When you look at the number of real numbers, you can always find new ones in both - you’ll never run out.
That being said, imagine (or actually draw) two number lines in the same scale. One [0,1] the other [0,2]. Choose a natural number n, and divide both lines with that many lines. You’ll get n+1 segmets in both lines.
When you let n run off into infinity, the number of segments will be the same in both lines. This is the cardinality of the set.
But for practical purposes of measuring a coastline, this approach is flawed.
Yes, you’ll always see n+1 segments, but we aren’t measuring thw number of distinct points on the coastline, but rather its length.
If you go back to your two to-scale number lines and divide them into n segments, the segments on one are exactly two times larger than on the other.
This is what we want to measure when we want to measure a coastline. The total length drawn when connecting these n points (and not ther number!) as their number runs off towards infinity.
If between 0 and 1 are an infinite number of real numbers, then between 0 and 2 are twice infinite real numbers, IIRC my college math. I probably don’t.
Infinities don’t care about the actual numbers in the set, but about the cardinality (size). Obviously the numbers between 0,1 and 1,2 are different but have the same size.
But 0,1 and 0,2? Size is unintuitive for infinities because they are … infinite. So the trick is to look for the simplest mathematical formula that can produce a matching from every number of one set to every number in the second. And as somebody has said, every number in 0,2 can be achieves by multiplying a number in 0,1 by 2. So there is a 1 to 1 relation between 0,1 and 0,2. Ergo they are the same size.
Here’s the proof: for each number between 0 and 1, double it and you get a unique number between 0 and 2. And you can do the reverse by halving. So every number in the first set is matched with every number in the second set, meaning they’re the same size.
Aren’t the number of real numbers and the number of integers also infinite? But they aren’t considered equal. The infinite for real numbers is considered larger.
Yes, the number of Intergers is ℵ0, the number of real numbers ℵ1, and this is what people generally mean with some infinities are bigger than others. Infinities can also be seem bigger than another, but be mathematically equal. The number of natural, real and rational numbers are all infinite, and might seem different, but they are all proven ℵ0.
Claypidgin was talking about the real numbers between [0,1] and [0,2], which are both ℵ1 infinite. Some infinities are indeed bigger than others, but those 2 are still the same infinity.
The cardinality of the two intervals [0,1] and [0,2] are equivalent. E.g. for every number in the former you could map it to a unique number in the latter and vice versa. (Multiply or divide by two)
However in statistics, if you have a continuous variable with a uniform distribution on the interval [0, 2] and you want to know what the chances are of that value being between [0,1] then you do what you normally would for a discrete set and divide 1 by 2 because there are twice as many elements in the total than there are in half the range.
In other words, for weird theoretical math the amount of numbers in the reals is equivalent to the amount of any elements in a subset of the reals, but other than those weird cases, you should treat it as though they are different sizes.
Sure, the length of the intervals is easily compared. But saying
there are twice as many elements in the total than there are in half the range
is false. They are both aleph 1. In other words, for each unique element you can pick from [0,2], I can pick a unique element from [0,1]. I could even pick two or more. So you can’t compare the number of elements in the two in a meaningful way other than saying they both belong to the same category of infinite.
This is the whole crux of the coastline problem, isn’t it?
Isn’t it a bit like saying “there’s obviously more real numbers between 0 and 2 than between 0 and 1”? Which, to my knowledge, is a false statement.
It isn’t.
When you look at the number of real numbers, you can always find new ones in both - you’ll never run out.
That being said, imagine (or actually draw) two number lines in the same scale. One [0,1] the other [0,2]. Choose a natural number n, and divide both lines with that many lines. You’ll get n+1 segmets in both lines.
When you let n run off into infinity, the number of segments will be the same in both lines. This is the cardinality of the set.
But for practical purposes of measuring a coastline, this approach is flawed.
Yes, you’ll always see n+1 segments, but we aren’t measuring thw number of distinct points on the coastline, but rather its length.
If you go back to your two to-scale number lines and divide them into n segments, the segments on one are exactly two times larger than on the other.
This is what we want to measure when we want to measure a coastline. The total length drawn when connecting these n points (and not ther number!) as their number runs off towards infinity.
If between 0 and 1 are an infinite number of real numbers, then between 0 and 2 are twice infinite real numbers, IIRC my college math. I probably don’t.
In math they’d both be equal
Discrete math typically teaches that some infinities are greater than others.
Yes, but those are both the same infinite according to math, so no, they’re still equal.
? But they’re not the same infinity according to math.
Infinities don’t care about the actual numbers in the set, but about the cardinality (size). Obviously the numbers between 0,1 and 1,2 are different but have the same size.
But 0,1 and 0,2? Size is unintuitive for infinities because they are … infinite. So the trick is to look for the simplest mathematical formula that can produce a matching from every number of one set to every number in the second. And as somebody has said, every number in 0,2 can be achieves by multiplying a number in 0,1 by 2. So there is a 1 to 1 relation between 0,1 and 0,2. Ergo they are the same size.
Here’s the proof: for each number between 0 and 1, double it and you get a unique number between 0 and 2. And you can do the reverse by halving. So every number in the first set is matched with every number in the second set, meaning they’re the same size.
They are literally both ℵ1 though?
Aren’t the number of real numbers and the number of integers also infinite? But they aren’t considered equal. The infinite for real numbers is considered larger.
Yes, the number of Intergers is ℵ0, the number of real numbers ℵ1, and this is what people generally mean with some infinities are bigger than others. Infinities can also be seem bigger than another, but be mathematically equal. The number of natural, real and rational numbers are all infinite, and might seem different, but they are all proven ℵ0.
Claypidgin was talking about the real numbers between [0,1] and [0,2], which are both ℵ1 infinite. Some infinities are indeed bigger than others, but those 2 are still the same infinity.
The cardinality of the two intervals [0,1] and [0,2] are equivalent. E.g. for every number in the former you could map it to a unique number in the latter and vice versa. (Multiply or divide by two)
However in statistics, if you have a continuous variable with a uniform distribution on the interval [0, 2] and you want to know what the chances are of that value being between [0,1] then you do what you normally would for a discrete set and divide 1 by 2 because there are twice as many elements in the total than there are in half the range.
In other words, for weird theoretical math the amount of numbers in the reals is equivalent to the amount of any elements in a subset of the reals, but other than those weird cases, you should treat it as though they are different sizes.
Sure, the length of the intervals is easily compared. But saying
is false. They are both aleph 1. In other words, for each unique element you can pick from [0,2], I can pick a unique element from [0,1]. I could even pick two or more. So you can’t compare the number of elements in the two in a meaningful way other than saying they both belong to the same category of infinite.
This is the whole crux of the coastline problem, isn’t it?