md65536

I'll try to fix my wording... Can inflation cause a time-like interval between 2 events to become space-like? Or does "the spacetime manifold is fixed" mean that nothing can change whether an interval is space-like, time-like, or light-like? and If our universe is flat, is it possible to find 2 points in spacetime such that no single lightcone (of any 3rd spacetime point) contains both, given inflation? Addendum: Can the BB be considered an event, with spacetime coordinates and a light cone? Treating it as a normal event might be the source of my confusion...

Several things I don't get: 1. Can we say that every event in the universe is causally connected to the big bang? 2. Can inflation cause 2 time-like events to become space-like? In a flat universe, inflation could cause 2 locations to not be in any single light cone*, right? And thus there is no single event that could be causally related to both. So does that mean that inflation prevents various locations in the universe from having a common causal source (and thus the answer to question 1 would be "no")? Or does it mean that inflation allows 2 events that are not in any single light cone, to have a common causal source? I think what I'm asking is "Does inflation destroy causal relationships, or does it preserve them?" * If spacetime is open, it might be possible to have an infinite inflating region of space contained within an earlier light cone... if I got that right. See http://edge.org/conversation/next-step-infinity

That's an interesting conjecture! If you can demonstrate that it's true, then relativity will be in trouble and you'll have a base to begin proving the rest of your statements. All it needs is a bit of evidence or logic to show that it's true. Meanwhile you have several people trying to prove to you that it's logically not true, given an acceptance of a constant speed of light in all inertial reference frames. I suppose that if you can show that universal simultaneity is true, it should suggest a way to punch a hole in their arguments. It took 18 pages, but I have a good feeling that we're getting close to the start of a productive conversation! Keep it up! Almost there!

One can only vote once on a particular post. So it is not someone; it is someones. I voted, to show that I wasn't one of the original 2 votes. Also I can only make one negative vote per day... I often find myself faced with a "you have reached your quota of negative votes" message. I don't think that it indicates that people are "mad at you". I myself try to vote + on any message that is helpful, that makes me "get" something or is enlightening or educational. I try to vote down posts that are negatively helpful or damaging, including misleading arguments, misinformation, spiteful insults, etc. I think the negative votes are mostly indicative that your students here in this thread are not getting your lesson. I myself think that you're thisclose to disproving relativity, and dismantling and rewriting all of science as well as philosophy and a lot of history as well. Except of course for just a few questions that remain still unanswered. But I also believe that SR is correct. See, when a student has already decided that the teacher is wrong, the student is not going to learn. Even with the unusually low student-to-teacher ratio here in this thread, and the amount of attention devoted to teaching the lesson, and the endless repetition of the lesson, it remains endlessly unlearned. But I wonder, is it rewarding to teach with persistence to a problem student, because the lesson can still benefit others? Or is it foolish to try to teach a student who refuses to accept the lesson? it's "either" I agree that it's helpful to learn about the diagrams in order to understand them. These concepts aren't going to make sense unless one lets go of some preconceptions from classical physics (including assumption of universality of simultaneity). Also, it's difficult to learn relativity from scratch by looking at diagrams alone. With a little understanding of the basics of relativity, the diagrams make a lot more sense. Understanding the diagrams makes some of the concepts of relativity easier to understand. Learning a bit from one helps with the other; they go hand in hand. Without one, the other alone can be quite counterintuitive.

Oops I got stuck thinking of real spacetime intervals with imaginary distance. But I've long left behind knowing what I'm talking about, so I'll leave this thread until I figure some of it out.

I think the thread so far is better here; speculations doesn't tend to be treated very seriously. The discussion seems to have evolved to "Traveling at the speed of light is prohibited by SR, but what implications does relativity have for hypothetical FTL speeds?" I think this is useful because it shows the problems with the concept within SR, which are not there in classical physics, and would be ignored with open speculation. Okay this next part belongs in speculations! And I'm in way over my head. I picture imaginary magnitudes as negative distances. Treating yourself as a point observer, these would be distances with a magnitude that projects "into you", not along normal spatial axises, but along imaginary axises. My intuition says that even a point particle at an imaginary distance, say equivalent to [math]\sqrt{-d^2}[/math], would take up area in your imaginary field of vision, and look like a spherical shell of radius d at a distance of d (I guess you'd intersect its surface). But uh... it's inverted or something. Anyway it would get bigger the farther it is from you, until it took up half the imaginary universe at a distance of infinity. Yes, my head exploded, and that was the result.

The Lorentz factor would be imaginary, as would the result of the Lorentz transform, no? Do you mean that intermediate imaginary numbers are fine, because the space-time interval can still come out to be real, even with imaginary time and spatial coordinates? Time dilation and length contraction wouldn't apply, and time paradoxes could be constructed, but the math still works out and describes a situation that can be made sense of, and is compatible with SR even if it corresponds to no observed phenomenon?

Can this be extended to other properties in general, or does it apply only to length? Would you say that 'Realism asserts that reality consists of only what is obvious'? Or are there "natural reasons" that are not obvious (just none for length)? But if gravity "curves the trajectories of objects" then the obvious ontological question is, "What is IT that gravity curves." I must be mistaken somewhere, because I thought your argument for why 'GR takes "spacetime" and "makes something of it"' is that GR says that spacetime is curved. My understanding of that is that if something can be curved, it must be an entity. You are saying that the trajectories of objects are curved. Does this mean that the trajectories of objects are entities? Where have I gone wrong?

I think though the observables would also be imaginary. I think you'd get imaginary distances. But maybe that itself allowed with GR? And objects inside black hole event horizons??? (It's easy to consider real-world examples that are outside the scope of SR simply by considering gravity. GR doesn't violate SR but it handles situations that SR doesn't.) I always thought that a violation of causality was a violation of SR. Certainly you can get paradoxes (http://en.wikipedia.org/wiki/Special_relativity#Causality_and_prohibition_of_motion_faster_than_light)... Or can you allow violation of causality without violation of SR through interpretations like alternate realities and MWI? And now that I think about it, saying "I don't see how that's possible" is not good evidence that it's impossible. However, you probably wouldn't be able to pass a car that you can't catch, by speeding up, nor would you be able to travel faster than light by speeding up. But if say your car were able to "jump" ahead without changing speed, it might be possible. But I still don't think it's possible. If you're only trying to understand observed reality or relativity, I think it's safe and beneficial to assume that causality cannot be violated. If you want to change our understanding of reality, perhaps there will come a reason to reevaluate it.

One, it's impossible. Two it doesn't make sense. The Lorentz factor would then have an imaginary number. This doesn't correspond to reality. If you can make sense of how your idea would work, you'd have to describe it with something that is not special relativity. To me it sounds something like "If I can't catch up to the car in front of me no matter how fast I go, then why don't I just pull in front of them and not worry about catching up?" If the amount of energy it takes to reach c is infinity, what would it take to go even faster? Because photons aren't observers in a frame of reference. Length contraction is described in terms of a particular frame or observer. In the twin paradox example, a traveler approaching c moving from earth to a mirror 1 LY away would see the distance between earth and mirror contract. The earthbound twin would be at rest with the earth and mirror and not observe any length contraction between the earth and mirror. Addendum: Photons are like point-particles without any size. I don't know if it's fair to consider them subject to length contraction, but if you did you might say that they're already infinitely length-contracted, according to any observer. But uh... don't quote me on that! This kind of involves imagining photons as ordinary matter, and will probably lead to more false conclusions than useful ones. Perhaps it is just better to say: Photons are point-particles and so wouldn't be affected by length contraction.

Einstein figured out a lot of stuff based on imagining moving at the speed of light, but it was more about figuring out the problems with this than suggesting it's possible. See http://www.pitt.edu/~jdnorton/Goodies/Chasing_the_light/ -- not sure if the linked information is any good but I couldn't find anything else easily. Because the speed of light is invariant and relative to any observer, even if you travel at 0.9999c relative to something else, light will still move away from you at c. If you try to catch up, it will just keep traveling at c relative to you. Another way to think of this is that you're always at rest relative to yourself, and any light you observe is moving relative to you. As v approaches c, the Lorentz factor approaches infinity, which means that lengths contract to (approaching) 0 in the line of motion. Any distance becomes vanishingly small. You're still traveling at a speed of (approaching) c, just over a tiny distance. The Lorentz transform "breaks down" at v=c... there is a division by 0. This is okay because traveling at the speed of light is impossible anyways. There are differences between how a moving object behaves, and how light behaves. You can imagine "being" light, but you can't imagine that you're still yourself! Light doesn't experience the universe as we do. It does not receive incoming light or information. However, I think that considering light's behavior using "v approaching c" is probably okay. I think you can say that photons do not age. This would mean: - There is no way to define a clock for a photon, in which a photon could experience time. - A photon is exactly the same at the end of its journey, as it was when it started.

Let's consider this in terms of the twin paradox, so that we can synchronize clocks at a single location and make sense out of it. Suppose you left earth with a velocity that approaches c, and you get bounced back off a mirror one light year away. The earthbound twin would observe that you reached the mirror after one year, but it would take one year for that observation to return to earth. Over 2 years, the earthbound twin can watch you approach the mirror (assuming we have additional signals to mark your progress, otherwise I think you'd be dimmed and redshifted to invisibility), and then after 2 years, you return at what approaches the same instant that observations of you reaching the mirror reach the earth. The total time is 2 years, the total distance is 2 LY, your speed is measured as approaching c. The earthbound twin can calculate that "your time comes to a stop" as the Lorentz factor approaches infinity and the elapsed time according to your clock approaches 0. In this case, the earthbound twin has aged 2 years and the traveling twin aged essentially nothing. Now from your perspective as the traveling twin, as the Lorentz factor approaches infinity the distance to the mirror approaches 0 and the time to reach it (traveling at a speed approaching c) also approaches 0. The return journey is the same. The distance to your destination has length-contracted to nothing. You travel essentially no distance in no time. Your time continues passing at the usual rate, but the entire journey happens in an instant. Your speed is c calculated using limits. You'd also calculate that the earthbound twin has aged 2 years in that instant. As usual everybody is in agreement about what happened. You can only come to a conclusion like "you travel non-zero distance in 0 time" if you mix up frames of reference and use distance from one and time from another, or something like that. Also, light has no "frame of reference" but we can use "limits as v approaches c" instead of "v = c" to avoid problems. For example, in the above example at v=c it wouldn't matter if you traveled a rest distance of one m and back, or of a billion LY and back. Calculated using only "distance = 0 and time = 0" for the traveling twin, the rest distance is indeterminate. The earth twin could have aged 2/c seconds, or 2 billion years, or any value.

I seem to be mistaken. There is a good discussion about it here: http://www.physicsfo...ad.php?t=118941 I feel inclined to argue about limits... that 0 times infinity is indeterminate because it uses the "limit as x approaches zero" instead of 0 itself... but I think that is the point. In my example I'd used "infinity" as if it were a number, and that doesn't make sense. I should have said "an arbitrarily large number times 0 is still 0", or I may have gotten away with "an infinitely large number times 0 is still 0". The point is that an infinitely large number is a number (is that valid?) while 'infinity' is not. So if we want to treat infinity as a number, we have to do so using limits. That being the case, the question "How many times would I have to take away 0 from some quantity until it is all gone?", might validly be answered with "infinity", meaning "Not any finite number", meaning "No matter how many times you take away 0, it will not be enough to eliminate that quantity." Your link suggests a clue to the discussion though. The word "indeterminate" seems to indicate a number that can be anything, but it's still a number. "Undefined" seems to indicate the same, but also includes the possibility that the result is not a number at all (or that it's impossible or doesn't make sense). I'm in over my head, but I googled it and found this: http://mathforum.org....divideby0.html 'Why is 0/0 "indeterminate" and 1/0 "undefined"?' So I was wrong, but there's some way forward with this. 0/0 appears not to be "undefined". The solution to "0/x = 0" for x would be indeterminate. It is the same reasoning that x could be anything in 0 = 0*x, while 1 = 0*x is not true for any number x. Addendum: I was wrong; 0/0 = 1 is as valid as 0/0 = 0. However I still think that 0*(0/0) might still be 0! Multiplication by 0 doesn't make an undefined thing into 0, but it should make an indeterminate number into 0. Asking "what is 0/0?" in terms of the example I've been using, is like asking "How many times can I take 0 away from 0 until it is gone?" An answer of "0" works, but so does an answer of "1". Or 10. Or any number. It is indeterminate. Asking "What is 0*(0/0)?" is like asking "Given the task (of repeatedly taking away 0 from 0 some indeterminate number of times to end up with 0), if I repeated this 0 times, how many times would I have taken away a 0?" Not the best of wording... but the answer is 0.

I don't think that provides any definition. If I have some quantity, and I ask "How many times can I take away 0 before the quantity is all gone?", there is no answer. It's not any finite number, and it's not infinite either (infinity times 0 is still 0... you could take away nothing forever and the original quantity will remain unchanged). If you plot 1/x, it approaches -infinity as -x approaches 0, and +infinity as +x approaches 0. I've always thought of "1/0 is undefined" as meaning that it could be any value from -infinity to +infinity... it has no definite value. But the above example suggests that it really means "no possible value". Edit: Your example, now that I read the whole thing, sounds more like "How many times can I divide 0 into separate piles" or something, and that is more like 0/x, which is defined: it is 0. x is unknown or can be any non-zero value... but I don't think that makes it undefined. Your question is basically "solve for x" where "x has infinitely many solutions." I don't see how you could possibly figure that, given that x/0 is undefined. However, I think it might be possible to try to argue that 0/0 = 0. If you have x/0 is undefined in the sense that it could be any value and so isn't defined, but you multiply it by 0, the result will be 0 regardless of what x/0 might represent. In a sense, multiplying by 0 can restore definition??? But, given your example above, I'd say 0/0 or 0*x/0 is still undefined. x/0 does not mean "can be any number". I'd say it can better be described as "cannot be defined as a number." Multiplication is not defined for something that is not defined, so 0 * q is not 0 for all possible things q outside of the realm of numbers! Perhaps there is some operation that can be performed on undefined things and result in a number, but basic math operations ain't it. Anything divided by 0 is undefined and any math operation you perform on the result will also be undefined. Ain't no mathmertician so I reckon someone else'll explain this better.

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If you have a model (like "pushing gravity") that one says accounts for gravity, then it can only do so by predicting that gravity works as we observe it working. You can discredit the model by finding ways in which the model and observations do not agree. You're not going to get anywhere by assuming the model works differently from gravity (eg that instead of working like gravity, a pushing force would instead work like air pressure). So you can safely assume that any model of gravity is going to work "like gravity" unless it states otherwise. It works through walls. It works in air and in a vacuum. A pushing force vs. a pulling force isn't going to differ there. Also... gravity does work differently indoors. For regular "pulling" gravity, the walls of a building pull (negligibly) on each other. In a more extreme example of "indoors", the inside of a spherical shell contributes 0 gravitational force everywhere inside the shell (off topic but integration can show this) while to the outside everywhere it contributes a gravitational attraction towards the shell. Also... with GR you can treat gravitational acceleration as a type of inertial motion through curved spacetime (rough interpretation of http://en.wikipedia.org/wiki/General_relativity#Geometry_of_Newtonian_gravity). You don't need to treat it as a force at all, I believe, in order to model it. So if it is modeled by a force, whether it's pushing or pulling doesn't matter (I am guessing); either should be workable???. I don't think you would need to invent anything new (like an aether) to model it. I think it would be silly to argue either for or against a "pushing force" model by assuming the existence of an aether and/or to assume that it behaves at all like air pressure. For or against a pushing force, I think it makes more sense to treat gravity behaving like gravity, not like something else that behaves very differently.

Label the circles 1, 2, 3, and the squares A, B, C Draw them on a rectangle of paper. Connect 1-A, 2-B, 3-C then 1-B, 2-C, all with straight lines. Connect 2-A with a curved line going around B, and then 3-A with a curved line going around C. Connections 1-C and 3-B are missing. Connect 1 to the middle of the left edge of the paper. Connect C to the middle of the right edge of the paper (going up around 3). Roll the paper into a cylinder, and 1-C will be connected. Connect B to the middle of the top edge. Connect 3 to the middle of the bottom edge (going around C). Take the cylinder, and bend it into a donut. 3-B is now connected. On a donut-shaped world, this would be possible with utility lines underground and not crossing. But that's not a plane. In a donut-shaped universe, this would be possible with straight lines extending beyond the edges of the flat paper (and thus it's on a plane???). It may seem unfeasible, but it's not really much harder than making an apple pie from scratch.

How about this variation: There are 3 doors, only one of them good. There are 3 guards, and you can ask one of them a question that can be answered with the guard pointing to one door. One guard always tells the truth. One guard always lies. One guard is insane but consistent, and alternatively acts truthful or lies (you don't know which will be first), with each "atomically evaluable" part of your question. I'm not sure is this variation is consistent. Assume the insane guard would evaluate the "parts" in order that you speak them (or reverse order, would be equivalent, but she wouldn't "optimize" or reorder your expression). Also assume that if the guard is evaluating a part truthfully, she treats any previously evaluated parts as the truth. That is, the guard "acts honest" and conveys the previous lie rather than turning a lie into a truth (which is what the consistent liar does). Assume that if you ask a question that can't properly be answered, the guard will just give you a dirty look and no answer. What question would you ask? (If this variation doesn't make sense, I could try to specify exactly how the insane guard is to answer a question.)

Cuz I says so! Literally, I have chosen (and specified adequately I think) the particular details of the example to arrive at the paradox. If the guests can't be compressed at all and they entirely fill the finite space of such a hotel, there is no more spatial room for any new guests*. If they can be compressed indefinitely, then their spatial aspects don't matter at all, and the variation of the hotel is equivalent to the original hotel. In between is a hotel that can take an infinite number of additional guests one way, but not another way (though a simple change would allow both ways). I guess it should be noted that these guests by definition are not all the same. Each of the rooms is a different size, and each of the guests fills their starting room so each guest is a different starting size. * ... of non-zero size. I suppose I should also specify that each guest has a finite size. Note that a spatially full hotel is full, no matter how small a potential new guest is. If the rooms are sizes 1, 1/2, 1/4, 1/8 etc, and so are the guests, then every guest in the hotel has a non-zero size or volume. The hotel is 2 m^3 and the guests take up exactly 2 m^3 and if you add any non-zero positive fraction to two you get more than two.

Well, there is no last room, but I'd better fix my wording. To fill each of an infinite number of rooms that take up in total a finite space, it would require that the guests can have infinitesimal size. I should have simply said "Suppose each guest can be compressed but that there are limits to how much a particular guest can be compressed." Note that as it's set up (with room n+1 half as big as room n), and allowing guest n to be compressed to fit in room 2n, we must have that an arbitrarily high value of n will allow for an arbitrarily high compression ratio. So we must allow for compression to infinitesimal size. Please consider the problem with the correction that "but not to an infinitesimal size" is removed. The paradox still stands. Not all the guests can keep being compressed indefinitely. Yes, assumptions must be made in any of the Hilbert hotel variations in order to allow them to be possible, but the impossibility of the paradox isn't based on an assumption. You can assume the best case for every assumption. There is no "hidden factor" like money or time. You can assume that an infinite number of people can be checked in in a finite time, for example.

No, that's a misleading way of putting it. If "photons get dragged along with a frame" makes sense at all, then it's not one particular frame it happens to, but all inertial frames. Since this has already caused more confusion than good, I think this explanation should be retracted. What I meant by it was something like how the moon seems to "follow you" when you move. AND it seems to follow each individual who moves. BUT from your perspective, it doesn't seem to follow anyone else who moves independently of you. YET if you imagine someone else's perspective, you should imagine the moon seeming to follow them! If that's too confusing, then certainly my explanation will not be useful. A similar (but not equivalent) thing happens with light. The difference between the moon and light is that with the moon it's only an illusion due to its great distance; move far enough (say a million killometers) in a straight line and the moon will not stay in the same place in the sky. With light, no matter how far or fast you move, the light will always be moving (not stationary as with the moon example) at the same speed relative to you. With light it's not just an illusion. A simpler way of reasoning about this is that, instead of imagining light "being dragged along" with moving inertial frames but only from the perspective of observers in such a frame, it is simpler to just realize that any inertial frame is at rest according to observers in that frame. The above explanation tries to explain the same idea, but in a convoluted manner. This is the root cause of most SR paradoxes: Light signals moves at c in your rest frame, and the same signals move at c according to anyone else's rest frame, even though these frames are not at rest relative to each other. This is the resolution of most SR paradoxes: Time dilation and length contraction ensure that all of this happens consistently. It's only a matter of how complicated the details are for a given example. Check out some videos such as this: They should help visualize the kinds of relativistic effects (namely aberration) that are required to fully explain your paradox. So, again, your paradox might be restated as such: Two objects moving together naively seem to have gravity (and light) from the other "come from behind" (thus slowing the objects) due to the delay in the travel time of gravitons and photons. Yet, in the rest frame of the moving objects, there is no movement at all, and those signals should seem to come from the side, so there must be no forward/backward acceleration. An external observer who sees the the objects moving, will see them subject to aberration, and will not observe signals between the two "coming from behind". However, complicated details come up when you try to precisely describe the two objects' relative motion while accelerating from one inertial frame into a new one. Aberration (a result of length contraction and time dilation) resolves the paradox, ensuring that the photons we may have thought would seem to "come from behind" according to each object... will not. If we work out the details, we'll find that the 2 synchronized objects will always see each other "directly to the side" while in an inertial frame, consistent with the simpler example of treating them at rest relative to each other (while in an inertial frame). If you use instantaneous gravity you should use instantaneous transmission of light. I would not recommend this route. If any information is transmitted faster than light, you'll derive all sorts of contradictions that involve violations of causality. That is to say, the gravitational pull of an object will appear to come from the same place that light from the object appears to come. You will not be "pulled toward an object's current position" while "seeing the object in its past position". Relativistic effects also happen at Newtonian speeds. The original paradox involves two objects slowing each other down due to delayed gravitational influence. If they are moving at Newtonian speeds, any angle (off of perpendicular) of incoming gravitons, according to any observer, will be negligible. The angle of aberration (for the same observer) will be correspondingly negligible. You must neglect both, or neither, or you'll derive "small inconsistencies" where there are none.

Well done owl! I'm genuinely impressed. This is how I feel right now: I gave it my best shot and you sailed through with ease, unable to be confined by my oppressive need for "right thinking" (as I judge it) in others. I keep getting pulled into your threads because most of the others I'm interested in seem to die (sometimes my fault), while you manage to keep a lively debate going. I was worried I might kill this one, but you quickly proved me wrong. So I'll bow out now, and instead silently cheer you on from now on. I disagree with what you say, but I'll defend to the death of the thread your persistence in saying it.

Oooof. You would have been fired from my university. "(A and B) or Not (A and B)" is a tautology and is true. "Not (A and B) implies not B" is not true in general. Let A be "Objects exist". Let B be "Objects have properties independent of observation and measurement". "If the latter" ("they don't") is equivalent to "Not A and B", which is equivalent to "Not A or Not B". Not B would mean "Objects don't have properties independent of observation and measurement." Not A would mean "Objects don't exist", in which case their existence doesn't depend on how they are observed. In this case, your argument is invalid. Now maybe this is all pedantic and what you meant was what you'd get if you cleaned up the logic and resolved ambiguities with what I think you meant: "Either (all objects that exist have one or more properties independent of observation and measurement), or not. Assume: not (all objects that exist have one or more properties independent of observation and measurement). Then there exists an object that doesn't have any properties independent of observation and measurement. Then there exists an object that has no properties or has at least one property that is dependent on observation OR measurement." If this isn't the correct interpretation it can be corrected, but it's not possible to deduce your original conclusion as it is. If you used proper logic, you'd see the logical problem with your argument. You're basically trying to say that if one property of any object is dependent on observation, then all properties of all objects are. You are leaving out SO MANY possibilities, which cannot fit into your false dichotomy. Here for example, you jump to "all its properties" with no justification. I disagree with your statement. But I am not an idealist. I agree that there are things beyond science's ability to observe and measure. This includes for example things that we can infer the existence of, but aren't practically able to observe, or aren't able to measure (if you want to make any distinction). So I don't even have to limit myself to questioning the existence of things that are not even theoretically possible to observe. Personally I believe that there are objects that have some properties that are invariant and thus don't matter how they are observed, AND other properties that depend on how they are observed. This means that I believe it is possible that these objects have a real existence outside of my mind. Therefore I am not a subjective idealist, by definition. I do not fit into your false dichotomy. Hey out of curiosity, why is it that you no longer teach Logic and the Scientific Method? I'm impressed with your immovable resolve to not give in on any aspects of your arguments. It would be a super useful attribute for a student to have, to reach a point where there is nothing new to learn that makes any difference. Please stop using it to insult others.

This puzzle deals with a variation of Hilbert's infinite hotel... http://en.wikipedia.org/wiki/Hilbert_hotel. In the simplest example, the hotel can be full but you can make room for another guest by moving every guest one room over. By induction, you can repeat this infinitely many times, and accommodate an infinite number of new guests. You can also accommodate a countably infinite number of guests all at once by moving every guest in room n to room 2n, freeing up all the infinitely many odd numbered rooms. All the above stuff should be clear to anyone familiar with the paradox. Variation: A hotel with an infinite number of rooms can take up a finite volume of space. For example, if the first room takes up 1 m^3, and each room n+1 takes up half the volume of room n, then the space of all the infinite hotel rooms is 2 m^3. Suppose that each guest is exactly as large as their room. In this case, the hotel is full and it doesn't matter if you can find an empty room or not; the hotel spatially is not infinite and cannot fit any additional guests with non-zero volume. UNLESS... the guests can be compressed! Say each guest in room n can be squished to half their size and fit into room n+1. Then you can free up a room. So here is my paradox. Suppose each guest can be compressed but not to an infinitesimal size. Suppose for any guest in room n, they can be compressed only enough to fit into room 2n. Then, if the hotel starts full but uncompressed, you can free up an infinite number of rooms and accommodate an infinite number of new guests all at once by moving every guest in room n to room 2n. However, if you had instead freed up rooms one at a time by moving every guest in every room n to room n+1, it is clear you could not do this an infinite number of times. So how is it that you can accommodate an infinite number of guests arriving all at once, but not if they arrive one at a time? Is there a flaw in my logic? How do you resolve the paradox?

Well that answers my question satisfactorily. It implies that a realist accepts the reality of "all natural phenomena" independent of frame of reference, and any exception makes one an idealist. These are not generally accepted definitions of the words realist and idealist. Just as with "time", you are making up your own definitions for things. I have no problem with that; it's just that I was confused by when you were using words I thought you meant some accepted definition. (The false dichotomy is either your "all or nothing" separation between realism vs. idealism, or your arbitrary categorization of specific aspects of reality as subjective or objective. I'm not sure which because you haven't explained which, if any, aspects of reality the categorization depends on.)