I am not a mathematician, however I have an interest and have come up across some of the infinity proofs etc. I done some research around sizes of infinity, especially popular math type videos (Matt Parker, Numberphile etc) but there are a few things that are especially missing for me, so I would love for someone to point out the flaws in my thinking here (as I am sure there are).
Cantors argument is to prove that one set cannot include all of the other set, therefore proving uncountability, but I never really understood why this works only for eg. decimal numbers and not integers, for which as far as I am seeing the same logic would apply. Quite a lot of the examples given take numbers randomly in order to prove the point, they are very pretty and just paint a picture (take wikipedia's page as an example)
Take a set of binary strings, you make a grid and show that the number is not therefore in the enumeration, therefore not being covered by it. If I were to make a mapping function that just turned the row number into a binary representation (1 => 1, 0, 0..., 2 => 0, 1, 0, 0... etc) then used cantors argument, when I get the number that is not in the set it should be readable as a number, therefore showing where it is in the set, and therefore proving that it is, in fact, in the list. I think the contradiction here is that to read off the number would require it to terminate at some point, which technically it wont as its infinitely long, and therefore you could never actually map it back, but at the same time the enumeration never really ends either, so trying to prove that one is not in the other is confusing as to where you can draw that line. All that we are proving in this case is that one is expanding faster than the other (IE its in the list, you just haven't got there yet).
Likewise the traditional proof that decimals cannot be done can be mapped by just flipping the number? As pointed out there are an infinite number of 0's in front of an integer, we just don't bother writing them out. so 1 => 1, 0, 0, 0... 101 => 1, 0, 1, 0, 0, 0... If we then pick a number that is not on this list then we can likewise map it back to the index that it should be at?
