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 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
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?