How Many Finite Integers are There?

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An integer is a whole number, like 1, 3, 99 and so forth.  Non-integers are fractions, such as 1/3, 5/4, and 99.5

No matter how large a finite integer is, you can always add one to it, without reaching infinity. 

This fact can be expressed as “Let X = X+1,” in repeating form, so that X continues to increase endlessly.

Simply stated, it is, in effect, counting from one, toward infinity, but never getting there.

As a consequence of this recursion, it is stated by experts that there must be an infinite number of finite integers.

But this makes no sense.  It sounds like a self-contradiction.  How can we make sense of it?

Let’s start with the title of George Gamow’s seminal (and short) book, 1,2,3. . .Infinity.  The title itself brings up the question, how high do we have to count to reach infinity?

The answer, of course, is that no matter how high we count, by ones, we never reach infinity.  If we count by twos, we get the same result, never getting there.  If we count by millions or trillions, the same.  If we multiply by the square of the previous number—if we raise any finite number X to the finite power of Y—no matter what we do with finite numbers, we never, ever, reach infinity.

But the paradox remains.  How can there be an infinite number of finite integers, without ever reaching infinity?  It seems nonsensical.

Let’s look at the problem from a different angle.  Let’s consider a line segment of finite length.  You can easily draw one on a sheet of paper with a pencil and a straight-edge ruler.  Any convenient length will do.

Next, let us begin repeatedly dividing that line.  First, divide it in half, and then divide the half into halves, then one of those halves into half, and so forth.  What you get are line segments of ½, ¼, 1/8, 1/16 and so forth.  How many times can the line be divided?  You can quickly see that (in principle) there is no end to it.  You never get a line segment so small that to divide it again would result in a length of zero.

But wait.  If you could (but you can’t) divide the line an infinite number of times, you would get a length of zero.  In fact, according to geometry, any finite line segment consists of an infinite number of geometric points, each of which has a length of zero.  Confused?  Yeah, me too.

Perhaps this will allow us, then, to (at least in principle) count from one, two, three, . . . all the way to the end of the line, to infinity.  Will it?

No.  Here is why.  To count, point by point, from one end of the line to the other, one would have to bear in mind that the length of each point is zero.  Therefore, any finite number of zeros adds up to zero.  Five times zero is zero.  Five kazillion times zero is still zero.  Since zero is the length of the first (endmost) point, one never gets past the first point.  Traveling a length of one point, two points, ten, a million or any finite number of points, never gets you past point one.  Never.

However, you can get to point infinity.  You do it by moving a finite length, either part way along the line, or all at once.  Each finite length of the line contains an infinite number of points, but never a finite number of points (which as we said, is zero length).

If you have trouble picturing this, you are in good company.  Nobody can fully understand it.  Some people think they can, but they are just too embarrassed to admit that we are as smart as they are, because they have degrees and titles, and what good are those if they are no smarter than you and me?

In my humble (haha) opinion, there is a way out of the paradox.  It is to make a distinction between the words, “infinite” and “endless.”  In the dictionary, they are the same, but in math and geometry, they have different implications, subtle but important.  The word, “infinite,” carries the implication that you got there, you got to infinity, simply by moving any finite distance along a line.  Any such movement travels an infinite number of points.  You got there.  But “endless” means you are never done, you always have further to go, as in “Let X = X + 1.” 

Is there any practical point to all this?  Scientists think so, and so do philosophers.  You see, there is a question as to whether space is continuous, or grainy.  Is space analog, or digital?

The ancient Greeks gave this some thought, and they came up with an ingenious answer to the question of space.  It’s grainy, that is, there is a unit of space that is of finite size, but so unimaginably small that, because of the very nature of space, no smaller size exists. It is called, the Planck Length.

How did they figure this out?  They took the example we used, of a line segment with an infinite number of zero-length points, and realized that, if space were like that, we could never move.  No matter how many points of length we might move, the distance moved would add up to zero.  Even if, somehow, we could move an infinite number of points at a time, then the finite distance we moved could be any distance at all, since an infinite number of points includes a line of one inch, one meter, or a kazillion miles.

Only by moving a finite number of Planck Lengths can we ever move at all.

So, how many finite integers are there?  We never get to the answer.

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