246 Logic as a Tool and Kronecker Cantor’s first paper on infinite sets was published shortly before he turned 30 The notion of infinity has always been a tricky and elusive topic that very easily leads to paradoxes such as those of Zeno of Elea (c 490–430 BC) Because of that, most mathematicians (including the genius Gauss) had previously preferred to work only with “potential” infinities (e.g., for every natural number there is a greater one) rather than actual “completed” infinities (e.g., the set of all natural numbers) Cantor was the first to deal explicitly with actual infinite sets, perform operations on them, and compare them; although regarded as a matter of course in mathematics today, this was quite revolutionary then It is therefore not surprising that Cantor’s theory of infinite sets and transfinite numbers was originally regarded as counter-intuitive, even unbelievable, such that it encountered strong resistance from many authoritative mathematicians of the time such as Kronecker and Poincaré, and later Weyl and Brouwer For instance, Poincaré referred to his ideas as a “grave disease infecting the discipline of mathematics,” while Kronecker went even further and personally attacked included Cantor, describing him as a “scientific charlatan” and a “corrupter of youth.” Some strong philosophical and theological objections against Cantor’s set theory were also raised against it, by some influential theologians and philosophers For example, much later Wittgenstein still regarded it as “utter nonsense” and “laughable” and complained about its poisonous effect on mathematics Russell’s discovery in 1901 of the paradoxical set of all sets that are not elements of themselves certainly contributed to the strong suspicion and even plain rejection faced by Cantor’s set theory for a long period Possibly because of the harsh reaction of his contemporaries, Cantor had recurring bouts of depression from 1884 until the end of his life in a mental hospital The practical power of his theory was gradually recognized however, such that in the early 1890s he was elected President of the German Mathematical Society He taught at the University of Halle in Germany from 1869 until his retirement in 1913 In 1904, the British Royal Society awarded Cantor its Sylvester Medal, the highest honor it confers for work in mathematics, while David Hilbert strongly defended his theory by famously proclaiming: “No one shall expel us from the Paradise that Cantor has created.” 5.3 Mathematical Induction and Peano Arithmetic This section is devoted to two related important topics which are specific to the arithmetic on natural numbers, where logical reasoning plays a crucial role The method of Mathematical Induction is a very important reasoning technique in mathematics which cannot be extracted from the purely logical rules of Natural Deduction for first-order logic Even though it is not a rule of purely logical reasoning, Mathematical Induction is an indispensable reasoning tactic for proving universal statements about natural numbers, so I present and explain it here in some detail