Levi Recognizes the Axiom of Choice in Set Theory

Date 1902

Beppo Levi acknowledged and criticized the axiom of choice in set theory and attempted to justify and carry out infinite choices.

Locale University of Turin, Italy

Key Figures

• Beppo Levi (1875-1961), Italian mathematician
• Giuseppe Peano (1858-1932), Italian mathematician and logician
• Felix Bernstein (1878-1956), German mathematician
• Georg Cantor (1845-1918), German mathematician

Summary of Event

The notion of infinity has been a perennial problem in both mathematics and philosophy. A major question in (meta) mathematics has been whether and in what form infinities should be admitted. The debate between Gottfried Wilhelm Leibniz and Sir Isaac Newton on the use of the “infinitesimal” in seventeenth century differential calculus is only one of many possible examples of this problem. In most pre-twentieth century mathematics, infinities occur only in what might be called a “potential” (inactual, implicit) form. Potential infinity can be illustrated readily with a simple example from limit theory. There is no question of an actually infinitely great or infinitesimally small quantity to be computed or otherwise arrived at. As noted set theoretician Abraham A. Fraenkel has remarked, many nineteenth century mathematicians such as Carl Friedrich Gauss and Augustin-Louis Cauchy considered infinity in mathematics to be largely a conventional expression showing the limits of ordinary language for expressing pure mathematical concepts.

In addition to the less problematic notion of potential infinity, an equally old problem of “actual” infinity had long been faced by speculative philosophers and theologians such as Saint Augustine, Saint Thomas Aquinas, René Descartes, and Immanuel Kant, among many others. A major thematic front throughout nineteenth century mathematics and logic was precise clarification of the boundary and relations between actual and potential infinities, in foundational as well as applied mathematics. One of the earliest such efforts was that of Bernhard Bolzano, whose Paradoxien des Unendlichen (1851; Paradoxes of the Infinite, 1950) cataloged a large number of extant and novel conundrums as well as unclear and unusual properties of actual infinities in mathematics. A particular emphasis, to recur repeatedly in later set theory and logic, was the apparent paradox of the equivalence of an infinite set to a proper part or subset of itself, implying different levels or kinds of infinity where previously only one infinity was supposed. The term “set” was first employed in Bolzano’s text as an important mathematical concept.

The first major developments in the formation of a consistent and comprehensive theory of actual infinities in mathematics were primarily the work of Georg Cantor. Between 1873 and 1899, Cantor sought to lay the foundations for a new branch of mathematics—called the theory of aggregates or sets—that would not only formalize the mathematically acceptable concepts of infinity but also serve as the foundation for every other mathematical theory and discipline employing infinities. Cantor’s set theory did not begin with philosophical speculations about infinity, but with the problem of actually distinguishing finitely and infinitely many “specified” points, such as the points of discontinuity in the theory of functions. Yet, despite many strenuous efforts, Cantor, as well as Richard Dedekind, Giuseppe Peano, and many other mathematicians, blurred or passed over unawares the distinction of choices implicitly or explicitly made by some kind of a rule or algorithm from which a given denumerable or indenumerable (uncountable) point set was defined.

In particular, although the first n members of a subset were seen clearly as selected specifically by a selection or generative rule, neither Cantor nor anyone else explained how such a rule could actually be extended to define likewise the entire (actually infinite) subset. This problem was further complicated by the fact that many mathematicians of the era frequently made an infinity of arbitrary selections, not only independently but also where each given choice depended on the choices previously made. These questions not only undercut the proof method of using “one-to-one” mappings or correspondences between different sets but also underscored the then-formulating foundational questions of whether “in the last analysis” mathematical entities such as sets are discovered or defined/created.

In 1882, particularly, Cantor argued to Dedekind and others that his means of extending the (infinite) sequence of positive numbers by introducing symbols for infinity, such as ∞, ∞+1, and ∞+2, was not merely conventional but a legitimate number choice. Perhaps the simplest example of Cantor’s transfinite numbers is the model suggested by Zeno’s paradoxes of motion. Here, a runner uniformly traverses a road divided into intervals. Although the number of intervals is ∞, the time taken to traverse them is finite (hence the paradox). If the first interval is designated the ωth interval, the subsequent intervals will be the ω+1th, ω+2th, and so on. These numbers ω, ω+1, ω+2, and so on, are the transfinite ordinal numbers first designated by Cantor. Also called into question was Cantor’s related continuum hypothesis, which asserts that every infinite subset of the real numbers either is denumerable (countable by means of a finite procedure) or has the degree of infinity of the continuum. In Cantor’s sense, the continuum is defined by the assumption that for every transfinite numbert, 2^t is the next-highest number. Equivalently, the continuum hypothesis proposes that there is no (transfinite) cardinal number between the cardinal of the set of positive integers and the cardinal of the set of real numbers.

These and other related developments left early twentieth century set theory with a network of fundamental, interlinked, and unsolved problems, all of which somehow involved the notions of selection or choice, linking well-known finite mathematics with the newer mathematics of actual infinities. Nevertheless, very few, if any, mathematicians explicitly considered all these questions from this viewpoint. Peano, an Italian mathematician responsible for the first symbolization of the natural number system of arithmetic, was also coming up against similar inconsistencies in his investigations of the conditions for existence and continuity of implicit functions. In 1892, one of Peano’s colleagues, Rodolfo Bettazzi, investigated conditions under which a limit point was also the limit point of a sequence. Bettazzi underscored the same underlying issue as Peano.

The third mathematician to consider this problem of how to justify or actually carry out infinitely many choices was Peano and Bettazzi’s colleague Beppo Levi. Levi was completing his dissertation research, inspired by the set theory work of French mathematician René Baire, who had developed the novel notion of “semicontinuity” for point sets. In 1900, Levi published a paper extending Baire’s investigations of fundamental properties satisfied by every real function on any subset of the real numbers. Without proof, Levi proposed that every subset 1 is equal to the union of subsets 2 and 3 minus subset 4, where 2 is any closed set and 3 and 4 are “nowhere-dense” sets. Levi also asserted that every uncountable subset of the real numbers has the power of the continuum, essentially Cantor’s continuum hypothesis.

In another dissertation of 1901, a student of Cantor and David Hilbert, Felix Bernstein, sought to establish that the set of all closed subsets of the real number system has the power of the continuum. In 1897, Bernstein had given the first proof of what is known as the equivalence theorem for sets: If each of two sets is equivalent to a subset of the other, then both sets are mutually equivalent. In his 1901 work, Bernstein remarked that Levi’s 1900 results were mistaken. As a response, in 1902 Levi published a careful analysis of Bernstein’s dissertation in which Bernstein’s use of choices in defining sets came into sharp and explicit critical focus.

In Levi’s broad analysis of then-extant set theory, he questioned Cantor’s assertion that any set can be well-ordered. Well-orderedness is the property whereby a set can be put systematically into a one-to-one correspondence with elements of another set. Levi pointed out that even though Bernstein had openly abandoned the well-ordering principle, Bernstein had nevertheless employed an assumption that it appeared to be derived essentially from the same postulate of well-orderedness. This questionable assumption was Bernstein’s so-called partition principle; that is, if a set R is divided or partitioned into a family of disjoint (nonintersecting) nonempty sets S, then S is less than or equal to R. Following Levi’s proof that Bernstein’s partition principle was valid only whenever R was finite, Levi critically remarked that the example was applicable without change to any other case where all the elements s of S are well-ordered or where a unique element in each s can be distinguished. This statement is, essentially, a summary of what would be explicitly termed the axiom of choice.

Significance

Although Levi’s 1902 paper proved a catalyst for subsequent work by Bernstein, Hilbert, Ernst Zermelo, and many other mathematicians, the direct response to and recognition of the paper was very limited. Basically, although Levi explicitly recognized the axiom of choice embodied in the work of Cantor and Bernstein, he rejected its use in its extant form. This led ultimately to Zermelo’s 1904 publication that proved the well-ordering principle by use of the axiom of choice.

In his 1910 seminal paper on field theory foundations in algebra, Ernst Steinitz summarized the widespread attitude toward the axiom of choice in algebra, topology, and the like. Thus, although explicit examination of the axiom of choice was largely ignored for some time, from 1916 onward the Polish mathematician Waclaw Sierpinski issued many studies of implicit as well as open applications of the axiom of choice. Although Levi in 1918 offered what he called a “quasi-constructivist” improved alternative to the axiom, most mathematicians considered his alternative too limited and unwieldy. In 1927, American logician Alonzo Church sought unsuccessfully to derive a logical contradiction from the axiom. No alternative was developed until 1962, when two Polish mathematicians, J. Mycielski and H. Steinhaus, proposed their axiom of determinateness. Since then, it has been shown that a number of other propositions are equivalent to varyingly weaker or stronger forms of the axiom of choice, as originally recognized by Levi and positively employed as such by Zermelo.

Bibliography

Dauben, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite. Cambridge, Mass.: Harvard University Press, 1979. A good rendering of Cantor’s original set theory papers. Undergraduate-level discussion.

Fraenkel, Abraham A. Set Theory and Logic. Reading, Mass.: Addison-Wesley, 1966. Intermediate level. Includes math background material to the debate that arose about the axiom of choice as well as examples of the axiom’s contemporary use.

Halmos, Paul R. Naive Set Theory. Princeton, N.J.: D. Van Nostrand, 1960. An introduction to Cantorian and contemporary intuitive, nonaxiomatized set theory for the general reader.

Kanamori, A. “The Mathematical Development of Set Theory from Cantor to Cohen.” Bulletin of Symbolic Logic 2, no. 1 (1996): 1-71. An account of the history of set theory from its beginnings, with an emphasis on the heritage that current set theory has retained and developed.

Rubin, Herman, and Jean E. Rubin. Equivalents of the Axiom of Choice. Amsterdam: North-Holland, 1985. Although technical, this book includes accessible discussions of most of the subsequent theorems considered equivalent to the Levi-Zermelo axiom of choice.

Suppes, Patrick. Axiomatic Set Theory. Princeton, N.J.: D. Van Nostrand, 1960. Presents the basic principles of more rigorous set theory in terms of the Zermelo-Fraenkel and Newmann-Bernays-Gödel axioms.

Van Heijenoort, Jean, ed. From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Cambridge, Mass.: Harvard University Press, 1967. Includes many English translations of works by Peano, Russell, Zermelo, Hilbert, and others relevant to set theory. A good introduction.