Does Infinity Exist?

by Robert Arvay

While infinity is a valid mathematical concept, that is not the same thing as saying that it is real in the physical sense.

Even in the abstract mathematical sense, infinity presents some paradoxes.  For example, in mathematics, it is said that there are an infinite number of positive, finite integers.  That is a paradox, because it implies that there can be an infinite series of finite integers (or anything else).  Why is that a paradox?  If every item in the infinite series is finite, then how can the series be infinite?  Infinity can never be reached, but only approached.  (Even THAT is a paradox, because how can one approach something, yet never get any closer to it?  One is always an infinite distance short of infinity.)  It has no finality.  By definition it has no end.  If one can never get there, is “there” even a “there?”  The semantics are maddening.

How does this apply to the physical world?  In the physical sense, it is speculated, by cosmologists, that the universe may extend forever in all directions.  In the hypothetical, infinitely large universe, we must ask, is that possible?  If the universe is indeed, infinitely large—physically—then infinity has been reached.  It physically exists.  Paradoxically, the endless has been both reached, and yet, can never be reached.

Consider this thought experiment, a test of that principle.  Imagine that you are in a spaceship, traveling in a straight line, in an infinitely large universe.  No matter how far you travel, you will never travel the infinite distance that defines the universe’s reach.  You will never reach the end, because by definition, there is no end.

But wait.  What if you travel at an infinitely fast speed?  What would happen then?

To answer that, let us first look at another, paradoxically infinite universe, one that is both infinite and finite at the same time.  Let us consider a ruler that is twelve inches in length.  Why is this both finite and infinite?  It is infinite, if you measure its length by the number of geometric points along its length.  There are an infinite number of such points.  A geometric point results if one divides the ruler in halves an infinite number of times.  This results in a point having a length of zero.

So, if one begins at one end of the ruler, and begins traveling toward the other end, point by point, then it would take an infinite amount of time to get to the other end.  Indeed, it would take an infinite amount of time to get anywhere along the ruler.  We know this simply by multiplying the speed of movement by zero.  Any amount, any number of points, times zero, is zero.  Since each geometric point is of zero length, then one could never move even as far as one point along the line.  So, one could never move any finite length, if one moves zero length at a time.  To move at all, one would have to move an infinite number of points at a time, all at once;  that is, at infinite speed.

But wait again.  If one moves an infinite number of points all at once, then he has moved a finite distance—but what is that distance?  Is it one foot?  One inch?  One mile?  A billion miles?

Of course, the distance moved would be arbitrary.  One could never specify how far that distance would be.  It could be ANY finite length.

Returning to our spaceship, what would happen if we increased the speed to infinity?  How far would we travel?  If we travel an infinite distance, in an infinitely large universe, then where would we be?  Some random place?  Back where we started?  Outside the universe?  More than one place at one time?  Everywhere?  Where?

What all this tells us is that our intellect is far too limited to understand an infinite reality.

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