The universe is accelerating, ie the galaxies and moving farther and farther away from each other into infinite expansion.
But could our model of the universe as infinite be as flawed as the old view that the world is flat? Perhaps, as Einstein suggested, space-time is curved, connecetd and compacted.
Yes some irrational numbers can be evaluated, but this evaluation does not allow you to have complete knowledge of the series.
Proof that the probabilty of choosing sqrt(2)/2 at random from [0,1] is zero:
Let s[sub]n[/sub] be the nth binary digit of sqrt(2)/2.
The probability that the nth digit of the number we choose is equal to s[sub]n[/sub] is 1/2.
Therefore the probability that the first m digits of the number we choose are all equal to the first m digits of sqrt(2)/2 (given the way we’re choosing the numbers; i.e. analogously to coin-flipping) is (1/2)[sup]m[/sup].
As m goes to infinity, (1/2)[sup]m[/sup] goes to zero. Therefore the probability that all of the digits of the number we choose are equal to the corresponding digits of sqrt(2)/2 is zero. Therefore the probability that the number we choose is sqrt(2)/2 is zero.
QED.
(I don’t have to have “complete knowledge of the series” to prove that the probability of choosing the series is zero. I don’t have to know what each digit of the series is, I just have to know that each digit is well-defined. The nth digit of sqrt(2)/2 is just the nth digit of sqrt(2)/2, and that’s all I need to know for the proof.)
I’m stillnot yet 100% convinced, so I’ll have to think on this one a bit before I answer it.
NEat, Math Geek. Very intuitive, IMO.
Yes, good show. MC, do bear in mind that Math Geek’s proof never referenced the value of any specific bit of s, so it’ll work for any real number in binary.
It is a good proof, but I still need to think about it a little more and perhaps revise my posistion a little, but one thing I would say is that it only works on irrational numbers that can be evaluated.
What do you mean by “can be evaluated”?
to unhijack this thread…
No there probobly wouldn’t be an identical earth, as the veiw of the stars effects us, constelations and such, so there not only would have to be an identical earth, but an identical galaxy and an identical set of galaxys around it and ect ect ect ect as far as ‘we’ can see
MC, it should work on any real number that can be constructed. If it worked on every real number… I don’t think it can.
Can it? 
I personally wouldn’t say another Earth, maybe another planet that is similar to Earth and supports life forms, but definately not an exact duplicate of Earth. Just like no two people are alike, unless they came out of the same mother at the same time, it’s plausible to assume that there are no two planets alike and if someone else wrote the same thing before me, I wouldn’t know because I didn’t read the replies…at least not all of them.
It does work on every real number. All we have to know is that every real number between 0 and 1 has a well-defined binary expansion, and we’re good to go.
Cool.
'How so? Scientists report that earth is a very rare example of a planet, so it would be unlikely that there would be infinite amounts of earth. Also, how would it be able to devolop so that the “duplicate” is doing everything you are doing in real time?:rolleyes:
But what’s the point? 1 x 1 x 1… is still just 1.
Given: The universe is infinite…
The earth is smaller than the universe, and clearly not infinite.
The earth is made of a finite number of ‘particles’, by virtue of being not infinite.
There are only a finite number of ways of arranging a finite number of particles. This is a very very very high number, with large number of particles. Infinity is still higher.
Of the ways of arranging particles, a smaller set will exist that compose ‘viable’ planets. ie: the magma is on the inside, the air on the outside.
Of all the possible ways to arrange a ‘viable’ planet, a smaller set will closely resemble our world.
This is both a finite number, and a much smaller number than any of the previous finite numbers.
A smaller set still could be described as “A combination of particles that, in every meaningful way, resembles our own Earth.”
Given that the universe is infinite, and given that we know this can happen once, the odds of it happening again are VERY VERY high, IMNSHO.
Granted, this could be the HHHHHHHHHTHHHHHHHHrepeating world, with an infinite number of heads and a single tails. But what are the odds of THAT? q;}
In fact, try the opposite now that I think about it. State categorically that there is NO other planet out there identical to ours, again granting that the universe is infinite. I can’t argue that… can you? Infinite universe… infinite space, infinite material, infinite energy… and in all that space this CAN NEVER HAPPEN TWICE. Explain why?
On sets of numbers (odds, evens, etc.):
These are abstract ideas, not physical entities. The OP referred to physical entities being identical. I don’t think the sets of numbers argument contradicts the OP.
the universe is not infinite. it is quite finite. by all estimations, at least. it is probably between 8 and 20 billion light years in radius, though you can never get to the edge, as you can’t travel at the speed of light. the space in which the universe exists might be infinite, but what is that besides a mathematical idea?
just because something is infinite doesn’t mean that something else must exist in it at all, let alone more than once. there are no negative numbers, for instance, in the set of numbers greater than 0. there is only one 2 in the same set. why do you think it must be that there is something exactly like our planet, given infinity? why couldn’t it be, say, infinite cheese after some distance from the earth (that is not a set of numbers)?
Radius? What does that mean with respect to the universe? I thought the universe was a cone.
NUMBERS and THINGS are two entirely different things. Numbers are mental constructs; things actually exist. You can have two identical things, such as two ashtrays. You can’t have two ‘mental constructs of an ashtray’. Once you’ve created the word ‘ashtray’, any other word you use to mean ‘ashtray’ is the same thing. A rose by any other name etc. “Two” is just the name we gave to the concept 2. All ‘2s’ are “Two” regardless of where they are. But I can still carve two chunks of wood into a ‘2’ shape, and have 2 "2"s. If that makes any sense.
To say that another earth can’t exist, because we’ve already named THIS earth, on the basis of these numerical arguments, is just foolish. q;}
Yes I know the universe isn’t really infinite. I’m not sure how big it would be actually… if it’s expanding from a central point in all directions at the speed of light, aren’t the outer edges moving TWICE the speed of light compared to each other? There’s a whole other debate in this one…
I believe we’re also assuming the laws of physics are the same throughout the universe. This negates certain possibilities, such as finding an infinite amount of cheese somewhere. Cheese is made of matter, matter has mass. Therefore any amount of cheese will produce gravity. An infinite amount of cheese would collapse upon itself and produce a black hole. Maybe that’s what black holes really are, but I doubt it. q;}