I mean, Cantor said so, not I. But an easy example
Imagine a list of all whole numbers. 1, 2, 3 on up and up. Obviously this list is infinite - numbers do not end.
Now imagine a list of all real numbers - that is, all numbers plus their decimal amounts between each while number. 1, 1.1, 1.11, 1.12, 2, 2.1, and so on. This list is also infinite - but it is also inherently larger than the infinite list of only whole numbers. It has more numbers.
I think Cantor would say you need a proof for that. And I think he would say you can prove it via generating a new real number by going down your set of real numbers and taking the first digit from the first number, the second from the second, third from third, etc. Then you run a transformation on it, for example every number other than 1 becomes 1 and every 1 becomes 2. Then you know that the number you’ve created can’t be first in the set because its first digit doesn’t match, and it can’t be the second number because the second number doesn’t match, etc to infinity. And therefore, if you map your set of whole numbers to your set of real numbers, you’ve discovered a real number that can’t be mapped to a whole number because it can’t be at any position in the set.
Some will say this proves that infinities can be of unequal sizes. Some will more accurately say this shows that uncountable infinities are larger than countable infinities. But the problem I have with it is this: that we begin with the assumption of a set of all real numbers, but then we prove that not all real numbers are contained in the set of all real numbers. We know this because the number we generated literally can not be at any position in the set. This is a paradox. The number is not in the set, therefore we don’t need it to map to a member of the other set. Yet it is a real number and therefore must be in the set. And yet we proved it can’t be in the set.
I’m uncomfortable making inferences based on this type of information. But I’m also not a mathematician. My goal isn’t to start an argument. Maybe somebody who’s better at math can explain it to me better.
That’s like saying am infinite number of feathers is lighter than an infinite number of bricks. Neither is heavier than the other - they’re both infinitely heavy.
You’re measuring a quality of the two objects, not the quantity, which might make a difference. I’m just sharing something I learned that I think is cool:)
It’s an interesting concept, for sure, don’t get me wrong. It’s intuitive to see the scenario of “different infinities” as being different sizes and believe it makes sense, but it doesn’t pan out. It’s weird because infinity is used in regards to numbers, but it’s not a number itself. It’s more the antithesis of a number - it’s everything. It’s a tool we use to interact with the concept of something that specifically can’t be measured. Measuring implies limits or bounds, but something that is endless has neither.
So saying there’s an infinite number of this or that is more akin to the “riddle” of if 100lbs of feathers weighs less than 100lbs of bricks. The trick is they both weigh the same, even though our brain might not intuitively realize that, just like infinities. Ultimately, it’d be more accurate to say there’s infinities within infinities, which is another tricky concept all on its own.
If you say so :)
I mean, Cantor said so, not I. But an easy example
Imagine a list of all whole numbers. 1, 2, 3 on up and up. Obviously this list is infinite - numbers do not end.
Now imagine a list of all real numbers - that is, all numbers plus their decimal amounts between each while number. 1, 1.1, 1.11, 1.12, 2, 2.1, and so on. This list is also infinite - but it is also inherently larger than the infinite list of only whole numbers. It has more numbers.
I think Cantor would say you need a proof for that. And I think he would say you can prove it via generating a new real number by going down your set of real numbers and taking the first digit from the first number, the second from the second, third from third, etc. Then you run a transformation on it, for example every number other than 1 becomes 1 and every 1 becomes 2. Then you know that the number you’ve created can’t be first in the set because its first digit doesn’t match, and it can’t be the second number because the second number doesn’t match, etc to infinity. And therefore, if you map your set of whole numbers to your set of real numbers, you’ve discovered a real number that can’t be mapped to a whole number because it can’t be at any position in the set.
Some will say this proves that infinities can be of unequal sizes. Some will more accurately say this shows that uncountable infinities are larger than countable infinities. But the problem I have with it is this: that we begin with the assumption of a set of all real numbers, but then we prove that not all real numbers are contained in the set of all real numbers. We know this because the number we generated literally can not be at any position in the set. This is a paradox. The number is not in the set, therefore we don’t need it to map to a member of the other set. Yet it is a real number and therefore must be in the set. And yet we proved it can’t be in the set.
I’m uncomfortable making inferences based on this type of information. But I’m also not a mathematician. My goal isn’t to start an argument. Maybe somebody who’s better at math can explain it to me better.
That’s like saying am infinite number of feathers is lighter than an infinite number of bricks. Neither is heavier than the other - they’re both infinitely heavy.
You’re measuring a quality of the two objects, not the quantity, which might make a difference. I’m just sharing something I learned that I think is cool:)
It’s an interesting concept, for sure, don’t get me wrong. It’s intuitive to see the scenario of “different infinities” as being different sizes and believe it makes sense, but it doesn’t pan out. It’s weird because infinity is used in regards to numbers, but it’s not a number itself. It’s more the antithesis of a number - it’s everything. It’s a tool we use to interact with the concept of something that specifically can’t be measured. Measuring implies limits or bounds, but something that is endless has neither.
So saying there’s an infinite number of this or that is more akin to the “riddle” of if 100lbs of feathers weighs less than 100lbs of bricks. The trick is they both weigh the same, even though our brain might not intuitively realize that, just like infinities. Ultimately, it’d be more accurate to say there’s infinities within infinities, which is another tricky concept all on its own.