Russell’s Paradox is a famous conumdrum of logic related to set theory in mathematics. It is sometimes illustrated by the following example: suppose you have a library of books, with two master index books that cover every book in the library including themselves. The first index book lists all the books in the library that somewhere in their text mention their own existence. The second index book lists all the books that do not. The paradox is that there doesn’t seem to be any way to account for the second book. If it doesn’t mention itself, it ought to, but if it does then it shouldn’t. The case of the first book is less impossible but still puzzling: it could plausibly either list itself or not and there’s no way to choose between the two possibilities.
A time travel paradox problem I recently read about seems startlingly reminiscent of Russell’s Paradox. The September 2002 issue of Scientific American had an article about the possibility of time travel via wormholes. It cited the following problem: suppose a billiard ball is approaching the entrance to the wormhole. Suddenly out of the other end of the wormhole comes the billiard ball as it will be in the future. The “second” billiard ball collides with it’s past self, knocking it off it’s path so that it will never enter the wormhole. How then can it have traveled back in time to interfere with itself? The article illustrates another possibility that seems equally paradoxical. Suppose the billiard ball would have just missed the entrance to the wormhole, but suddenly it’s future self appears out the other end, collides with the past billiard ball, and knocks it into the wormhole. That could happen, but why did it, when it could equally possibly have simply never done so in the first place? (The famous Predestination Paradox of so many science fiction stories).
The resemblence seems obvious to me, and I wondered if anyone has ever tried to apply some of the proposed solutions to Russell’s Paradox to the time travel dilemma?
The time traveller’s paradox you mention is actually something of a theoretical resolution of yet another paradox which is the famous “Grandfather Paradox” of time travel. (You know the story, you go back in time and kill your grandfather, so whodunit if you couldn’t have been born?). One interesting thing that has occurred is that “time travel to the past” solutions seem to have the peculiar property that an initiatiation period and an ending period of possible time travel occurs (when the wormhole is constructed constitutes the EARLIEST possible time one can travel back to). Some theoretical time travelists set up a “Grandfather Paradox” with these time-travel machines using purely mechanical objects (say a particle goes back in time to the point it was sitting just before it left and detonates a bomb… KABLAMO! what goes back in time? Grandfather Paradox!)
The question really comes down to causality. There are two ways causality could work. One way is you go back in time and you enter and alter an “alternative” universe (something like the many-worlds interpretation of quantum mechanics). You can murder your “grandfather” fine because your “grandfather” in this universe isn’t your “grandfather” in the other universe. An alternative way the universe (s?!) might be set up is it is self-contained and self-consistent and you are predestinaltionally forbidden causally to “murder your grandfather”… that the universe sets itself up in such a way that prevents you from doing something that violates fundamental causality.
What a challenge for theorists! Can they prove from a priori conditions (that is ACTUAL time travel solutions) which interpretation is correct? Turns out, they can’t. Amazingly, it seems that for ANY solution that is proposed where causality is violated an equally good explanation is provided for where causality is NOT violated. For example, instead of the particle going back in time and blowing itself up, it might blow up, send a piece of shrapnel through the wormhole which comes back out and triggers the detonation. There’s no way to say which solution is the one which is actually theoretically favored. It seems that for every solution that violates causality, there’s another equally plausible one that doesn’t and there’s no way to choose between the two of them.
This is what you describe as a “paradox”, but it’s really simply a yet unsolved conundrum about which way the universe is. It is either causal or it’s not and it’s impossible right now for us to know which way it is without going back in time and doing such a thing. Unfortunately, the only time travel solutions that are known right now require ridiculous… on the order of magnitude of GALACTIC… amounts of mass to acheive the desired effect and we aren’t going to find out soon. If some does eventually build a time machine to the past, they will be able to go back and time and try to alter history. Then they can witness if the worldlines are self-consistent or not and they will know for certain what kind of universe we inhabit ([HIJACK] potentially… if we are truly self-contained it might be hard to prove that for certain as the experiment done to alter history will have failed and you really haven’t falsified anything but that that PARTICULAR case didn’t result in any perceivable changes of world lines. That might not mean that it is IMPOSSIBLE to do so. Perhaps it is just very hard? However if an alteration is witnessed, then the time traveller will know for certain causality has been violated (assuming they can trust their observations) and it will be up to them to convince the rest of us… Contact-like… that what they witnessed was actually true. But now I’m getting truly wicked in this area [/HIJACK].)
So, to answer your question, I can see how these self-containing sets might be an intriguing comparison, but really it’s not so much a PARADOX as simply a statement that we don’t know the correct way to interpret the solution of ANOTHER related paradox.
For time travel paradoxes, how come I’ve never heard this possibility…
Ok, say you have the Grandfather Paradox. IF it were possible to travel back and kill your grandfather, wouldn’t you create an endless loop?
A) You go back and kill your grandfather.
B) Without your grandfather, you wouldn’t exist to kill him.
C) Since you couldn’t go back and kill him, he lives, you’re born.
D) Since you’re born, go back to A
I myself believe that you go back kill your grandpa, therefore you don’t exist. Thats it. None of that not existing so you can’t go back stuff. You kill him, your gone, hes still dead. It just makes it easier.
Yeah, Joel, this is what makes it a paradox. It is contradictory when taken to its logical conclusion. It’s not so much an “endless” loop as it is an impossibility. Since it is impossible to not kill your grandfather and kill him in the same sequence of events, it therefore must be the case that there is some inherent contradiction in the premises (either that you’re not allowed to go back in time, you’re “not allowed” to kill your grandfather, or that the person you kill isn’t your “grandfather” at all). Hawking proposed that the first solution to the paradox was the correct one. “The laws of nature always conspire to prevent time-travel to the past”. This has been disproven on theoretical grounds (there are perfectly well-behaved physical systems that we can model that WOULD allow for time-travel to the past, even though they may be utterly impractical and never observed). The two remaining resolutions are what constitute the possibilities in time-travel. Either you don’t kill your grandfather because the universe is self-consistent or you don’t kill your grandfather because it’s not the same “grandfather”. We don’t know which interpretation is correct.
You are, however, touching on another issue of time travel: the so-called “djinn” particle. Let’s say you go back in time and procreate with somebody. You have a child and that child grows up to become you. Then you go back in time and procreate to create yourself. You would be a so-called “djinn” particle that is basically a worldline that “loops” back in on itself. This is a pretty weird thing to consider. It is totally outside of the universe unlike “normal” particles as there really is nothing that “creates”. This is a concept Richard Gott jumped on and used as a possible explanation for the existence of the universe itself. Hey, if primal cause seems like such a conundrum (who created the universe and who created the who that created the universe?) if the universe, djinn particle like, is the cause of its own existence time travel to the past could be a sort of primal cause.
Right now, Rich Gott’s theory is entirely untestable speculation… but it’s an intriguing one nonetheless. You can read about it in his pop science book, Time Travel in Einstein’s Universe.
One view: if time travel will ever exist, it exists already. You theoretically shouldn’t be able to kill him, no matter how hard you try. If you succeed, try killing your grandmother’s mailman. This is similar, I suppose, to the possible solution to Russell’s paradox, a proposed set of axioms that prevent the set of all sets from being a set. Just can’t do it.
Actually, I take that back. The set of all sets is ruled out as well. Sets are (or should be) stratified into levels that correspond to their construction, the elements of the set must (or should be) be less that the set itself. A “set of all sets” is too comprehensive and doesn’t tell you anything, like an infinity of all infinities. It contains elements greater than itself. To speak of sets of all sets, possibly, perverts the idea of a set to begin with, or is a different thing from a set altogether.
Russell’s not-very-incredibly-deft semantic abuse is to hypothesize something which exists, i.e., the second book, then claim it cannot exist.
I don’t agree the book exists, or could exist. So there’s no paradox.
Just because he can write a sentence claiming there’s a book which is both painted all blue and painted all red, doesn’t mean his claim has meaning. It doesn’t require the world to explain his statement. It does prove he was a good showman.
Pure rhetoric, and verbal slight-of-hand. Which Russell was quite accomplished at.
Time travel has far more deep and difficult problems, dealing with the physics, time, and the nature of reality. I.e., not rhetoric, and bona fide scientific issues.
JS Princeton, about the part where you said…
“You are, however, touching on another issue of time travel: the so-called “djinn” particle. Let’s say you go back in time and procreate with somebody. You have a child and that child grows up to become you. Then you go back in time and procreate to create yourself. You would be a so-called “djinn” particle that is basically a worldline that “loops” back in on itself. This is a pretty weird thing to consider.”
That couldn’t be possible. Before going back in time, you have to be born first, which means that you have to have an independant father, or, that is to say, your father couldn’t be you. Going back and being your own father would mean that you created yourself.
About the book paradox. Doesn’t this assume that both books are 100% correct? I don’t expect the books to be completely correct even if they were normal books like an index of science fiction books and an index of fantasy books. Only in the world of pure logic are the Russell books impossible.
That wasn’t his claim, though, and to reduce it to such is (please forgive me, no offense meant) a superficial reading. Paradoxes absolutlety exist, otherwise we wouldn’t have a name for them. To dismiss Russell’s paradox, or Zeno’s paradox for that matter, as a mere “sematic game” is sophistry, to a degree, on the level of “it doesn’t make sense so let’s quit talking about it”. No, it doesn’t make sense, but it doesn’t make sense within a logical system that has proven itself so useful it helped make the computers we’re using and should never not make sense. Solving paradoxes leads to breakthroughs in thought, like Zeno’s paradoxical challenge to the school of thought proposed by the Pythagoreans led to the development of calculus.
Being your own father would make a closed circle.
All right, let me try again.
Dad comes along, meets Mom.
They have a kid, you.
Now since you already have a dad, you couldn’t go back, have sex with your mother, and become your own dad.
And if you say, what if dad was you all along, I’ll have to say, no. If Mom never has sex with anybody, she has no kid. And since she has no kid, you wouldn’t be around to go back and have sex. Thus we’re back to, if you exist, then it’s not you who’s your father.
However, you’re still stuck thinking about a linear cause in time that no longer applies. These particles are totally self-consistent within the universe. There was no “cause” to them (there is no “dad” at all), there is only you and you alone, looping back on yourself.
One of the problems with complicated djinn particles is that if they have any chance to decay, they will be completely decayed from the “endless loop” phenomena of the particle going back in time and being itself. This independent decay in the reference frame of the particle is troubling to some who work with time travel because it seems to indicate the particle is NOT the same as it was when it began. The solution to this problem is either the “many worlds” interpretation or to have nature conspire so that decay does not occur. It is not clear which one of these conditions is true.
On further examination of my example, I realize that actually you aren’t QUITE a djinn particle. This is because you aren’t the same person that is conceived by yourself (you are two different people at that point in proper time). The classic example of a djinn particle is a watch that is given to someone at a certain point with the instructions: “Here is a watch that I was told to give to you. Someday you will go back in time and meet a person. Tell them exactly what I tell you and give them the watch.” The person follows the instructions. In the future, they end up going back in time to the very point where they meet themselves and hand themselves the watch and say to themselves, “Here is a watch that I was told to give to you. Someday you will go back in time and meet a person. Tell them exactly what I tell you and give them the watch.” Etc. The watch is a djinn particle. There is no “existence” for it outside of its looping worldline. Djinn particles are easier to explain in terms of elementary particles like photons or positrons… they get confusing when you talk about macroscopic objects like watches.
It is impossible for you to make a djinn particle, however, there is no way for you to know at this point what, if any, particles that are around you are djinn particles (they may just be on the upside of a loop that swings back arouind and creates itself). A djinn particle needs to come into the normal “stream” of time by means of a time-travel to the past and so would appear to show up at some point (it was not in the observable universe before the earliest event in its time-loop) and would no longer be around after some point (it will not be in the observable universe after the latest point in its time-loop).
It may seem contradictory to consider these particles, but they are totally self-consistent solutions. The question as to whether these particles actually exist or (“how” they exist) is much more difficult. We don’t know of their existence until we actually see them. In this way they are kind of like the universe, no?
Ok, say I buy a watch, in oh say, 1992. Say I travel back to 1991 and give myself the watch before I buy it. Ok, there, I’ve just created an endless loop, right?
But if I have sex with my mother, BEFORE I was born…
Ok wait, I’m just repeating myself, I’ll try a different approach.
Actually, I have to get going now, so I’ll post again later. All I have time to say, is that I tried using the GOOGLE search to look up Djinn particles, but it can’t find any web sites that talk about them.
My favorite example of a “Djinn particle” – although I’d never heard that term before – is with something like a piece of music. Imagine going back in time to 1800 and handing a young Beethoven a copy of his Ninth. So now who wrote it? Beethoven didn’t, because he was inspired by a copy he’d already seen. And the copy you take back to him wasn’t written by Beethoven either, as it is a descendant of his copy-inspired original. So the piece is now without an author; it just exists.
It’s brain-twisting possibilities like this that make me want to agree with Hawking: hopefully the Universe has the decency to prevent these kinds of things from happening.
I’ll disagree with JS Princeton on two points: First, he claims that a time machine could not access any event prior to its own creation, and second, he states that all known time travel solutions would require Galactic masses. Actually, there are a few solutions (such as the Alcubierre solution, although he didn’t directly apply it to time travel) which suffer from neither constraint.
The biggest constraint on time machines is not that they require too much energy, but that they require too little: Specifically, they all seem to require the use of something with negative energy. So far as is known, this is impossible, and if it is possible, it requires quantum effects. Since we don’t yet have a single theory which can reconcile quantum mechanics with general relativity, we’re still in the dark on this one.
No, you’re missing the point of how a djinn particle works. You DON’T buy the djinn watch at all. In 1991 you appear and give yourself a watch. Then later you travel back to 1991 and give yourself the watch. That’s the loop. No initial buying required (or allowed).
The way you illustrated it your paradox is grandfather-like… it violates causality …OR… since 1992 you have had two watches (one given to you in 1991 by yourself one that you buy in 1992).