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

280 god and infinityby many others, notably Leopold Kronecker (1823–91). This opposition may havecontributed to Cantor’s declining health in later years, including a series of nervousbreakdowns that began in 1884. Cantor died on January 6, 1918 in a mental institution. 15It is thanks to Cantor that we know how to apply basic mathematical operations,which we use with finite sets, to infinite sets, including addition, multiplication, exponentiation,the relation “greater than,” the equivalence of sets by counting, and so on.This, in turn, meant that he could give an explicit procedure for constructing differentkinds of infinity and for testing their internal consistency. Cantor’s fundamental claimis that these operations can be transferred from their foundations in finite sets to infinitesets. Cantor coined the term “transfinite” for the cardinal number of infinite sets.Rather than assuming that the infinite is in direct contrast with the finite, Cantor treatsan infinite set in a direct analogy with how he treats finite sets.Let’s start with the simplest example of an infinite set, the set of natural numbers{1, 2, 3,...}. Cantor chose a 0 (“aleph-null”) to represent its transfinite cardinal number.He thought of this set as a given whole, and not just an incomplete sequence of unendingfinite numbers ascending in scale. In other words, Cantor actually distinguishedbetween an unending but finite sequence of elements, such as the sequence 1, 2, 3,...,a sequence that is potentially infinite but always, in fact, finite, and the complete infinitesequence of these numbers thought of as a whole, that is, the set {1, 2, 3,...}. He calledthe potential infinite a “variable finite” and symbolized it as ∞; the actual infinite hesymbolized by a 0 , as we saw earlier. Thus, ∞ never reaches completion, never becomesa 0 . To think about ∞ is to think of an ever-increasing series of numbers continuingforever without reaching an end. To think about a 0 is to stand outside this series subspecie aeternitatis and to consider them as a single, unified, and determinate totality.Cantor then extended what we know about counting finite sets to infinite sets: allinfinite sets whose elements can be put in a one-to-one correspondence with the naturalnumbers will have the same cardinal number, a 0 . We call these sets denumerably orcountably infinite. This leads to some surprising results. For example, recall Galileo’sparadox about square numbers. Because there is a one-to-one correspondence betweenthe natural numbers and the square numbers (technically a bijection), it means that theset of square numbers is countably infinite; 16 it has the same cardinal number, a 0 ,asthe set of natural numbers. Similarly, the set of even numbers is equivalent to the setof natural numbers, as is the set of odd numbers. 17Let’s go further. We can generate infinities that are “bigger” than the set of naturalnumbers, although all of these still have cardinal number a 0 . In the case of finite sets,the ordinal and the cardinal numbers of a set are the same. It turns out, however,that they are not the same for infinite sets! We start, as before, with the infinite set of15 Note: Unfortunately there is a textual error in previous publication of portions of this essay in which the texthere reads, erroneously, “uncountably infinite.”16 While still a student, Cantor discovered that the set of all rational fractions (i.e., the quotient of two naturalnumbers such as 4/7) is denumerably infinite, and the same holds for the set of all “algebraic” numbers, suchas √ 3. All these sets have the same transfinite cardinal number, a 0 .17 ω + 1isnot equal to 1 + ω; the latter is equal to ω. Thatis,1+ ω = ω because all the symbol 1 + ω meansis that we add one element to the elements that taken endlessly but thought of as a whole are the set of naturalnumbers whose ordinal number is ω. The former, ω + 1, means adding to a given infinite whole, namely,{1, 2, 3,...}, a new element 1, thus forming the set {1, 2, 3,...,1}. This means that {1, 2, 3,...,1}, taken asa whole, is not equivalent to the set {1, 2, 3,...} taken as a whole. Thus, ω + 1 is not equivalent to ω.