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| Author |
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| 25 new of 85 responses total. |
dang
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response 46 of 85:
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Oct 9 21:31 UTC 1997 |
No, one infinite is not "bigger" than the other. The real problem is that
it takes a very strange mind to understand the concept of infinity. :) The
easy answer to your question is "You can't fill it up with marbles, of any
size, so don't worry yourself about it." Yes, if you had an infinite number
of marbles (where would you keep them?) and an infinite amount of time, and
could put them in the universe at an infinite speed, then you could fill it
up. However, you don't have any of the three, so don't bother trying. Or
something like that. ;)
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valerie
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response 47 of 85:
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Oct 10 02:23 UTC 1997 |
This response has been erased.
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rcurl
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response 48 of 85:
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Oct 13 20:56 UTC 1997 |
It is even more interesting. It is easy to show that there are the same
number of integers as there are rational fractions, in the sense that you
can associate an integer with every rational fraction you can generate.
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dang
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response 49 of 85:
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Oct 20 17:44 UTC 1997 |
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.
:)
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rcurl
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response 50 of 85:
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Oct 20 19:13 UTC 1997 |
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.
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willow
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response 51 of 85:
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Oct 25 23:07 UTC 1997 |
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. :)
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rcurl
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response 52 of 85:
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Oct 26 16:42 UTC 1997 |
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?
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dang
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response 53 of 85:
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Oct 26 21:13 UTC 1997 |
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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valerie
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response 54 of 85:
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Oct 28 05:08 UTC 1997 |
This response has been erased.
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rcurl
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response 55 of 85:
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Oct 28 20:11 UTC 1997 |
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.... :)
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srw
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response 56 of 85:
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Oct 30 05:14 UTC 1997 |
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).
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srw
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response 57 of 85:
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Oct 30 05:16 UTC 1997 |
(that is to say, all countable infinities are equal)
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rcurl
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response 58 of 85:
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Oct 30 07:12 UTC 1997 |
...but one is still eight times as many as the other...though equal.
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srw
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response 59 of 85:
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Oct 30 18:58 UTC 1997 |
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.
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dang
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response 60 of 85:
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Nov 4 17:58 UTC 1997 |
(except where n=0. Then you have some strange stuff. Usually called
undefined.)
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aruba
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response 61 of 85:
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Mar 3 09:18 UTC 1998 |
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.
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lilmo
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response 62 of 85:
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May 28 21:46 UTC 1998 |
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.
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rcurl
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response 63 of 85:
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May 29 04:41 UTC 1998 |
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.
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dang
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response 64 of 85:
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Jun 23 18:45 UTC 1998 |
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.
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rcurl
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response 65 of 85:
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Jun 24 04:43 UTC 1998 |
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.
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dang
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response 66 of 85:
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Jul 8 19:04 UTC 1998 |
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.)
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rcurl
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response 67 of 85:
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Jul 8 19:19 UTC 1998 |
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....
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srw
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response 68 of 85:
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Jul 12 05:38 UTC 1998 |
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 :-)
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rcurl
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response 69 of 85:
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Jul 12 16:47 UTC 1998 |
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.
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dang
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response 70 of 85:
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Jul 16 17:16 UTC 1998 |
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
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