Post updated on 2018/04/05, 19:25 HRS IST:
See the sections added, as well as the corrected and expanded PDF attachment.
As usual, I got a bit distracted from my notes-taking (on numbers, vectors, tensors, CFD, etc.), and so, ended up writing a small “note” on the title question, in a rough-and-ready plain-text file. Today, I converted it into a LaTeX PDF. The current version is here: [^].
(I may change the document contents or its URL without informing in advance. The version “number” is the date and time given in the document itself, just below the title and the author name.)
(However, I won’t disappoint those eminent scholars who are interested in tracing my intellectual development. I will therefore keep the earlier, discarded, versions too, for some time. Here they are (in the later-to-earlier order): [^][^][ ^ ].)
This PDF note may look frivolous, and in some ways it is, but not entirely:
People don’t seem to “get” the fact that any number system other than the real number system would be capable of producing a set consisting of only distinct numbers.
They also don’t easily “get” the fact that the idea of having a distinct succession numbers is completely different from that of a continuum of them, which is what the real number system is.
The difference is as big as (and similar to) that between (the perceptually grasped) locations vs. (the perceptually grasped) motions. I guess it was Dr. Binswanger who explained these matters in one of his lectures, though he might have called them “points” or “places” instead of ”locations”. Here, as I recall, he was explaining from what he had found in good old Aristotle: An object in motion is neither here (at one certain location) nor there (in another certain location), Aristotle said; it’s state is that it is in motion. The idea of a definite place does not apply to objects in motion. That was the point Dr. Binswanger was explaining.
In short, realize where the error is. The error is in the first two words of the title question: “How many”. The phrase “how many” asks you to identify a number, but an infinity (let alone an infinity of infinity of infinity …) cannot be taken as a number. There lies the contradiction.
BTW, if you are interested, you may check out my take on the concept of space, covered via an entire series of (long) posts, some time ago. See the posts tagged “space”, here [^]
When they (the mathematicians, who else?) tell you that there are as many rational fractions as there are natural numbers, that the two infinities are in some sense “equal”, they do have a valid argument.
But typical of the modern-day mathematicians, they know, but omit to tell you, the complete story.
Since I approach mathematics (or at least the valid foundational issues in maths) from (a valid) epistemology, I can tell you a more complete story, and I will. At least briefly, right here.
Yes, the two infinities are “equal.” Yes, there are as many rational fractions as there are natural numbers. But the densities of the two (over any chosen finite interval) are not.
Take the finite interval . There are number of distinct natural numbers in them. The size of the finite interval, measured using real numbers, also is . So the density of the natural numbers over this interval is: .
But the density of the rational fractions over the same interval is far greater. In fact it is so greater that no number can at all be used to identify its size: it is infinite. (Go, satisfy yourself that this is so.)
So, your intuition that there is something wrong to Cantor’s argument is valid. (Was it he who began all this business of the measuring the “sizes” of infinite sets?)
Both the number of natural numbers and the number of rational fractions are infinities, and these infinities are of the same order, too. But there literally is an infinite difference between their local densities over finite intervals. It is this fact that the “smart” mathematicians didn’t tell you. (Yes, you read it here first.)
At the same time, even if the “density” over the finite interval when the interval is taken “in the gross” (or as a whole) is infinite, there still are an infinite number of sub-intervals that aren’t even touched (let alone exhausted) by the infinity of these rational fractions, all of them falling only within that interval. Why? Because, notice, we defined the interval in terms of the real numbers, that’s why! That’s the difference between the rational fractions (or any other number-producing system) and the real numbers.
May be I will write another quick post covering some other distractions in the recent times as well, shortly. I will add the songs section at that time, to that (upcoming) post.
Bye for now.