[QUOTE=Frylock]
What he does do, though, is “assume” (in a loose sense) that you can do so, specifically for the purpose of showing that this assumption can’t actually hold. It’s a reductio argument. But “assuming” p for a reductio is by no means basing one’s proof on an assertion that p is the case! Almost exactly the opposite!
[/QUOTE]
First of all, I have no idea what your bracketed term " (in a loose sense) " means, and I would love you to elucidate.
Secondly, I am well aware of what constitutes a reductio ad absurdum argument, and my view is that Cantor’s proof isn’t one, or at least not a valid one. A reductio ad absurdum argument shows that a sequence of legitimate steps or deductions leads to a contradicton. What I am discussing is the fact that Cantor’s famous proof involves an illegtimate step. Specifically, it asserts that, given an infinitely long list of numbers, one can construct a new number by taking the first digit of the first number, the second digit of the second number and so on. This sounds seductive and plausible, which is why is tends to pass without remark or objection. However, this procedure is only legitimate and meaningful for a finite list of numbers. If you are starting with a list of numbers that you assert has an infinite number of members, each of which can be expressed as an infinitely long decimal (hence the use of an ellipsis at the right-hand end of each number in Anne Neville’s list), then the procedure described is meaningless.
Cantor’s proof as it is traditionally presented uses an analogy in which the first part corresponds to easily understood, easily visualised everyday life, but the second part does not. This is where the illegitimate step creeps in, usually unnoticed. If you show me ten numbers written down as decimals, sure, I can note the first digit of the first number, the second digit of the second number, and so on for the complete list. This is something no-one has trouble either visualising or understanding. If one wants to, one can actually perform this task and, by altering each of the noted digits in some specific way (for example by adding 1), one can create an eleventh number that is distinct from each of the ten numbers originally on the list. Cantor’s proof, as traditionally presented, and as presented earlier by Anne Neville, just assumes that one can perform the same operation on an infinitely long list of infinitely long decimal expressions, and uses this step within a reductio ad absurdem argument. What I am saying is open to qurstion is whether this assumption holds. I maintain that it does not.
One cannot perform this operation (taking the first digit of the first number… and so forth) on an infinitely long list. One way to realise this is to appreciate that one cannot count along to the ‘infinity-th’ digit of an ‘infinitely’ long number. Infinity is not a term that refers to a location, either on a long decimal expression or in any other sense. It is the term we use to denote ‘non-finite’.
Another way to realise this is by reference to my earlier thought experiment about a diagonal line. On my list of ten numbers I can construct a diagonal line (passing through the nth digit of each of the countable numbers on the list), and if I want to I can rotate this diagonal line through 90 degress and tell you where the end points lie after this rotation. I cannot do this on Cantor’s list. On Cantor’s list, the ‘diagonal line’ cannot be created in the first place. If it were possible, Cantor would be able to rotate this line through 90 degrees and tell me where the end points or defining point now lie.