An Historical Account of Set-theoretic Antinomies Caused by the Axiom of Abstraction

The beginnings of set theory date back to the late nineteenth century, to the work of Cantor on aggregates and trigonometric series, which culminated into his seminal work Contributions to the Founding of the Theory of Transfinite Numbers [2]. Although this work was published in 1895 and 1897, in two parts, the theory of sets had been established as an independent branch of mathematics as early as 1890 ([9], p. 1), from Cantor’s earlier work. Despite initial reservations against the representations of infinite collections as single entities, Cantor’s set theory gained acceptance by the mathematical community once its usefulness was manifested in various areas, including analysis and geometry. As mathematics was making tremendous strides using the theory of sets, several paradoxes emerged from Cantor’s loose definition of a set—around the turn of the century—that threatened to undermine the foundations of mathematics being built upon Cantor’s theory. In his Contributions, Cantor defined a set to be “any collection into a whole (Zusammenfassung zu einem Ganzen) M of definite and separate objects m of our intuition or our thought” ([2], p. 85). To the naturally finitistic mind this definition appears innocuous and mathematically appropriate, but an application of this definition to infinite sets can be especially pernicious. Cantor’s definition gave rise to a principle asserting that certain formulas define sets, often called the Axiom of Abstraction. This axiom states that for any formula, φ(x),there is a set containing precisely the elements a for which φ(a) is true. In other words, for every formula φ(x) in one free-variable ∃A∀x(x ∈ A ↔ φ(x)). A form of this axiom was used by Frege in the first volume of Grundgesetze der Arithmetik, published in 1893 ([10], p. 10), and later, a paradox caused by the axiom undermined most of his theory immediately before the second volume was published. This paper will be concerned with a historical description of the prominent paradoxes that resulted from the use of Cantor’s definition and the Axiom of Abstraction near the beginning of the twentieth century. The distinction between ‘paradox’ and ‘contradiction’ will be used as in [6]. Thus, a contradiction will occur when a statement and its negation can both be proved true, but a paradox will be defined as “an argument which ends in a contradiction although all of its premises and modes of reasoning are prima facie acceptable” ([6], p. 321), with the further stipulation that “the one who discovers it give up a premise or mode of reasoning that he has previously accepted as correct” ([6], p. 321). In this case, the Axiom of Abstraction will be the abandoned premise, given up for either Zermelo’s Axiom of Separation or Fraenkel’s Axiom Schema of Replacement, which implies Zermelo’s Axiom. The three most important paradoxes discovered were the paradoxes of Burali-Forti, Cantor, and Russell, which will be discussed below. The discussion will proceed chronologically, at least to the dates traditionally assigned to each paradox, and consequences of these paradoxes will be assessed.