Here is a book lying on a table. Open it. Look at the first page. Measure its thickness. It is very thick indeed for a single sheet of paper — one half inch thick. Now turn to the second page of the book. How thick is this second sheet of paper? One fourth inch thick. And the third page of the book, how thick is this third sheet of paper? One eighth inch thick, etc. ad infinitum. We are to posit not only that each page of the book is followed by an immediate successor the thickness of which is one half that of the immediately preceding page but also (and this is not unimportant) that each page is separated from page 1 by a finite number of pages. These two conditions are logically compatible: there is no certifiable contradiction in their joint assertion. But they mutually entail that there is no last page in the book. Close the book. Turn it over so that the front cover of the book is now lying face down upon the table. Now, slowly lift the back cover of the book with the aim of exposing to view the stack of pages lying beneath it. There is nothing to see. For there is no last page in the book to meet our gaze.
An interesting (and very cool) twist on the nature of infinity. Or, should I say, infinities, for there is not just one.
There are a lot of curious things about allowing the existence of infinite things, which is one of the primary reasons why I do the kind of work that I do. I was fascinated by the notions that crop up. Like the fact that once you formalize the concept of a "problem" that can be solved and go on to formalize its difficulty, then in very short order you come to realize how tiny our intellect is. (Viz., as I have said too often, there are not only infinitely more problems that we cannot solve, there are infinitely many that we can't even adequately describe. This is very easy to prove, definitively, with easy math.)
But infinities have been bugging people for ages. Aristotle, that most famous of misguided philosophers, once wrote a version of Zeno's Paradox thusly: "In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead."
Obviously untrue, yes? Put in more familiar terms, you can never leave the room that you're in because you must first walk half the distance to the door. And then half again, and so on. But half of a finite quantity is always another finite quantity. Nevertheless, I still somehow manage to walk from my office to the coffee maker every morning, and the Boston Marathon is still run every year despite the algebraic obviousness that it can't actually happen at all.
Things like this have confounded scientists for centuries. There was a time when nobody could comprehend the idea of empty space, since they already knew of radio waves. As a result, they tried to prove (without success) of the existence of a material "ether" that transmitted the wave effects. And when confronted with proof that light is neither wave nor particle, but rather both, their minds rebelled.
And yet today, we're comfortable with the dual nature of matter (both waves as well as particles), and given proof that the ether cannot exist we now simply accept the fact that electromagnetic radiation propogates through empty space without any other medium, and the mathematical notion of limits has long since dispensed with any curiosity about the answer to Zeno.
Dare I say that there is an element of faith involved here?
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