ABSTRACT
Study of the concept of infinity dates back to the Greek philosopher Zeno of Elea (c. 450 B.C.). However, the modern era of the study of infinity was begun by the Italian physicist and mathematician Galileo Galilei (1564-1642). In 1638, Galileo published a Dialog Concerning Two New Sciences. His primary goal in Dialog was to establish the Copernican heliocentric theory of the solar system over the accepted and church supported Ptolemaic geocentric theory. For this, Galileo was charged with heresy by the Inquisition, forced to recant, and spent the last eight years of his life under house arrest. In Dialog, Galileo also discussed the concepts of infinite and infinitesimal. He observed that the set of natural numbers, N, properly contains the set of perfect squares, S = {1, 4, 9, 16, . . .}, and that there are an infinite number of elements in both N and S. He concluded that there were as many elements in S as in N. However, Galileo believed this conclusion was absurd, since this would mean it was possible to apply the terms “equal to,” “greater than,” and “less than” to infinite quantities. In 1691, the English mathematician Edmond Halley (1656-1742) published an article in the Philosophical Transactions of the Royal Society titled “On the Several Species of Infinite Quantity and the Proportions They Bear to One Another” in which he suggested that the phrases “twice as infinite” and “one-fourth as infinite” might be meaningful. In other words, Halley was suggesting that infinite quantities might have different “sizes” and be “comparable.”
