If I did the math right, approximately 1013(210118) light years.
Yeah, it's a bit of a ways.
If I did the math right, approximately 1013(210118) light years.
Yeah, it's a bit of a ways.
JoeTheJuggler
Penultimate Amazing
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Ah--thanks. I was thrown by the statement that if you travel in one direction some specific distance you'll find another Earth there.
It's just saying that if there an infinite number of things in the universe, that there must be another Earth (indeed, an infinity of other Earths).
But, as you say, that doesn't follow if the universe is merely infinite in size. Space time could be infinitely large and not have to have another Earth. In fact, it could be infinitely large and have nothing in it beyond what we already see--logically.
By analogy, if an ocean is infinite, it doesn't follow that there is an infinity of fishes.
Brian-M
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Let me try and explain my thinking....
You could make a 1:1 comparison with any set of infinite numbers. For example, comparing the set of integers with the set of prime numbers, you could match up 1 with the first prime number, 2 with the second prime number..... n with the nth prime number, and so on.
But comparing infinite sets it this way makes discussion meaningless. You can't actually count-up the number of numbers in each set to find out if the comparison is valid because the infinite nature of the sets makes this impossible.
An alternate method of determining the relative sizes of different infinites is required in order to make any meaningful comparison. My way of thinking of it is to picture each number in the set placed on an infinite number line, positioned according to its value, and a different number line for each set, like this:
1 2 3 4 5 6 7 8 9 10 11 12 ... (Integers)
1 2 3 4 5 6 7 8 9 10 11 12 ... (Even numbers)
You can't count up the entire length of an infinite number line to compare the number of numbers in them, but you can count up finite sections. You can compare the first ten places on the number line and determine that there are twice as many integers on that particular section of the number line then there are odd numbers. You can do the same with a section of a thousand places, a billion places, a googolplex of places, so why not follow this through to infinite places?
It can be demonstrated that any finite section of the number line has twice as many integers as it has even numbers*, so why wouldn't it be a valid inference that an infinite section of the number line has twice as many integers as it has even numbers?
* On average. Odd-length sections will have either half a number more or half a number less even numbers than exactly half the number of integers.
Or would it be more accurate to describe the set of integers as being twice as dense as the set of even numbers, rather than twice as large?
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Michael C
Graduate Poster
Yes, but we have already demonstrated that we can establish a one-to-one correspondence between the numbers on the two lines. So we may say that twice infinity = infinity. Or three times infinity = infinity. And so on.
Or would it be more accurate to describe the set of integers as being twice as dense as the set of even numbers, rather than twice as large?
We can use "density" to refer to the representation of numbers along a line, but I don't think the word has a meaning for the magnitude of a set. To go a bit further, think of the set of all fractions. There is an infinity of fractional numbers between 0 and 1, so if we look at the numbers stretched out along a line, the fractions are infinitely more densely distributed than the integers. Nonetheless, it is possible to set up a one-to-one correspondence between the elements of the set of fractions and those of the set of integers.
The set of all fractions and the set of all even numbers are countable because we can make a unique association between each element of the set and one of the integers. Things get even more interesting when we reach uncountable sets. For instance, the set of real numbers is uncountable. Cantor formulated his famous proof of this in 1891. So there are different sorts of infinities.
I'm pretty sure, though I wasn't paying close attention, that they were talking about an infinite number of universes at this point, though they did not make it very clear. If they are correct, they gloss over the fact that there will be a huge number (well, infinity, probably) of planets that are almost the same, plus many more that are nothing like our earth.
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ddt
Mafia Penguin
Your way of thinking about "densities" is not very fruitful, as not all sets lend themselves to be pictured on a line (or an area, or ...).
The standard way to compare infinite sets is by establishing functions between them.
An (infinite) set A has smaller or equal cardinality ("size") than a set B, if you can make an injective function (aka one-to-one function)
f: A -> B
Injective means that:
f(x) != f
if x != yTwo sets A and B have the same cardinality if you can make a bijective function (aka one-to-one correspondence
f: A -> B
Bijective means that the function is injective and surjective (aka onto), i.e., every element z of B is the image of an element x of A (so f(x) = z).
Then it's easy to see that the set Z of all integers has the same cardinality as the set N of natural numbers. Define the function n2z by:
n2z
= n/2 if n is evenn2z
= -(n+1)/2 if n is oddand it's easy to see this function is a bijection.
The set Q of rational numbers can be shown to have the same cardinality as N too. Ditto for the set A of algebraic numbers.
The set R of real numbers is "bigger", though. Even the set (0,1) of real numbers between 0 and 1 is "bigger" than N. This is Cantor's famous diagonal argument. The proof goes by contradiction.
First, we observe that every real number has a unique decimal expansion - well, there's the issue of real numbers whose expansion can end in 999... or 000..., but you'll see that these doubles won't bother us.
Now suppose there is a bijective function n2r: N -> (0,1). Then we construct a real number x by defining it's decimals - x_n being the n-th decimal:
x_n = 4 if n2r
has 5 as its n-th decimalx_n = 5 otherwise
Now obviously, the number x is different from any n2r
, as it differs from n2r in the n-th decimal place. Therefore, n2r is not surjective and therefore not bijective.Ergo, the real interval (0,1) has a bigger cardinality than N, and obviously, the complete set of real numbers R has a bigger cardinality than N. However, R has the same cardinality as its subset (0,1): just think of the tangens function to establish a one-to-one correspondence.
No. For example, there's no way to do it for the integers and the real numbers. There will always be lots of real numbers left over, however we do it. So, we say that there are more real numbers than integers.
Yes. So, there are as many prime numbers as integers.
There's no need to count them one by one, as long as we describe the correspondence clearly enough and we show that all members of both sets are accounted for. If we say, "every number gets paired with its double" or "the integer n gets paired with the nth prime number", that's pretty clear.
That could work for comparing sets of integers. What if we want to compare sets whose members are not numbers, so that they have no numeric value with a position on a number line? We could perhaps assign numbers to them in some way (if there aren't too many of them), but then the size of a set would depend on how we label its members, which is just weird. The size of a set shouldn't depend on the names of its members. Renaming things can't possibly change how many of them there are.
See also this somewhat related post of mine from a couple of years ago.
You could look at it this way, I suppose: Any finite section of the number line, no matter how large, is merely an infinitesimal fraction of the entire, infinite, number line, so it's quite a stretch to extrapolate from finite sections to the whole thing.
Dancing David
Penultimate Amazing
It only takes a monkey five minutes to type the scripts for Glenn Beck, he uses a word salad generator!
Ethan Thane Athen
Illuminator
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I liked the mathematician who said that you can only count so far (though that's further than any number we've got to yet - at least in terms of working out it's exact digit construct) and then the next number is zero ie you go back to the beginning.
He seemed to have pulled that theory out of his ass because he didn't like infinity.*
*No doubt I'm shortchanging him and he's done years of research and come up with some proof....
He seemed to have pulled that theory out of his ass because he didn't like infinity.*
*No doubt I'm shortchanging him and he's done years of research and come up with some proof....
His name is Doron, which tells you all you need to know if you've ever chanced across one of the longer-running threads in R&P...
Ambrosia
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That'll be the possible number of ways of arranging all of the particles in the observable universe then.
BenBurch
Gatekeeper of The Left
Trite? Check!
Ignorant? Check!
Right-wing? Check!
Racist? Check!
Good job. And a minimum of words too, sort of a haiku.
LandR
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http://www.wolframalpha.com/input/?i=2^%2810^118%29
Dave Rogers
Bandaged ice that stampedes inexpensively through
I don't think anyone's answered this yet, though it deserves a response.
Try rearranging the terms and changing the values, and you'll see that neither of these can be correct. First of all:
Infinity + 1 = ?
The only value this can have is infinity. The same is true for adding any finite number to infinity. Therefore,
Infinity + N = Infinity, for any finite value of N.
Therefore,
Infinity - Infinity = N, where N can take any finite value.
Secondly:
1 + 2 + 3 + 4 + ....... = Infinity.
2 + 4 + 6 + 8 + ....... = Infinity.
Since each term in the second expression is twice the corresponding term in the first, then the sum of the second is twice the first. Therefore,
Infinity x 2 = Infinity
The same is true of any multiplier of the first series; therefore,
Infinity x N = Infinity, for any finite, positive value of N.
Rearranging, we get:
Infinity / Infinity = N, where N can take any finite, positive value.
So in fact both of these sums give indeterminate answers.
It depends on the nature of the universe. If it's infinite and contains an infinite number of particles, then there is only a finite number of ways those particles can be arranged, and there is therefore an infinity of completely identical Earths, plus another infinity where the only difference is that I forgot to capitalise Earth in that last clause. If it's finite but unbounded, which is what I think you meant, then, after travelling a certain distance in what appeared to be a straight line, one would circumnavigate the universe and arrive back at one's starting point.
Dave
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I thought calculations involving infinity were essentially meaningless as infinity is a concept rather than an actual number to place in equations.
Having taken a peak at the web-site, this is the one they were talking about, but they rather confusingly referred to an infinite number of universes (each being one possible arrangement of the particles), within the one infinite universe.
Dave Rogers
Bandaged ice that stampedes inexpensively through
Yes, I assumed it was something like that, but I couldn't resist the opportunity to answer a question with "It depends on the nature of the universe".
Dave
JoeTheJuggler
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xnm=xn * m
So the answer is 210 * 118=21180. Since the base is 2, I don't know how many zeroes might be in the number (you can only do the trick of using the value of the exponent to know how many zeroes if the base is 10 or a multiple of ten). There are something like 200 digits in the number.