I realized that this might be confusing…
The point is that when you have a list like:
012345678910
123456789101
234567891011
345678910111
456789101112
567891011121
678910111213
Then you add another interval:
024681012141
246810121416
468101214161
681012141618
810121416182
101214161820
If you mix these two intervals together (weave them together every other), you’ll have:
1.) 012345678910
2.) 024681012141
1.) 123456789101
2.) 246810121416
1.) 234567891011
2.) 468101214161
The problem here is that when you converge this series at infinity, you have to get past an infinite amount of zero’s to even begin the 1’s:
1.) 012345678910
1.) 024681012141
1.) 036912151821
1.) 048121620242
1.) 051015202530
1.) 061218243036
etc…
So… it is impossible to count an infinity before you begin counting another infinity, and this makes it impossible to count all of the reals in one dimension with one list. No diagonal argument needed here!
BUT! for each of these sequences, you can still keep them countable by adding additional lists!
List 1.
1.) 012345678910…
2.) 123456789101…
3.) 234567891011…
4.) 345678910111…
5.) 456789101112…
etc…
List 2.
1.) 024681012141…
2.) 246810121416…
3.) 468101214161…
4.) 681012141618…
5.) 810121416182…
By using this technique, you can effectively square an order of infinity and still keep it countable. You can run the diagonals for these lists by adding another dimension to each list. List 1 list 1.1, list 1.2, list 1.3 etc… List 2, list 2.1, list 2.2, list 2.3 etc…
Or as was mentioned earlier, you could run the diagonals with a simple algorithm and still accommodate a counting sequence.