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.
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.