Most difficulties with the diagonal method revolve around the definition of the word "denumerable." As mentioned previous, the diagonal method is supposed to show that the rational numbers are denumerable, while the real numbers are not. But there is no way to understand the meaning of the term denumerable, until after you have applied the diagonal method.
This game with the definition is a rather nasty piece of dialectic. At first glance the definition appears to be quite intuitive. A set is denumerable if you are able to count the members of the set. For example, the set of three pies is denumerable. It is finite, you can sit in the kitchen and count the pies, one, two, three--you're done.
When we speak about infinite sets, the counting becomes more difficult. Some might go as far as to say that you cannot count an infinite set. For that matter, the fastest super computer known to man has only a finite amount of memory, and would come to a crashing halt well before it finished the job of counting to infinity.
The idea of counting to infinity is completely out of the realm of our experience. We can only imagine the process with the highest degree of abstraction, but without even a hint of trepidation, the transfinite theorist has no problems saying that you could "count" an infinite set by placing it in a one-to-one correspondence with the Natural numbers.
Again, I should emphasize, that it is only a matter of speculation that anyone could perform such a trick. But I am willing to accept such a definition. It is easy to see that if I started a list of the squares, I would not run out of squares before running out of natural numbers. I can create a one-to-one mapping of the sets with the function y = x^2:
1 2 3 4 5 6 ... 1 4 9 16 25 36 ...
Note, my intuitive understanding of this definition is based on the ability to begin a list.
Okay, in the final step of the proof, we learn that the real numbers are not denumerable. This is where the circularity of the definition comes into play. In this final part of the proof, we are told that a set is not denumerable because it fails the diagonal test. My intuitive definition of the term is thrown off guard, I can start a list of real numbers, but I am told the list is not denumerable because I cannot finish it.
As you see, I am now in a circular definition. The diagonal method shows that a set is not denumerable, but the definition of non-denumerable is based on the sets failing the diagonal method.
Although presented as a simple proof, the definitions of the terms in transfinite theory will catch any thinking student in a serious dialectical bind. The proof tries to extend simple intuitive notions, such as counting, into a transfinite realm, where the definitions suddenly begin to change meaning.
The intuitionists are correct in complaining about the poor terminology used in this theory; however, they are not correct in dismissing the theory. The theory has a poorly defined, unintuitive vocabulary. The goal of the transfinite theorist should be to improve the vocabulary used to describe the theory, so we can figure what they are saying.
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