37 new of 85 responses total.
That is called countably infinite. Ie, you can assign one integer for every item in your set. However, there is another kind of infinite, which can be considered "bigger." It's uncountably infinite, and it means you cannot assign one integer to each member of the set. The real numbers, for example, are uncountably infinite. Assign 1 to 0. Then pick as small a decimal as you can think of. Assign 2 to that. No matter how small it is, there are an infinite (uncountable, even) number of numbers between 0 and that number. Now that I've completely confused everyone, I'll go. My work here is done. :)
Then there are sets open or closed below (above). The set [0, 1] of all real number in the interval 0 and 1 includes 0 and 1, and is therefore closed above and below. However the set (0,1] is open below, and 0 is not a member of the set. This set *has no smallest number*, as no matter how small a number you choose, you can always take 1/2 (say) of that. Both sets are, of course uncountably infinite.
I like that better than the original answer, which implied don't ask. If I didn't HAVE to know I wouldn't ask. Call me fixated but I needed an answer that made"sense" to me. :)
Back to #45, however - the universe is not infinite. It apparently started with the "big bang". Therefore it has a boundary at at 12 billion light years, give or take a few. Of course, what constitutes a "boundary" to the universe is unclear, since apparently there is nothing, not even space, beyond that. Or, we need to ask, what is the "medium" in which the Big Bang occurred?
The current guess, as of when I last took Modern Physics (last year) was that the universe is finite, but unbounded. That means that it has a limited amount of space in it, but there is no real edge. It's the 3D equivilant of a sphere, which is a finite but unbounded 2D surface: You can go forever on the surface of a sphere, and never come to the end, but it has 4*pi*r^2 surface area.
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That's because the universe *is* implausible. Our definition of what's "plausible" comes from our only being able to sense low velocities, a trivial range of wavelengths, an equally trivial range of temperatures, and experiencing only a trivial range of substances. How can you expect to find the "truth" plausible if you are so deprived sensorally? It used to be "intuitive" that the earth is flat (and, turtles all the way down....). Of course, what we are slowly coming to observe, are such implausibilities as black holes, time dilation (a clock in orbit in a satellite *does* appear to run slower (to have lost time after it returns) than the same clock on earth), superfluids (that exhibit zero viscosity) and superconduction (conductors with zero resistance), quarks, quantum linkage (events that appear to be linked though separated and space with no signal between them), even light interference (an illuminated surface that can be turned dark by shining a light on it). None of these are "intuitive". Just go with the (quantum) flow, Valerie.... :)
Right, Rane just beat me to saying that. By expecting things to remain plausible, you are tacitly assuming that nature remains the same when you change the scale of things. There is plenty of scientific evidence that it does not. When you go faster and faster, at first, the rules of normal speeds continue to apply, but the closer to the speed of light you get, the more they are actually different. By being familiar with only slow speeds (relative to "c"), you are able to extrapolate from a very limited sense of what is plausible. Same goes for sizes, as you get smaller and smaller. Quantum effects become more important. However, at the sizes of things that we are most familiar with, quantum effects are quite implausible. For small things, like electrons in orbit, they are crucial though. As you approach the edge of the universe, you also approach the beginning of time (the big bang). The edge is receding from us at the speed of light. You can no more reach that edge, than you can go faster than light or back in time. But yes, most evidence shows the universe to be finite. There is an edge. Now, to go back to Carol's question of resp:45. It seems to be about physics, but any question involving the concept of infinity is also a methematical question. Dang confused us for a bit with the comments on uncoutable infinities. That's an interesting topic, but not germane to Carol's question. If you stacked up an infinite number of marbles in an infinite universe, it would be a countably infinite number. That is so, because (regardless of how they were packed) they could be numbered in a rank order. Perhaps arbitrarily define your starting point and number them based on their distance from it. We have a countable infinite number of integers, and we can put them in a 1:1 relationship with the marbles. We can do it again for the marbles that are bigger, too. So there are exactly the same number of marbles when they are bigger. This idea of using a 1-to-1 pairing of things is central to the way infinite numbers are dealt with in mathematics. Now for the big question. Which is bigger - the infinite pile of small marbles, or the infinite pile of large marbles? Assuming the question refers to the mass, and that the marbles all have the same density. I think they are the same size. Here's my logic: The volume of the 1" marbles is exactly 8 times the volume of the 1/2" marbles. Therefore the mass is also 8 times greater. We can split the pile of small marbles into 8 equal piles. Lets do it by first numbering the pile, and then usin the value of the remainder when dividing the number of each marble by 8 to decide which smaller pile it goes into. Each of the 8 piles of small marbles can also be put into a 1:1 relationship with the pile of large marbles. Therefore we can associate with each marble in the large pile, 8 small ones, having the same aggregate mass. We can do this forever, matching every large marble agains every small marble (8 at a time).
(that is to say, all countable infinities are equal)
...but one is still eight times as many as the other...though equal.
It is as many times the other as you like. Any countable infinity can be placed into a 1:1 relation with another one that is n times as large (or small) for any integer n. This means that you can multiply or divide any countable infinity by any integer and still have the same amount ... a countable infinity. I prefer to stick to the statement that they are equal, because that carries some meaning.
(except where n=0. Then you have some strange stuff. Usually called undefined.)
Well, you get into trouble when you think of infinity as a number.
Mathematicians instead talk about sizes of sets, which they call
"cardinalities". You can compare cardinalities, as srw did in proving that
the two sets of marbles had the same cardinality as that of the positive
integers (that cardinality is usually represented either by a lower-case
omega or the Hebrew letter aleph with a 0 subscript ("aleph nought")).
I don't think it's a valid question to ask whether the masses of the two sets
of marbles are the same, though, because there you're talking about numbers
rather than sizes. The best you can do is say that both masses are infinite
and leave it at that.
BTW there are lots of infinite cardinalities; those of the integers and of
the reals are just the two best known. I don't know them the way a
mathematical logician would, but I know they get quite esoteric.
It was one of Hilbert's problems to prove "the continuum hypothesis", which
states that there is no cardinality which is larger than that of the integers
and smaller than that of the reals. Unfortunately, the continuum hypothesis
was proved to be undecidable; you can't prove it one way or the other given
the normal axioms of logic. So there might be no cardinalities in the middle,
or there might be 1, or 17, or a countably infinite number, or an uncountable
number.
Physics quibble! Re #56: the edge of the universeis not receding from us at the speed of light. In fact, despite the fact that the universe is undenibly finite, there is *no edge*. The easiest way to explain this is to step back one dimension, to a 2-D world. Think of our univers, not as the infinite sheet of Einstein, but as the suface of a balloon. Finite, of course, we (as 3-D beings) can even measure its suface area. Yet, I defy you to find an "edge" (not counting the blowhole). ^^^^^^ surface (both times) So, in theory, one could head off into space in a straight line, and end up back where you started. In Stephen Hawkings excellent book, _A_Brief_History_of_Time_, however, he states that even if you could travel at the speed of light, and start at the beginning of time, you would come back to your starting point only at the end of time, when the universe collapses on itself.
Let's see - and I any wiser...(no....). I've heard the expression that the universe is "finite but unbounded", but I've never understood the practical import of that. We *appear* to be at the center of the universe since its furthest observable limits are the same distance in all directions from us, but I've also heard/read that this is true no matter where you are in the universe. Another conundrum. If I were at the furthest reach we can observe - and looked further - would the furthest reach from there be just as far away? Would I see *our nebula* in that direction? That doesn't seem logical, as then we could estimate the "size" of the universe. Oh well...I'll sleep on it.
Well, sure we could estimate the "size" of the universe. That's what finite means. It's the unbounded part that results in the "center of the universe" effect. Return to the 2-D anology. You are 2-d creature on the surface of a sphere. At any point on the sphere, the "furthest" away point is on the opposite side, and it always appears to be the same distance away.
But you can reach it - and return home - by proceeding in the straight line on a sphere. Can we do that in this unbounded universe? That does not seem reasonable.
Well, obviously, no one knows. Theoretically, yes we can. The problem comes with the amount of time it takes. Because the universe is expanding (the radius of the sphere is increasing) it's possible that you could never reach that "opposite" point without going faster than light, before the universe "ends", however it ends. (BTW, why doesn't it seem reasonable? It always has to me, so I'm curious.)
Because the geometry would be too coincidental to be exactly lined up with the point at which you started, after proceeding an an arbitrary direction. Or is there only one such straight line along which this can be done (there are an infinite number on a sphere)? It doesn't seem to me relevant that one can't exceed the speed of light. *Conceptually* I can jump anywhere, instantaneously, even onto the furthest galaxy. Of course, it won't be where I see it now, but I just figure out where it really is, and then jump....
But due to the warping of space time, it won't be *when* you see it now, either. If you could jump to a spot near what we might think of as the edge of the universe, you would find that it is a much younger universe there. In the limiting case. at the edge of the universe, the big bang is still going on. (that should clear it right up :-)
Quite right, but that doesn't stop me from doing it conceptually. If the big bang is spread out around the edge of the universe, it must be very dilute, and now just a sort of hiss, not a bang.
resp:67 You're right, there would be a fairly limited number of geometries that ended up with that effect. However, if you can conceptually jump anywhere, I can conceptualize the universe as a hypersphere. :P
In resp:69 you said "hiss not a bang." Well, due to the compression of time, the universe there is only a few seconds old, so if you squeeze all that hissing into a few seconds, it's probably more like a bang.
It is only a few seconds old, but it is spread over the outer "surface" of the universe, which dilutes it greatly.
If you lived on the surface of a sphere, and started walking in what seemed to you to be a straight line, you would eventually end up at the point "opposite" of where you started. (Think of where you started as the North Pole: Any direction you walk is south, along a meridian, and you eventually end up at the South Pole.) If you keep going in the same direction, you end up back where you started. Is there any particular reason that some directions would work, and others would not? In the real universe, there are stars, so, for practical reasons, not every direction is suitable. But conceptually, there is no reason any direction ought to be better than any other. "Unbounded" just means "without an edge". The Universe can be of limited volume while still being unbounded for the same reason that the surface of a sphere can be (and is) unbounded (where is the "edge" of a sphere?) and yet of limited (and, indeed, calcuable) area.
It is the consequences of those general questions I've been asking about. A reiteration of the theory doesn't help. I would like an explanation of how earth could be still "straight ahead" if we went in a straight line to the "edge" of the universe and beyond (by edge, I mean the point to which the universe has expanded since the Big Bang. Of course, I am approaching this conceptually so that I can reach any point instantaneously as measured by earth time.
Because the universe is curved in the fourth space dimension, just as the surface of a sphere is curved in the third, a path which seems to be straight to us, is actually curved. Personally, the only way I keep track of the discussion is by bouncing back and forth between the 2-D sphere surface, and the "real world", hoping the extensions to the analogy still fit. Does this help?
Rane, according to this theory, the universe does not have an "edge" to which it has expanded since the big bang. It is expanding in 3 dimensions, certainly, but the edge is in the fourth (or higher) dimension. In the sphere equivilant, the surface of the sphere has no "edge", because the expansion is coming in the third dimension, namely the radius of the sphere. Certainly, the surface area of the sphere is increasing, but it has no "edge" to which it has increased.
Did it have an "edge" at a couple of nanoseconds into the Big Bang? They speak of its "size" at that time. By the way....what was outside the universe at that time? :) [I understand the sphere analogy, but I don't understand the universe....]
The Universe has a "size", just as the surface of a sphere does, and it can even be measured, by looking at the curvature of space (or the surface), soemthing that can be done indirectly by measuring the angles of large triangles.
Presumably, no it didn't have an edge a couple of nanoseconds after the Big Bang, it was just a really small hypersphere. (Incidentally, at that point it likely was a hypersphere, rather than the bizarre shape it now. :) I don't understand the universe either, but I sort of understand the basics of one of the current theories about it.
It does sort of dawn on me that at a couple of nanoseconds we are not talking about the space and time we "know" today. I'll live with that.
As to what is outside the universe, think of it this way. There isn't anything, in the two dimensional sense, outside the surface of the sphere. Likewise, there isn't anything in the 3 dimensional sense outside our universe. There *is* outside, but only in higher dimensions.
Now what's this about the universe being *10* dimensional?
Actually, it is theorized that the Universe originally had 26 dimensions, of which I think four were time axes, but they quickly collapsed into the three space and one time dimensions with which we are familiar. The 10-dimension stage was of longer duration than any other stopping points, I believe.
I don't know about the 26 dimensions. I think you are referring to string theory (and superstring theory). Here is some interesting text I found on this subject (source: http://www.lassp.cornell.edu/GraduateAdmissions/greene/greene.html where the author was talking about incompatibilities between general relativity and quantum echanics at the scale of elementary particles....) -----begin quote----- String theory solves the deep problem of the incompatibility of these two fundamental theories by modifying the properties of general relativity when it is applied to scales on the order of the Planck length. String theory is based on the premise that the elementary constituents of matter are not described correctly when we model them as point-like objects. Rather, according to this theory, the elementary ``particles'' are actually tiny closed loops of string with radii approximately given by the Planck length. Modern accelerators can only probe down to distance scales around 10^(-16)cm ( 10^(-17) in) and hence these loops of string appear to be point objects. However, the string theoretic hypothesis that they are actually tiny loops, changes drastically the way in which these objects interact on the shortest of distance scales. This modification is what allows gravity and quantum mechanics to form a harmonious union. There is a price to be paid for this solution, however. It turns out that the equations of string theory are self consistent only if the universe contains, in addition to time, nine spatial dimensions. As this is in gross conflict with the perception of three spatial dimensions, it might seem that string theory must be discarded. This is not true. -----end------ The author goes on to explain how this conflict can be resolved by assuming that 6 of the spatial dimensions are curled up at scales too small to be measured by experiment. I think physicists prefer the term "curled up" rather than "collapsed," although this may not be an important distinction. I found this site clearer than most on the subject, although still this is a very difficult subject.
Yes, "curled up" is what I meant, but I used a term that came more easily to mind, for those less familiar with that kind of physics. (like me! *grin*)
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