It’s called countable and uncountable infinity. the idea here is that there are uncountably many numbers between 1 and 2, while there are only countably infinite natural numbers. it actually makes sense when you think about it. let’s assume for a moment that the numbers between 1 and 2 are the same “size” of infinity as the natural numbers. If that were true, you’d be able to map every number between 1 and 2 to a natural number. but here’s the thing, say you map some number “a” to 22 and another number “b” to 23. Now take the average of these two numbers, (a + b)/2 = c the number “c” is still between 1 and 2, but it hasn’t been mapped to any natural number. this means that there are more numbers between 1 and 2 than there are natural numbers proving that the infinity of real numbers is a different, larger kind of infinity than the infinity of the natural numbers
It’s like when you say something is full. Double full doesn’t mean anything, but there’s still a difference between full of marbles and full of sand depending what you’re trying to deduce. There’s functional applications for this comparison. We could theoretically say there’s twice as much sand than marbles in “full” if were interested in “counting”.
The same way we have this idea of full, we have the idea of infinity which can affect certain mathematics. Full doesn’t tell you the size of the container, it’s a concept. A bucket twice as large is still full, so there are different kinds of full like we have different kinds of infinity.
It’s weird but the amount of natural numbers is “countable” if you had infinite time and patience, you could count “1,2,3…” to infinity.
It is the countable infinity.
The amount of numbers between 1 and 2 is not countable. No matter what strategies you use, there will always be numbers that you miss. It’s like counting the numbers of points in a line, you can always find more even at infinity.
It is the uncountable infinity.
I greatly recommand you the hilbert’s infinite hotel problem, you can find videos about it on youtube, it covers this question.
Natural numbers being infinite, how it be possible for the values between 1 and 2 to be “more infinite” ?
It’s called countable and uncountable infinity. the idea here is that there are uncountably many numbers between 1 and 2, while there are only countably infinite natural numbers. it actually makes sense when you think about it. let’s assume for a moment that the numbers between 1 and 2 are the same “size” of infinity as the natural numbers. If that were true, you’d be able to map every number between 1 and 2 to a natural number. but here’s the thing, say you map some number “a” to 22 and another number “b” to 23. Now take the average of these two numbers, (a + b)/2 = c the number “c” is still between 1 and 2, but it hasn’t been mapped to any natural number. this means that there are more numbers between 1 and 2 than there are natural numbers proving that the infinity of real numbers is a different, larger kind of infinity than the infinity of the natural numbers
Great explanation by the way.
I get that, but it’s kinda the same as saying “I dare you!” ; “I dare you to infinity!” ; “nuh uh, I dare you to double infinity!”
Sure it’s more theoretically, but not really functionally more.
It’s like when you say something is full. Double full doesn’t mean anything, but there’s still a difference between full of marbles and full of sand depending what you’re trying to deduce. There’s functional applications for this comparison. We could theoretically say there’s twice as much sand than marbles in “full” if were interested in “counting”.
The same way we have this idea of full, we have the idea of infinity which can affect certain mathematics. Full doesn’t tell you the size of the container, it’s a concept. A bucket twice as large is still full, so there are different kinds of full like we have different kinds of infinity.
When talking about infinity, basically everything is theoretical
It’s weird but the amount of natural numbers is “countable” if you had infinite time and patience, you could count “1,2,3…” to infinity. It is the countable infinity.
The amount of numbers between 1 and 2 is not countable. No matter what strategies you use, there will always be numbers that you miss. It’s like counting the numbers of points in a line, you can always find more even at infinity. It is the uncountable infinity.
I greatly recommand you the hilbert’s infinite hotel problem, you can find videos about it on youtube, it covers this question.
I thought the same but there is a good explanation for it which I can’t remember
I’m confused as well. Isn’t that like saying that there is more sand in a sandbox than on every veach on the planet?