PART I. GREAT ILLUSION OF TWENTIETH CENTURY MATHEMATICS
2.5.3
Introduction of Mathematical Formalism in Set Theory
If we model individual objects belonging to some community under investigation as abstract ur-objects (that means as objects emptied of their contents), we create a formal structure of the community under investigation. Independent study of such formal structures then relies merely on logical proofs (usually in predicate calculus) of various formal assertions from previously chosen formal axioms. It was basically in this way that David Hilbert (1862–1943) approached Euclidean geometry in his book Grundlagen der Geometrie, that appeared in 1903.20 Then he formally captured the structure of real numbers in similar way.21 In both these cases, Hilbert could not do without infinite sets, in particular Bolzano’s theorem about suprema. This blemish on the beauty of formal mathematics (only artificially repaired with the help of second-order logic) notwithstanding, in his lecture Grundlagen der Logic und der Arithmetik 22 Hilbert emphasised the necessity of a strict mathematical formulation of the language of mathematics and logic. Hilbert’s emphasis on axiomatisation of mathematical theories was just as important. Results in this or that mathematical theory must be obtained by purely logical arguments based on previously chosen axioms, that is, in a purely formal way, without reliance on intuition. He thus initiated the mathematicalphilosophical approach called mathematical formalism. The obligation to axiomatise mathematical theories advocated by Hilbert naturally concerns set theory too. On account of the non-actualisability of the set of all sets, only corpuses of sets can be axiomatised (using predicate calculus). If that can be done, the above-mentioned blemish upon axiomatisation of real numbers could be removed. Ernst Zermelo (1871–1953) undertook this task and he published such an axiomatization in Mathematische Annalen.23 However, he neglected to include an axiom corresponding to the condition (g) required of corpuses of sets. This was remedied by Dimmitrij Mirimanov (1861–1925) in a paper published in 1917.24 The same was also achieved by Adolf Fraenkel (1891–1965) in Mathematische Annalen25 and Thoralf 20 Vopěnka refers to the second, extended edition, David Hilbert, Grundlagen der Geometrie (Leipzig: Teubner, 1903). Grundlagen der Geometrie first appeared in 1899. [Ed] 21 In the Czech original, Vopěnka refers here to David Hilbert, “Die Theorie der algebraischen Zahlkörper,” Jahresbericht der DMV 4 (1894/95): 175–546. The usual reference, used by Vopěnka in Chapter 12, is David Hilbert, “Über den Zahlbegri↵,” Jahresbericht der DMV 8 (1900): 180–194. [Ed] 22 David Hilbert gave this lecture at the third international mathematical congress in 1904 in Heidelberg. 23 Ernst Zermelo, “Untersuchungen über die Grundlagen der Mengenlehre,” Mathematische Annalen 65 (1908): 261–281. 24 Dimitrij Mirimanov, “Les antinomies de Russell et de Burali-Forti et le probléme fondamental de la théorie des ensembles,” L’Enseigment Mathématique 19 (1917): 37–52. 25 Adolf Fraenkel, “Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre,” Mathematische Annalen 5, no. 86 (1922): 230–237.
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CHAPTER 2. RISE AND GROWTH OF CANTOR’S SET THEORY
Skolem (1887–1963) at the congress of Scandinavian mathematicians.26 Axiomatisation of not only corpusses of sets but also of their subsets makes it possible to eliminate axiom schemata from formalisation of set theory This was the path taken by John von Neumann (1903–1957) who created and developed such an axiomatisation in the paper “Eine Axiomatisierung der Mengenlehre”.27 A more natural form was given to it by Paul Bernays (1888–1977) in a series of articles that started to appear in 1937 in Journal of Symbolic Logic under a shared name “A system of axiomatic set theory”28 and by Kurt Gödel (1906–1978) in a slender volume from 1940.29 This axiomatisation came to be called Gödel-Bernays axiomatisation. If set theory is to guarantee the consistency of mathematical theories that are modelled in it, then obviously this theory itself needs to be consistent. That means at least some axiomatisations of it need to be consistent. The question of consistency of various axiomatisations of set theory has been forsaken by mathematicians. After all, they could not do anything else. They had to content themselves with Cantor’s insight into the world of actually infinite sets not being a deceptive and collapsing mirage. Even though mathematical formalism failed to reach the Hilbert’s goal of guaranteeing the consistency of axioms of set theory, or at least of axioms for natural numbers, its entrance into set theory was an extraordinary event nevertheless. To wit, it was the second grand encounter and blending of two most important currents in mathematics: that of intuition and that of calculations. In the mathematics of intuition the task is to make results evident. To evidence something means to see it (it the widest sense of this word) and to know that we are seeing it. This current of mathematics was born in the geometry of Greek antiquity, and in the twentieth century it worked upon Cantor’s set theory although the agent who evidenced the results was not Zeus but the God of medieval and scholastic theology. In the mathematics of calculations the task is to gain the results by applying correct calculations with signs, that means calculations carried out according to various previously established fixed rules. This current emerged from ancient India in creating the arithmetical and algebraic calculus. Their first grand encounter occurred when René Descartes (1596–1650) introduced algebra into geometry. Their second encounter and blending occured when David Hilbert brought mathematical formalism into set theory. The entrance of mathematical formalism into set theory had similar impact on mathematics as in its own time the entrance of algebra into geometry. A 26 Thoralf Skolem, lecture “Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre,” published in Wissenschaftliche Vorträge auf dem Fünften Kongress der Skandinavischen Mathematiker in Helsingfors (1922). 27 John von Neumann, “Eine Axiomatisierung der Mengenlehre,” Journal für die reine und angewandte Mathematik 154 (1925): 219–240. 28 Paul Bernays, “A system of axiomatic set theory – Part I–III,” Journal of Symbolic Logic, Part I (1937): 2 (1): 65–77; Part II (1941): 6 (1): 1–17; Part III (1942): 7 (2): 49–104. 29 Kurt Gödel, Consistency of the Axion of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Princeton University Press, 1940).
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PART I. GREAT ILLUSION OF TWENTIETH CENTURY MATHEMATICS
whole new and extraordinarily stimulating body of problems arose in set theory. Up to that point, mathematicians looked for assertions that were true in set theory, but now their interest transferred to assertions such that could not be proved from the axioms of set theory. Some such assertions, or possibly also their negations, could be added as further axioms of set theory. They were usually identified by creating models (interpretations) within axiomatic set theory of theories in which they did not hold. The question whether these assertions are true in set theory does not concern the mathematics of calculations. The answer should be provided by the mathematics of intuition. All that has been achieved using the axiomatic approach in set theory could fill many volumes. Let us just recall, for the interest it holds, that it is possible to define a certain subset of the set of all natural numbers such that its existence is provable from the axiom of existence of an inaccessible cardinal number. However, it is not provable from the axioms of set theory without this additional axiom! In other words, the existence of this set of natural numbers excludes the existence of inaccessible cardinal numbers. In his work Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory, Kurt Gödel defined constructible sets and proved that they form a model of axiomatic set theory in which the axiom of choice and continuum hypothesis hold.30 Moreover, in this model the ordinal numbers are absolute.
30 Cantor’s continuum hypothesis is the assertion according to which the set of all subsets of natural numbers has the same cardinality as the set of all ordinal numbers smaller that the first uncountable initial ordinal number.
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