I think Cantor would be upset because you aren't following his rules. You are supposed to produce a list of numbers, not an algorithm for producing numbers. He wants a list so that there is an unambiguous answer to questions like, what is the 837th entry in your list? As you point out, your 837th entry changes with each iteration of your algorithm. Cantor needs that unambiguous answer so that he can construct a number not on your list by making the 837th digit in his constructed number different from the 837th digit of your 837th entry. In general, he will make the nth digit of his number different from the nth digit of the nth entry in your list. By giving him a moving target, he can't produce his number.
This is, of course, his famous diagonal approach which was also instrumental in Goedel's proof of his famous theorem. Incidentally, speaking of the diagonal approach, the Science News article had a neat stereo pair of squares populated with nearly the same array of random dots. The only difference between the two squares was that corresponding dots along one diagonal were different from those of the other square. I am a great fan of stereo images like this and I found the image provided by those squares was striking. Once you got the 3-D effect, the diagonal obviously stood out. But you couldn't see it distinctly. If you focused on it directly, that portion seemed to disappear and the rest of it sort of shimmered. I also found that as I moved the center of my vision around the square, portions of the shimmering diagonal would disappear. It seemed to me that it disappeared in places that wouldn't be accounted for by the blind spots in my eyes, but I'm not really sure about that. Anyway, I was curious as to whether or not you looked at that image in 3-D before your magazine got thrown out. Did you?
Now, back to your post. You asked whether in my opinion the rationals or the reals were being used up faster. You also asked me to think about that one for a while. Well, I did think about it and there are a few different aspects that are fun to think about. The net result of my thinking, though, is that I think the rationals are being used up faster. In spite of the fact that the rate of consumption of reals increases exponentially over that of the rationals as your algorithm grinds away, the difference between the number of significant digits of each only increases arithmetically. The reals always have twice the number of significant digits as the rationals. I think of it like a race between me and Bill Gates writing checks to spend our money. If we had to write each check with one more significant digit for the amount than the previous check, I'm sure I would run out of money first. Of course this is just meaningless whimsy. I was simply following your admonition to keep Jerry Bona's joke in mind.
"My personal opinion is that "size of infinity" is an undefinable term."
I would amend that to say "My personal opinion is that "size of infinity" is an undefinable term if the definition is to be consistent with the rest of the math".
"Now I would like to bring up a serious question about the concept of existence."
I am delighted. As I said, I was disappointed that Michael didn't want to get into metaphysics, but I don't think you can make this discussion pertinent to God and Science without doing so.
"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 my opinion: Yes.
"In other words can your mind create "real" objects?"
In my opinion: Yes
" How would you propose to prove it can not?"
Since I think it can, I don't think the burden of proof is on me. If I'm right, no such proof is possible.
"Just what is a "real" object anyway?"
Now there is the interesting question. I'm glad you asked me in a public forum because I am interested in what other people think of my ideas on this question. (Incidentally, I'm not sure why our correspondence shifted from private emails to public postings, but now that we are on this topic, I'm glad it did.)
In my view, there are two types of real objects: thoughts and things. You could debate whether thoughts are real, or whether thoughts are things, but however those debates might be resolved, what is left would answer your question.
In our ordinary experience, thoughts and things are different, so I categorize them in those two categories. I think most people would agree that thoughts exist, so I call them real for that reason. So now, your question becomes "Just what are thoughts and just what are things?"
My answer to the first of these is that thoughts are knowledge directly available to my consciousness. If other people have such knowledge and such consciousness, which I suspect they do, then they should be able to recognize what I am talking about.
Before I give you my answer to the second question, I'll give you Bertrand Russell's answer: He says "Things are those series of aspects which obey the laws of physics."
Now, to give you my answer, I have to jump headlong into metaphysics. (None of this should be new to long-time readers of this forum.) In my view, there is only and exactly one thinker in all of reality. All of us humans who each think we are an independent thinker, are in actuality a sort of time-shared subset of the experiences of that one thinker. In other words, we are that thinker. It's just that when we (I, it, us, you, she,... the pronoun doesn't matter when there is only one antecedent.) are thinking, most all knowledge is unavailable to us. The only easily accessible knowledge is the relatively tiny amount associated with the particular brain that seems to be doing the thinking at that moment in that circumstance. It's just like a single time-shared program running in a big computer having access only to its local data.
So what we have, in reality, (literally) is this one thinker thinking thoughts. A subset of those thoughts are a consistent set comprising what seems to us (that misleading pronoun again) to be a physical universe populated with, among other things, independently thinking people. The fact that that subset is consistent, as you have proved, means that this physical universe must obey the laws of physics. This, then, makes Russell's definition of 'thing' make sense.
I think this view is consistent with what mystics have been trying to articulate for thousands of years. And, I think this view is consistent with what science seems to be uncovering about the true ontology of reality.
Anyway, I'm glad you asked. It has been fun to answer.