The concept of infinity is such an intrinsically paradoxical concept that some mathematicians prefer to reject its validity as a mathematical concept. Some prominent ancient mathematicians like Eudoxus and Archimedes preferred to use what is termed the “method of exhaustion” in solving problems in geometry, to avoid having to break figures into infinite number of indivisibles. Isaac Newton introduced the concept of the limit, the underlying concept to his new mathematics of calculus (“method of fluxions”). The success of the limit method finally nailed the coffin to those methods relying on “indivisibles.”

Late in the nineteenth century, Georg Cantor (1845-1918), a German mathematician who had studied medieval works on the concept of infinity began independent research of his own in the field. His studies of trigonometric series brought up the problem of infinite sets. Cantor’s quest for conditions which ensure that a function has only one trigonometric series representation led him to the discovery that a series need not converge to the function at every point of the interval being considered. Cantor’s research led him deeper and deeper into a world of more and more complex varieties of infinite sets. He encountered various paradoxical properties of infinite sets, but unlike earlier mathematicians in this field, he did not discard his results as useless because of their paradoxical nature.

Cantor encountered paradoxes of the same essential nature as Zeno’s. He demonstrated that a small segment of a line had just as many points as an infinite line! He found that he could set up a one-to-one correspondence between any point on a circle and points on an infinitely extended line! This can be done by drawing lines from the center of a circle through the points on the circle to the line. Thus, we are allowed to reason, logically, that the infinite line is not greater than the circle!

The preceding is the grounding principle of Cantor’s infinite sets: the whole standing in a one-to-one point correspondence with its part. Galileo had provided a simpler example of infinite sets when he corresponded the integers with their squares.

The set theory of mathematics runs into serious trouble when we begin considering Cantor’s monster sets. The Cantor set is almost impossible to visualize. One may construct a Cantor set by starting with an interval [0,1], and them remove, say, a middle third, and then from the resultant pieces remove another middle third and then perpetuate the procedure. The resultant pieces get continually smaller in length as we proceed, but each time we find that there are twice as many of them.

Now, consider a situation after we have repeated the procedure an infinite number of times. We find that there are still some points not removed! These are the Cantor sets. It can be demonstrated by summing a geometric series that the intervals removed from [0,1] have a summed up length equal to 1. Which means that the Cantor set has a total length zero! Yet, Cantor could paradoxically show that there are just as many points in the Cantor set as there are in the interval [0,1]!

Historians tell us that Cantor finally broke down and died in a psychiatric hospital from contemplating his daemon-spawn sets. Yet, he was able to show before he died that there are sets with more elements than the set of points on a line.

Cantor, unfortunately, did not appreciate the full significance of his work (and neither did his contemporaries who rejected his work on infinite sets). To show that a system of logic is paradoxical is to show that the system of logic is inconsistent and therefore “invalid.” That is, there is an aesthetic foundation to what we consider to be logical. Zeno had relied on methods which revealed the inconsistencies in his opponents’ reasoning to prove the falsehood of their conclusions. The unveiling of paradoxes to our set theory immediately raises questions about the validity of appearances in our physical world. Succeeding generations of mathematicians have not realized that Cantor’s work, if it had been properly understood, might have set the basis for a new non-set logic approach to mathematical thinking, which has as its first premise the realization that our “physical” world is the product of “free creation,” and that, therefore, the much revered Boolean rules of set logic are faulty contrivances which reflect no fixed immutable laws of logic or divine creation.

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