Hi Paul,
I think you are right. A lot of the quotes you gave sound very familiar. It is funny how the memory works; I think one's imagination adds "unknowable data" to make the gist of the experience make sense. I could have sworn it brought up the Axiom of Choice by name. Well, I apparently read it over a week ago and didn't take any notes; I was only reminded of it by your post. I suppose I can be forgiven for getting it a bit twisted.
Since reading your response last night, I have been thinking a little. In particular, about Cantor's method of determining cardinal equivalence of two sets. I would like you to think about the following pairing procedure between the set of natural numbers and the set of real numbers.
1. Begin by constructing a 10 by 10 matrix of cells. Across the bottom row, insert the digits zero through nine. In each column, insert the digits one through nine in the empty cells.
2. One can now construct a real number corresponding to each and every cell via the following procedure:
a) The real numbers will consist of one integer above the decimal and one below the decimal.
b) The real number in the cells across the bottom will have the inserted digit as the integer above the decimal and zero as the integer below the decimal.
c) The real number in all the other cells will have the integer at the bottom of the column as the digit above the decimal and the digit in the cell as the digit below the decimal.
3. That collection contains exactly 100 real numbers. They can be counted and they can be paired to the first 100 members of the set of natural numbers.
4. Now construct a 100 by 100 matrix of cells. Across the bottom row, insert the digit pairs zero-zero through nine-nine. In each column, insert the digits zero-one through nine-nine.
5. one can now construct a real number corresponding to each and every cell via a procedure exactly analogous to that in 2 above.
a) The real numbers will consist of two digits above the decimal and two digits below the decimal.
b) The real numbers in the cells across the bottom with have the inserted digit pair as the two digits above the decimal and zero-zero as the two digits below the decimal.
c) The real number in all the other cells will have the digit pair in the bottom cell of the column as the digit pair above the decimal and the digit pair in the cell as the digit pair below the decimal.
6. Once again we have a countable subset of the real numbers (10,000 of them to be exact) and they can be paired to the first 10,000 members of the set of natural numbers. It is important to notice that this set also contains every real number contained in the collection laid out in the 10 by 10 matrix.
7. Now, you can continue this procedure with three, four, five, ... digits in a cell. At every step of the procedure the subset of real numbers can be counted and paired with the proper subset of the natural numbers.
8. As the procedure is continued, there is no real number which will not be picked up so long as we are allowed to continue forever: i.e., if the procedure of pairing the odd numbers to the natural numbers can be deemed accomplishable, one must conclude that the real numbers can also be so paired.
Now I believe Cantor would get very upset with the fact that I am rearranging the numbers at every step, but I don't think one could argue that the rearrangement is significant to the count as, each time I add some more real numbers, I am always including the real numbers already considered in my count which is the actual central issue here. Now with your complaint about members of the set being used up faster, in the procedure I just laid out, which ones are being used up faster? The real numbers or the natural numbers? Think about that one for a while!
With regard to things being counter-intuitive, I think it would be good to keep Jerry Bona's joke in mind. My personal opinion is that "size of infinity" is an undefinable term. If it is infinite, it just never stops and if you were smart enough you could come up with a pairing program which would pair off any infinite set with the natural numbers. Of course, that is no more than an opinion.
Now I would like to bring up a serious question about the concept of existence. I know you are aware of the fact that your mind can present you with illusions which are not controllable in any conscious way. What I mean here is that you cannot eliminate the illusion by knowing it is an illusion. We generally "know it is an illusion" because we can prove it is "inconsistent with actual reality". Suppose your mind were to create a particular illusion which just happened to have no inconsistent elements at all. Does that make it "consistent with reality"? In other words can your mind create "real" objects? How would you propose to prove it can not? Just what is a "real" object anyway?
Have fun -- Dick |