this doesn't make sense to me, smart people? (infinity)

[daviddem]

Sort of. The steps are:
1) Assume that the real numbers between 0 and 1 are countable, just like the natural numbers. If we can prove this wrong, and the real numbers between 0 and 1 cannot be counted then by obvious extension the entire set of real numbers also cannot be counted. But for now, assume the real numbers between 0 and 1 are countable.

2) Since you assumed they are countable, it implies that you should be able to come up with a list that includes them all, just like the list
0
1
2
3
.
.
.
represents the list of all natural numbers.

So let's say that our list of all real numbers between zero and one is:

0.a11a12a13a14...
0.a21a22a23a24...
0.a31a32a33a34...
0.a41a42a43a44...
.
.
.

where aij is a figure 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9. For example if the first number in the list is 0.5678000... then a11=5, a12=6, a13=7, a14=8, a15=0 and so on.

3) Now let's show that whatever this list is (for whatever possible aij) we can come up with a real number between zero and one which is not included in this list. Since this number is not included in the list, it ensues that this list cannot be the list of all real numbers between 0 and 1. And since we can do this for any aij and hence for any list of real numbers between 0 and 1, this proves wrong our assumption that the real numbers between 0 and 1 are countable: no list can represent all of them, since for every possible list, I can find a real number between 0 and 1 that is not included in it.

A number not included in the above list is 0.b1b2b3b4... where:
b1 can be any figure not equal to a11.
b2 can be any figure not equal to a22.
b3 can be any figure not equal to a33.
b4...
etc
This number is not in the list because by its structure it differs from every number in the list by at least one decimal place. So we just found a real number between 0 and 1 (actually a bunch of them) which is not in this supposed list of all real numbers between 0 and 1. So obviously the list was not really the list of all numbers as we wrongly assumed. And since we can do that for any list we like, it means that no list at all can include all the real numbers between 0 and 1. And so the real numbers are not countable as we first assumed.

[seven-deuce]

Sort of. The steps are:
1) Assume that the real numbers between 0 and 1 are countable, just like the natural numbers. If we can prove this wrong, and the real numbers between 0 and 1 cannot be counted then by obvious extension the entire set of real numbers also cannot be counted. But for now, assume the real numbers between 0 and 1 are countable.

2) Since you assumed they are countable, it implies that you should be able to come up with a list that includes them all, just like the list
0
1
2
3
.
.
.
represents the list of all natural numbers.

So let's say that our list of all real numbers between zero and one is:

0.a11a12a13a14...
0.a21a22a23a24...
0.a31a32a33a34...
0.a41a42a43a44...
.
.
.

where aij is a figure 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9. For example if the first number in the list is 0.5678000... then a11=5, a12=6, a13=7, a14=8, a15=0 and so on.

3) Now let's show that whatever this list is (for whatever possible aij) we can come up with a real number between zero and one which is not included in this list. Since this number is not included in the list, it ensues that this list cannot be the list of all real numbers between 0 and 1. And since we can do this for any aij and hence for any list of real numbers between 0 and 1, this proves wrong our assumption that the real numbers between 0 and 1 are countable: no list can represent all of them, since for every possible list, I can find a real number between 0 and 1 that is not included in it.

A number not included in the above list is 0.b1b2b3b4... where:
b1 can be any figure not equal to a11.
b2 can be any figure not equal to a22.
b3 can be any figure not equal to a33.
b4...
etc
This number is not in the list because by its structure it differs from every number in the list by at least one decimal place. So we just found a real number between 0 and 1 (actually a bunch of them) which is not in this supposed list of all real numbers between 0 and 1. So obviously the list was not really the list of all numbers as we wrongly assumed. And since we can do that for any list we like, it means that no list at all can include all the real numbers between 0 and 1. And so the real numbers are not countable as we first assumed.

10/10 Post.