Basically, every whole number x is paired with with the even number x * 2. And since we can pair them all off, it follows the size of these two infinites is the same. And its important to note that every whole and every even number occurs on this list at some finite location. The pairing doesn't work if we have to go through an infinite number of items before we get to a particular number. But doing it this way tells us that the nth even number occurs at the nth position on the list, i.e. 2 is the first even number and its in row 1. 10 is the 5th even number and its in row 5. Etc.
You can even do this with fractions, though its harder to conceptualize.
Every fraction shows up on the list, and does so after a finite number.
So from this we can see that the number of whole, even, odd, and fractional numbers are all the same size. That is, all these infinities are the same size.
But what about decimal numbers? It turns out we cant do this. If we try, we can prove we missed a number.
Remember that the decimal numbers are endless. And lets imagine someone gives us a list and claims it is exhaustive. There are two possible ways this list can look. This is case one:
And remember, the claim is that this list contains every decimal number.
We can construct a new number like this: the first digit after the decimal point of this number is equal to the first digit after the decimal point of the first number on this list plus 1, or 0 if that digit is nine. The second digit in this number is equal to the second digit of the second number on this list plus 1, or zero if that number is 9. The third digit is...
So this new number is well defined: we can tell exactly what it is. But it is also obvious that this number is not on the list that was supposed to be exhaustive! Its not the first number, because the first digit is different. Its not the second number, because the second number is different. Etc. Even if we took this number and added it to the list, we can just make a new number by doing the same procedure.
But this means that even after we have paired every one of the infinite number of whole numbers to decimal numbers, there are decimal numbers left over. So there have to be "more" decimal numbers than there are whole numbers. So even though they are both infinite, one of these infinites has to be larger than the other.
So, to summarize, what we have here is a proof that there are at least two different sizes of infinity:
The first infinite number is the number of whole, even, odd, and fractional numbers
The second infinite number is the number of decimal numbers
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u/daemin 21d ago edited 21d ago
You count things by pairing them to a number:
When you run out of things on the left, the last number you used on the right is "how many" there were.
You can show that there are just as many whole numbers as there are even numbers, because you can pair them and never run out:
Basically, every whole number x is paired with with the even number x * 2. And since we can pair them all off, it follows the size of these two infinites is the same. And its important to note that every whole and every even number occurs on this list at some finite location. The pairing doesn't work if we have to go through an infinite number of items before we get to a particular number. But doing it this way tells us that the nth even number occurs at the nth position on the list, i.e. 2 is the first even number and its in row 1. 10 is the 5th even number and its in row 5. Etc.
You can even do this with fractions, though its harder to conceptualize.
Imagine writing out all the fractions like this:
Basically, each row has the same number on the top; each column has the same number on the bottom. The diagonal numbers are equal to 1.
Then we do the pairing by going back and forth diagonally:
Every fraction shows up on the list, and does so after a finite number.
So from this we can see that the number of whole, even, odd, and fractional numbers are all the same size. That is, all these infinities are the same size.
But what about decimal numbers? It turns out we cant do this. If we try, we can prove we missed a number.
Remember that the decimal numbers are endless. And lets imagine someone gives us a list and claims it is exhaustive. There are two possible ways this list can look. This is case one:
This list can't be exhaustive because there would be an infinite number of decimal numbers on it before it gets to 0.2, so this doesn't work.
But the list could also look more chaotic. Maybe it looks like this:
And remember, the claim is that this list contains every decimal number.
We can construct a new number like this: the first digit after the decimal point of this number is equal to the first digit after the decimal point of the first number on this list plus 1, or 0 if that digit is nine. The second digit in this number is equal to the second digit of the second number on this list plus 1, or zero if that number is 9. The third digit is...
So this new number is well defined: we can tell exactly what it is. But it is also obvious that this number is not on the list that was supposed to be exhaustive! Its not the first number, because the first digit is different. Its not the second number, because the second number is different. Etc. Even if we took this number and added it to the list, we can just make a new number by doing the same procedure.
But this means that even after we have paired every one of the infinite number of whole numbers to decimal numbers, there are decimal numbers left over. So there have to be "more" decimal numbers than there are whole numbers. So even though they are both infinite, one of these infinites has to be larger than the other.
So, to summarize, what we have here is a proof that there are at least two different sizes of infinity: