Some of this is paraphrased so as to shorten it. I've only taken out detail I assume you guys would know.
Penrose
Cantor's argument proceeds by contradiction. Suppose that the set of all real numbers is countable. Then the real numbers between 0 and 1 are countable, and we shall have some list providing a one-to one pairing of all such numbers with the natural numbers, such as:
0 --> 0.10357627183...
1 --> 0.14329806115...
2 --> 0.02166095213...
3 --> 0.43005357779...
4 --> 0.92550489101...
5 --> 0.59210343297...
6 --> 0.63667910457...
7 --> 0.87050074193...
8 --> 0.04311737804...
9 --> 0.78635081150...
10--> 0.40916738891...
The digits on a diagonal have been bolded and for this particular listing they are:
1,4,1,0,0,3,1,4,8,5,1,...
The diagonal slash procedure is to construct a real number (between 0 and 1) whose decimal expansion (after the decimal point) differs from these digits in each corresponding place.
I don't understand why this has to be the case, and what the purpose of creating this diagonal slash is. If you are acquainted with some Computer Science, Penrose demonstrated the same argument to show that there is no Turing Machine which can determine if any other Turing Machine eventually stops. So an explanation of this argument would be much appreciated.
Penrose
For definiteness, let us say that the digit is to be 1 whenever the diagonal digit is different from 1 and it is 2 whenever the diagonal digit is 1. Thu, in this case we get the real number: 0.21211121112...
This real number cannot appear in our listing since it differs from the first number in the first decimal place (after the decimal point, from the third number in the third place, etc. This is a contradiction because what we are trying to prove, namely that there is no one-to-one correspondence between the real numbers and the natural numbers and, accordingly, the number of real numbers is actually greater than the number of rational numbers and is not countable.
This real number cannot appear in our listing since it differs from the first number in the first decimal place (after the decimal point, from the third number in the third place, etc. This is a contradiction because what we are trying to prove, namely that there is no one-to-one correspondence between the real numbers and the natural numbers and, accordingly, the number of real numbers is actually greater than the number of rational numbers and is not countable.
Why cannot the new real number appear on the list? Why is the criterion set that all digits in the new real number must differ from all other digits in the same place?
I must say that the argument makes intuitive sense simply because there is an infinite number of real numbers in any particular interval and thus they are not countable even within an infinitessimally small interval, while for rational numbers you can always find an interval small enough between which there will only be irrational numbers. Is my own reasoning correct or faulty with this?
