I don’t get for Cantor’s diagonalization proof, why do we need to use the diagonal digits to form the new number?
Would the proof work the same if we instead used the first digit of every number in the list?
Would the proof work the same if we instead used the first digit of every number in the list?
{0.22, 0.32, 0.33}
We now construct a new number in this way:
First digit of the first number is 2, so we take 3 which is different as our first digit.
First digit of the second number is 3 so we take 2 as our second number. Now we have 0.32
First digit of the third number is 3 so we take 0 as our third number.
We've constructed the number: 0.320 = 0.32 which is in the set already. So this construction method doesn't guarantee that it's a new number, only that it's different from the first.
With Cantor's Diagonalization argument we guarantee it's different from the first number since it's different in the first digit, then we guarantee it's different from the second since it's different in the second digit, etc. etc. it's different from all the numbers in our list.
To take the same set above as an example, we end up creating a number like: 0.318 which is definitely different from the numbers in the set
Two real numbers are different if in the same place they have different digits. [1] So the idea is to take your infinite, hypothetical list of all real numbers A[] and make a new real number X, where for all i, there's some digit where A[i] and X differ. Easiest way is to make X differ from A[i] at digit i.
[1] There's a subtlety here about repeating nines at the end of a number, but it is inessential.