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.
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.
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.
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.
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.