81–90 of 171 posts
Assuming the many worlds interpretation of quantum physics is true, then the number of atoms includes all combinations of locations, momentum, etc., and the real number of atoms is vastly vastly greater than combinations of just about anything else you might imagine. (Except for combinations of configurations of quantized spacial points!)
If 12megapixels can produce 10 to the power 86696638 images, and we came up with a way of enumerating those images, could we then build a function that given anyone of those images return the index of that image within reasonable time with current hardware. ie. "you have just taken 3999999987493th image"?
Earlier quoted context omitted.
> the total number of atoms in all those universes combined would be close Close?? Wouldn't it still be roughly 10 billion times smaller...?
When you're dealing with numbers on the order of 10^80 to 10^170, I think you're entitled to calling that "close".
But if differences become so large we cannot imagine the differences, then we could imagine there are no differences at all, so ... psychologically/subjectively there would be no difference?
Earlier quoted context omitted.
Math includes the idea of orders of infinity. There are infinite prime numbers, there are more positive integers, even more integers (positive and negative), even more rational numbers (A/B), and even more numbers (rational + irrational {e, Pi} etc)...
In what sense are there more rational numbers than prime numbers? They can be put into bijection with each other, so we generally think of them as being same infinity. There are more real nubmers, of course, by Cantor's diagonalization, so your basic point is true.
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.
Earlier quoted context omitted.
If that is the case was the universe once finite and then went infinite during the early (big bang) expansion? I don't understand how something could have expanded if it was always infinite in size. I'm not even sure the concept of expansion even makes sense. What is infinite + 1? It's just infinite. It seems more like the expansion is a distribution of internal things.
Math includes the idea of orders of infinity. There are infinite prime numbers, there are more positive integers, even more integers (positive and negative), even more rational numbers (A/B), and even more numbers (rational + irrational {e, Pi} etc)...
Earlier quoted context omitted.
In what sense are there more rational numbers than prime numbers? They can be put into bijection with each other, so we generally think of them as being same infinity. There are more real nubmers, of course, by Cantor's diagonalization, so your basic point is true.
The set of all prime numbers is contained within the set of rational numbers, but they are rational numbers that are not within the set of prime numbers. 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)) algebraical…
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?
Earlier quoted context omitted.
Math includes the idea of orders of infinity. There are infinite prime numbers, there are more positive integers, even more integers (positive and negative), even more rational numbers (A/B), and even more numbers (rational + irrational {e, Pi} etc)...
Within a segment of numbers I understand how there are more rational numbers than integers, but I don't understand it in the context of infinity. How can there be more rational numbers than integers when in both cases there are infinite amounts? Are there mathematical operations or concepts that depend on this (in the context of infinity, not subsets)?
Earlier quoted context omitted.
When you're dealing with numbers on the order of 10^80 to 10^170, I think you're entitled to calling that "close".
The ratio is 10^90 which is not small. The subtractive difference rounds to 10^170. In either case, I think its fair to say that 10^170 is unimaginably larger than 10^80. But if differences become so large we cannot imagine the differences, then we could imagine there are no differences at all, so ... psychologically/subjectively there would be no difference?
If 12megapixels can produce 10 to the power 86696638 images, and we came up with a way of enumerating those images, could we then build a function that given anyone of those images return the index of that image within reasonable time with current hardware. ie. "you have just taken 3999999987493th image"?
How would we enumerate all these several gazillion image possibilities?
Well. Let's say number one is all black. Every pixels and every channel is all zero in its value. And let's say the last image to be enumerated is all white. 255 for each pixel and each channel.
Every conceivable image is created in between these two ends. For example, image two is all black, but the last pixel has a value of 1 instead of 0 for its value channel.
Image 1840274917 has pixel 27581 slightly reddish.
Hey, wait a minute, you've just created an image format for describing the data within the image! The only space you're saving is that (given this format) you save space on darker images, because they're likely lower in the sequence.
But that's only because this specification demands that each image be the same exact size and can make assumptions based on that. A lossless format like PNG would be able to perform much better over a wider range of images. (Eg all white will be huge in our system, but cheap in PNG)
Earlier quoted context omitted.
In what sense are there more rational numbers than prime numbers? They can be put into bijection with each other, so we generally think of them as being same infinity. There are more real nubmers, of course, by Cantor's diagonalization, so your basic point is true.
The set of all prime numbers is contained within the set of rational numbers, but they are rational numbers that are not within the set of prime numbers. 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)) algebraical…
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)).