The simplest model in rough agreement with observations is called the Einstein-de Sitter Model, and it's flat with a zero cosmological constant. See http://www.britannica.com/science/cosmology-astronomy/Relati...
More general models are covered here: https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtr...
Now try to wrap your mind around this: someone that's one light year to the left is going to see a slightly different visible universe, also expanding, into the same infinite space. But if we look in their direction, we see the edge of our visible universe expanding into the void, but from their point of view looking in the same direction our edge is one light year short of their edge. So what's our edge expanding into?
That, must be a terrifying place to live in......
What about looking "up" and "down?" i.e. into the inside of balloon or away from its surface?
I have a hard time wrapping my head around the balloon surface analogy, because galaxies seem to be in all directions of each other..
Or, alternatively, if you think of time as a dimension, then we can call "up" the future and "down" the past. In that case, if you look down inside the past, you'll see that in that analogy the center of the universe corresponds with the big bang! And all the galaxies are equidistant from that point/moment in spacetime.
That's why gravitational waves are such a big deal: they allow us to look further. (Not further than the limit imposed by the speed of light, though.)
Cantor's diagonalization is simply demonstrating that same inequality by showing a number in set A is not in set B.
Just because you can map two infinity's to each other does not mean they are of the same size consider: Limit(0->inifinity) of (x - (x/2)) algebraically that's clearly Limit(0->inifinity) of X/2 which is infinity.
PS: What makes Cantor's diagonalization interesting is you can repeat it recursively an infinite number of times. This is more obvious in base 2.
Diagonalization isn't showing that a number in set A isn't in set B - that's obviously true for reals and integers, but it's also true for rationals and integers. It's showing that there does not exist a mapping from B to A where there's an element in B for each element in A.
We're obviously not using the same definition of "size". I generally think in terms of cardinality, what are you thinking of?
If every element in set A is in set B, and there are elements in set A left over it's larger because that's what larger means. {A,B} < {A,B,C}
There are countable and uncountable infinite set's. All countable set's have a bijection with N. However, there are more than two sizes of infinite sets. Real numbers < Imaginary numbers.
There are more than two sizes of infinite sets, but there are just as many real numbers as imaginary numbers for the same reason there's just as many integers as rational numbers.
Try reading this: https://en.wikipedia.org/wiki/Cardinality#Infinite_sets
Now, feel free to try and map the set of Real numbers to the set of irrational numbers. ex: e + ei.
2. f(x) = x + sqrt(2) if there exists an integer k>=0 such that x - k * sqrt(2) is rational; f(x) = x otherwise.
This function maps all real numbers to irrational numbers, 1-to-1.
So, after you define x+1 and x+i, now what? Also please bear in mind that "infinity" is neither real nor complex: if you have a well-defined mapping into complex numbers, then by definition, it never maps to infinity.
(Yes, there are some "functions" like y = 1/x that "maps to infinity", but it's simply mathematicians being lazy and abusing notations because everybody around them understands what's going on.)
There is a basic contradiction in set theory. Called Russell's paradox, there are two ways around it. First is ignoring it, aka everything builds from it's self nothing can become recursive. Or Zer's something or other that basically removed membership and equality and hides in the corner crying.
As such infinity is generally assumed not to exit in R. And causes all this all infinite set's map able to each other are equivalent size crap. It's also why real mathematicians laugh at the set guys.
But, sorry the way R was initially defined it included infinity and you only get to put it into 1 place on your mapping. Or as a math professor said, what angle is the highest number in R mapping to.
If you are interested, please read an actual math textbook. (Yes, they can be a giant time sink, but at least you'll learn the correct meanings of sets and functions.)
But, the absurd results are not generalizable outside of their assumptions sorry Axioms. Set theory being one of the most obvious cases.
It's sadly like a religion in many ways, follow enough false statements and you can prove anything. Yet, if you find a contradiction then don't actually accept at least one of your assumptions are false.
For a your mapping, see: http://math.stackexchange.com/questions/512397/is-there-a-si...
This is correct. However, the number of real-valued points between 0 and 1 is the same as the number of real-valued points between 0 and 2.
> The number of points between 0 and 1 is larger than the number of rational numbers.
This is also true because there are uncountably many real-valued points between 0 and 1 and countably many rational numbers.
But please, continue to argue with mathematical definitions that have been established for over a century.
[1]: https://en.wikipedia.org/wiki/Aleph_number#Aleph-naught
The reals, on the other hand, cannot be placed in a bijection with the natural numbers, and there are therefore "more" reals than naturals (i.e. there is an injection from the naturals to the reals, but not from the reals to the naturals -- any function from reals to naturals must have some pair x ≠ y with f(x) = f(y)).
Is the set of Real Numbers larger, smaller, or the same size as the set of points in a finite 2d object? Can you setup a bijection in either direction?
- normalise x and y coordinates in the shape into the interval (0, 1)
- interleave the bits of the normalised x and y coordinates
This gives you a bijection between points in (a) 2-dimensional space and a segment of the real line (which, in turn, has a bijection with the whole real line if you want to specify that).
Once again, cardinality in set theory is based on injections and bijections. If there is an injection from X into Y, then Y is at least as big as X. If there is a bijection between them (i.e. injections in both directions), then they are the same size.
(Also, bijections are inherently bidirectional.)
In short, infinities are complicated and intuition doesn't work well.
In slightly longer, you're talking about the difference of the rate of growth of two functions rather than actually about the cardinalities of the sets.
We define two sets as having the same cardinality when we can create a bijection between them. We can list the primes in order from smallest to largest and number them with the natural numbers. So we'll have 2 match up with 0, 3 with 1, 5 with 2, 7 with 3, etc. Every single prime number will correspond with a natural number AND every single natural number will correspond with a prime number, no exceptions. So they must be the same size. They are also the same size as the integers and the rational numbers but the set of real numbers is a bigger infinity.
The answer is no, as illustrated by the Hotel paradox[0], in which we have a infinitely many rooms and want to accommodate a (possibly infinite) number of guests.. To summarize: For any finite number of guests, you can always find an even-numbered room to correspond to that guest (assume the guests are numbered sequentially, then double their number and put them in the room that has that number.). This creates a one-to-one correspondence, which means that the sets are the same size.
You might say, 'well, that only works because we're dealing with a finite number of guests. But we're talking about infinity'. There are a few different ways of answering that question. In my opinion, the easiest way to look at it is to remember that there is no such number as 'infinity' - when we say 'infinity', we're really trying to express the concept of growing without bound. So, the above strategy (double the person's number) works for any arbitrarily large group. At no point does it stop working, even as the group size grows larger and larger, so we can say that the two sets have the same size.
On the other hand, we have no such strategy for putting every irrational number in one-to-one correspondence with the counting numbers. That proof is a little harder, and the analogy with the hotel guests breaks down, unfortunately, so it's a bit tougher to explain.
[0] https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand...
If that were true, why go to all the trouble, just show 1/2 which is not a natural number, or sqrt(2) which is not a rational number.
Cantor's diagonalization is proving that no mapping exists between the natural numbers and the real numbers in [0, 1]; that no matter what mapping you (try to) come up, there will be a number you would miss.
The primes and rationals have the same size (cardinality) as the natural numbers, namely countably infinite. See https://en.wikipedia.org/wiki/Countable_set#Formal_overview_...
There are an infant number of points between 0 and 1 and an infinite number of points between 0 and 2. The distance between 0 and 2 is larger. The number of points between 0 and 1 is smaller than the number of points on the unit circle AND they are a different class of infinity.
Try to make arguments from axioms and definitions rather than asserting things from intuition. Intuition is often a useful tool, but (1) it's not an argument, and (2) it's not very helpful once you step into the infinite realm. Incidentally, that's why I went for programming: it's like math, but with no infinity (unless you're using floats, but that's a much easier infinity).
But, I have had my fun poking people who don't really get set theory.
Did you mean reals here? There _is_ a (bijection) mapping between integers and rationals.
https://en.wikipedia.org/wiki/Cantor_pairing_function#Cantor...
But most people would disagree with you when you say "how can there be more rational numbers than integers..." because while we don't have a firm grasp of how many, we definitely would say that having the same cardinality means that there isn't some notion of "more".
I'm not sure what you mean by mathematical operations or concepts that depend on this.
> I'm not sure what you mean by mathematical operations or concepts that depend on this.
I suppose I meant to ask if there was any practical application of the concepts you described.