Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

from potential infinity to actual infinity 39nineteenth century, the best minds in philosophy, natural sciences, and mathematicsdid not accept the concept of actual infinity; only the idea of potential infinity (in thesense of Aristotle) was accepted. The towering minds of people such as Galileo Galilei,Gottfried Wilhelm Leibniz, Baruch Spinoza, Isaac Newton, and even Carl FriedrichGauss rejected actual infinity. It was rejected because they thought that this conceptincluded antinomies, such as the one Galileo Galilei discovered. However, BernardBolzano (1955) showed that most of the antinomies of actual infinity could be reducedto paradoxes, and they could be resolved in a logical sense. In fact, antinomies wouldhave been disastrous for mathematics, because they entailed the possibility of provingall mathematical conjectures, including those that are obviously, false.Georg Cantor revived the tradition of Nicholas of Cusa, to whom he even alludedin the endnotes of his Grundlagen. Cantor is in fact the first person who claimed –despite the Kantian philosophy that prevailed then – that actual infinity could be anobject of mathematical research. He said that the human ratio could create conceptualtools in order to discern its internal structure. Georg Cantor actually proceeded to do it.Here are a few important examples to show how Georg Cantor made infinity a subjectof rational research. These examples fall into three different categories: (1) rationaldiscernment of infinity, (2) real antinomies (logical contradictions), and (3) resolvingthe antinomies.1.4.2.4.1 Rational Discernment of Infinity. Let me start with the rational discernmentof the internal structure of infinity. Georg Cantor created a new kind of number. Hedefined the infinite number of the integers N as a new number that he called a 0 , or the firsttransfinite set, or the first cardinal number. This new number a 0 = N ={1, 2, 3,...,n}serves as a kind of mathematical measurement device for the internal structure ofinfinity. This means that he substituted the “bigger” and “smaller” relation (), andthat was used by Galileo to compare the integers with the numbers of the power two,and that comparison created the paradox that was seen earlier. Thus, an infinite setitself became a mathematical measuring device for other infinities.By relating a 0 = N ={1, 2, 3,...,n} to the rational numbers Q and the real numbersR, he could show that these sets N and Q, on the one hand, and R, on the otherhand, had a different cardinal number (“Mächtigkeit”). N and Q are countable infinities,whereas R is a noncountable infinity. 106 He called R the next higher cardinal number(“Mächtigkeit”) a ∗ and showed that a ∗ > a 0 . He also raised the question as to whether,between a ∗ and a 0 , there is another set of numbers with a specific cardinal number(“Mächtigkeit”), known as the continuum hypothesis. This hypothetical set he calleda c . He showed that this hypothesis could be formulated by the mysterious relationa c = 2 a0 . He also created an arithmetic for transfinite numbers, in which the commutativelaw was not valid. Then, he created higher orders of the a s , such as a 1 , a 2 ,a 3 ,...,a n , and even a n .With the help of this new concept of cardinal number (“Mächtigkeit”), he dealt withthe paradox already noticed by Galileo Galilei, that a subset of N, like the integers of the106 This distinction within infinity was partly anticipated by Bernard Bolzano. He spoke about different “orders”of infinity. However, he did not arrive at Cantor’s clear-cut notion of “Mächtigkeit” (cf. Bolzano 1955, §§ 21,29, 33).