how much of it with the actual bending of time? Could it be that our methods for measurement are flawed?

35.3 Summary: The notion that time can be stretched is called “time dilation”. A very compelling illustration is given. Suppose that we could observe a light flash bounce between two mirrors inside a transparent spaceship. An observer onboard the spaceship, and an observer on the ground would both measure the speed of light as the same rate, denoted c. Yet to the observer onboard the spaceship, the flash of light is only moving up and down (which it is – relative to the environment of the spaceship). However to the observer on the ground, the flash of light will actually be moving up and down, and horizontally as well (in a zig-zag pattern). This is because the entire spaceship, including the mirrors and the light flash, are moving parallel to the ground. To the observer on the ground, the flash of light will then be covering more distance than to the observer on the spaceship. If speed is equal to distance divided by time, and the speed of light is constant, and the distance stretches, then the time must logically be stretching, or dilating, as well. This occurrence can be expressed mathematically.
The text then contends that time is actually dilating, there is nothing unusual about the functioning of a clock itself, as indeed there is nothing unusual about clocks here on earth now, though they are all whizzing through space (because the earth on which they are is whizzing through space). Next the text explains that as a result of time dilation, according to the measurements we have made, as speeds approach the speed of light, time slows down and seems to approach zero. A mass at the velocity of light, c, takes an infinite amount of time to elapse one second, and zero amount of time to travel an infinite distance. Thankfully, the Law of the universe has set c to be a type of cosmic speed limit such that it is impossible to reach or exceed c. It is important to note that time dilation, to be perfectly accurate, most be taken into account even for small masses and slow accelerations. It is also important to note that as speeds approach c, mass approaches infinity, length approach zero. However light, which moves at c, will always be measured to be moving at c, regardless of your frame of reference.
Space travel is also discussed in this chapter. At high speeds it would be theoretically possible to spend 5 years traveling at 99.999 % of c, and return to earth only to find that 1100 years have passed for it and its inhabitants.
35.3 Comments: This chapter hasn’t mentioned it yet, but c is only the speed at which light moves in a “vacuum”. Light moves significantly slower in gas, even slower in liquid, and slower still through solids (if at all). This chapter coaxes another question. If Einstein is right, then why can light go the speed of light? Isn’t light made up of photons? Why are they exempt from the c speed limit? Well, if photons have zero mass, then perhaps it is not so absurd. I would also like to note that this section does not assert that time can be turned backwards. It can be sped and slowed (relatively), but there is nothing in this section that argues for bending and twisting time like a pretzel, or going back in time. In fact on page 845 it says that time accelerated astronauts will not be able to go backward in time.

35.4 Summary: Length contraction is related to time dilation. As was mentioned, as a mass approaches the speed of light, its length approaches zero, when measured by an observer who is not traveling relative to a common frame of reference. This contraction only takes place in the direction of motion. This phenomenon is related to time dilation, as distances will flatten to zero when a mass approaches them at c. In this way, one could travel an infinite distance in zero amount of time. Furthermore, if a mass approaches the speed of light, its length does not change relative to itself. Rather, the time it has to travel through in order to reach its destination is collapsing. However from the perspective of an outside observer, lengths are contracting.

35.5 Summary: In this chapter relativistic momentum is explained. The speed of any mass has a limit: c. However, momentum may increase without limit. This is illustrated by Einstein’s new equation for calculating momentum, which accounts for the Lorentz factor. Therefore as speed approaches c, momentum approaches ∞. With any progress toward c, objects appear to increase in mass, because they display an increase in inertia. Einstein however, seemed to think later in his life that mass is the same in all frames of reference, but space-time changes with speed.
35.5 Comments: It seems to me that if you accept Einstein’s initial premises, then all of this follows logically. To be honest, I think I might just have a lot of emotional struggles with accepting all of this.

35.6 Summary: Einstein asserts that matter and energy may be exchanged according to the expression E=mc2; E being energy, m being mass, and c being the familiar uninhibited speed of light. When fuel is used to produce energy, the system will afterward be found to be less some mass, in accordance with Einstein’s equation!

35.7 Summary: A desire to describe nature using standard forms and perspectives motivated Einstein to pursue a description of the phenomena he was examining, which came to be know as his theory of General Relativity. By thinking about a spaceship accelerating at g, the principle of equivalence became manifest, that “local observations made in an accelerated frame of reference cannot be distinguished from observations made in a Newtonian gravitational field”. A few thought experiments quickly establish this theory as reasonable, but Einstein elaborates on it by saying that, (in a free environment) because the trajectory of a beam of light relative to an accelerated mass is identical to the trajectory of an additional object relative to the same accelerated mass, space-time itself curves. He contends that matter and energy are two aspects of the same thing.
Euclidean geometry produces accurate results when dealing with objects on a flat surface. However, Einstein’s observations and calculations argued for a four dimensional warping of space-time. This is seen when the shortest distances between three planets form a triangle whose angles add up to greater than 180 degrees. The lines making up the legs of triangles such as this are called geodesics. The lines are technically not straight, but they are short. If this curving of space-time is positive throughout the universe, then the universe is warped like a sphere. If it is negative, then it is shaped more like a Pringles potato chip. This curvature of space-time causes what we perceive as gravity, and is proportional to mass. This principle is seen when we calculate the effect of large masses as they change position. It would seem that, rather than a sudden ceasing of gravitational force on other masses, a large mass that changes position creates gravity waves, similar to the waves made on the surface of a waterbed when a bowling ball is rolled across it.
Einstein’s calculations, when applied to planets in Our Solar System predicted precessing orbits. When Newtonian physics were used to predict the orbits of the planets it pretty much worked out – except for the missing 43 seconds of arc that Mercury displayed. Einstein’s equations had produced the right predictions.
Another confirmation of Einstein’s theory occurred in 1919 when a solar eclipse revealed that his predictions about the exact arc beams from distant stars display when passing near the Sun where dead on.
A third test confirms General Relativity, and it becomes clear that in General Relativity, the gravitational “red shift” is relative to location rather than speed (because c is a constant). It is also important to note that time is never sped or slowed for an object relative to itself (obviously). This is to say that though an object may pass through time more slowly than another relative to it, its lifespan is never extended from its own perspective.
35.7 Comments: I heard once that Einstein responded with a complete lack of surprise when his predictions about starlight arc were shown to be accurate. He said something about being certain from his equations alone; observation is secondary.

Concluding Thoughts:
This chapter has brought many questions and ideas to mind. For example, if God has no mass, then (if Einstein is right), He is not subject to time. Perhaps He can interact with this universe in the ontological now, all the while being atemporal and eternal. Maybe that photons have no mass somehow plays into a facet of God’s foreknowledge or omniscience.
This in turn raises thoughts concerning humans and the resurrection. If human souls have no mass, maybe it is so that, upon death (separation of body and soul), a soul zooms to resurrection day, upon which it meets its body and enters into temporality again. I understand this to be a much bigger scriptural and scientific topic to be researched than is feasible here, but it is interesting to be introduced to such musings.
I cannot keep from thinking about the implications of God stretching out the heavens. Could this skew our observations of the age of the universe? What if our solar system was accelerated altogether such that the age of our solar system is different than the age of another? I have heard some compelling arguments for a very young earth, and some for a very young solar system. But I have also heard about evidence that seems to support an older universe. Could it be that when God stretched out the heavens, galactic clocks spun at different rates relative to each other?

Consider:
Job 9:8 “[God] stretches out the heavens”
Ps. 104:2 “stretching out heaven like a tent curtain”1
Isa. 40:22 “He ... stretches out the heavens like a curtain and spreads them out like a tent”1
Isa. 42:5 “... God the Lord, who created the heavens and stretched them out”
Isa. 44:24 “I, the Lord, am the maker of all things, stretching out the heavens by Myself”
Isa. 45:12 “It is I who made the earth and created man upon it. I stretched out the heavens with My hands”
Isa. 48:13 “Surely My hand founded the earth and My right hand spread out the heavens.”
Isa. 51:13 “the Lord your Maker, Who stretched out the heavens and laid the foundations of the earth”
Jer. 10:12 “He has stretched out the heavens”
Jer. 51:15 “He stretched out the heavens”
Zech. 12:1 “the Lord who stretches out the heavens”

Please also see:
http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/measure_c.html
http://www.creationscience.com/onlinebook/FAQ111.html#wp1885480
http://www.cosmicfingerprints.com/

Lastly, I would like to leave the reader with the following thought. When God destroys nature, and recreates it (Rev. 21:2), will he do so with a new set of universal constants, quantities, and forces? Did the fall of Adam knock a lot of forces and constants off track? Humans have the opportunity to seek the answers, but let us give glory to God, above all.