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Sets and the Continuum Before 1907

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Sets and the Continuum Before 1907 Chapter 1 Sets and the continuum before 1907 For a fruitful discussion of Brouwer’s early work, a reasonable amount of knowl- edge of the development of set theory, as this took place during the second half of the nineteenth century, is required. We must keep in mind that Brouwer reacted to a new and very current branch of mathematics, which received a lot of attention during the period beginning just a few decades before Brouwer’s active life as a mathematician and philosopher. This first chapter contains background material, needed for a proper under- standing of the discussion about the foundational parts of Brouwer’s disserta- tion. The content will be familiar to many readers, since it consists mainly of basic set theory. For those readers a quick glance will suffice. After 1870 set theory developed rapidly into a separate branch of mathe- matics, despite resistance from mathematicians like Kronecker, but also from philosophers and even from theologians. This new development took place mainly through the work of Georg Cantor, who, for that reason, must be con- sidered as the founder of set theory. Before Cantor, use was made of certain collections possessing certain proper- ties, like the locus in geometry, but collections like that did not seem to have the status of a mathematical object with which and on which certain mathematical operations could be performed. Likewise collections with an infinite number of elements, although their existence had of course to be admitted, seemed to possess strange properties to such an extent (like the possibility of a one-one correspondence between a collection and a part thereof) that they too seemed to lack the status of mathematical object.1 1see [Galileo 1974], page 40, 41 where one of the participants in the discussion concludes, because of the possible one-one relation between natural numbers and their squares, that concepts like ‘equal’, ‘greater’ and ‘smaller’ are not applicable in case of infinite quantities. 1 2 CHAPTER 1. SETS AND THE CONTINUUM BEFORE 1907 To get an insight into the development and in the state of knowledge of set theory at the time of Brouwer’s public defence of his dissertation, the following persons and subjects are of importance, and will be discussed: 1.1 Cantor 1.2 Dedekind 1.3 Poincar´e 1.4 Zermelo 1.5 Schoenflies 1.6 Bernstein 1.7 The paradoxes An important predecessor in the development of set theory was Bolzano (1781–1848), whose main publication on this topic, Paradoxien des Unendlichen, was published posthumously in 1851.2 Bolzano was attracted by the seemingly paradoxical nature of the infinite. Without going into the details of Bolzano’s work, we just mention the fact that he very clearly saw the problems concerning the infinite, that he asked the proper questions, but did not take the steps and did not draw the conclusions, that Cantor did twenty five years later. Because Brouwer made use of Cantor’s results (and disagreed in many re- spects with these results), it is, for a proper understanding of the references that Brouwer made, of importance to give a survey of the content of Cantor’s work in a systematic way. This will be done in the first section of this chapter, despite the fact that it concerns rather basic set theory with which many readers will be familiar. 1.1 Cantor (1845–1918) Cantor started publishing on set theory in 1870, shortly after his Habilitation. From this time onwards the idea that a set can have a one-to-one relation to a proper subset of itself is generally accepted as a typical property of infinite sets (and is certainly no reason to turn away from those sets).3 The word set (German: Menge, French: ensemble) did not appear right from the beginning in the work of Cantor. In his earliest letters to Dedekind during the years 1873 and 18744 and in his paper Uber¨ eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, published in 1874, he employed the mathematically undefined term totality (Inbegriff) e.g. of the natural numbers or of the rational numbers. In the 1877-correspondence to Dedekind and in the 1878-paper Ein Beitrag zur Mannigfaltigkeitslehre, the term Mannigfaltigkeit was used to indicate such totalities, but now also set (Menge) appeared, and in 2[Bolzano 1851]. 3For Dedekind the possibility of a one-to-one mapping of a set onto a proper subset of itself became the definition of the infinity of a set, now known under the term Dedekind infinite. See under section 1.2. 4Cantor communicated his new ideas on sets and cardinalities with Dedekind in a series of letters after 1873; see [Cantor 1937]. 1.1. CANTOR (1845–1918) 3 the eighties of that century this term became the accepted one. We will from now on use only the term set. In the Beitr¨agezur Begr¨undungder transfiniten Mengenlehre5 the following definition of a set is given (§ 1): Unter einer ‘Menge’ verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objecten m unsrer Anschauung oder unseres Denkens (welche die ‘Elemente’ von M genannt werden) zu einem Ganzen. To indicate the magnitude of a set M the term power, cardinality or cardinal number (M¨achtigkeit, Cardinalzahl) is used: two sets have the same power or cardinality if a one-to-one mapping of one set onto the other is possible, under abstraction from the order and the nature of their elements. This cardinality with its double abstraction is indicated for a set M by M. Sets satisfying that mapping condition are then said to be equivalent (gleichm¨achtig). Consulted literature for this section about Cantor consists of the following material: – Briefwechsel Cantor-Dedekind, edited by E. Noether and J. Cavaill`es, Paris, 1937. This volume contains the Cantor-Dedekind correspondence between 1872 and 1882; the important 1899-part of this correspondence is contained in Can- tor’s Gesammelte Abhandlungen. – Cantor, Gesammelte Abhandlungen, edited by E. Zermelo, Berlin, 1932. From this volume in particular the following articles: – Uber¨ eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. (Journal f¨urdie reine und angewandte Mathematik6 1874.) – Ein Beitrag zur Mannigfaltigkeitslehre. (idem, 1878.) – Uber¨ einen Satz aus der Theorie der stetigen Mannigfaltigkeiten. (G¨ottinger Nachrichten, 1879.) – Uber¨ unendliche lineare Punktmannigfaltigkeiten. (Mathematische Annalen, 1879, ’80, ’82, ’83, ’84; M.A. Volumes 15, 17, 20, 21 and 23 respectively.) – Uber¨ verschiedene Theoreme aus der Theorie der Punktmengen in einem n-fach ausgedehnten stetigen Raume Gn. (Acta Mathematica, 1885.) – Uber¨ die verschiedenen Standpunkte in bezug auf das aktuelle Un- endliche. (Zeitschrift f¨ur Philosophie und philosophische Kritik bd. 88, 1890.) 5[Cantor 1897], also in [Cantor 1932], page 282 ff. 6Brouwer usually referred to this journal under the name of its editor Crelle, and we will follow this practice henceforth. 4 CHAPTER 1. SETS AND THE CONTINUUM BEFORE 1907 – Mitteilungen zur Lehre vom Transfiniten. (Zschr. f. Philos. u. philos. Kritik, 1887.) – Uber¨ eine elementare Frage der Mannigfaltigkeitslehre. (Jahres- bericht der Deutschen Mathematiker-Vereinigung, Bd.1, 1890–91.) –Beitr¨agezur Begr¨undung der transfiniten Mengenlehre. (Mathema- tische Annalen 1895, ’97, M.A. Volumes 46 and 49 respectively.) As far as possible, the papers mentioned above will be discussed in a chrono- logical order, which results not only in an overview of the state of knowledge on sets in the year 1907, but additionally in a survey of the origin and history of set theory. We must keep in mind that there was only one complete textbook on set theory available during the years of Brouwer’s preparation for his thesis.7 Brouwer drew most of his information directly from Cantor’s papers. In the first sections frequent use will be made of the Cantor-Dedekind cor- respondence; this in particular gives a good insight in the development of his ideas. It will be noticed that in the beginning Cantor often made use of terms and concepts which were only properly defined in a later stage of the develop- ment. 1.1.1 The discovery of non-equivalent infinite sets In a letter to Dedekind of 29 November 1873,8 Cantor considered the possibility of the construction of a one-to-one relation between the natural and the real numbers. He did not know a definite answer to this problem yet, and asked Dedekind to think it over. Intuitively the answer should be no, but there was justified doubt: Auf den ersten Anblick sagt man sich, nein es ist nicht m¨oglich, denn (n) besteht aus discreten Theilen, (x) aber bildet ein Continuum; (...) W¨are man nicht auch auf den ersten Anblick geneigt zu behaupten, p dass sich (n) nicht eindeutig zuordnen lasse dem Inbegriffe ( q ) aller p positiven rationalen Zahlen q ? But this last statement is easy to prove, just as the theorem of the equivalence between the natural numbers and the algebraic numbers. In a following letter to Dedekind of 7 December 1873, Cantor proved in an ingenious way that the assumed equivalence between the natural numbers and 7The first general textbook on set theory was Schoenflies’ Bericht ¨uber die Mengenlehre, [Schoenflies 1900a], published in the Jahresbericht der Deutschen Mathmatiker-Vereiniging; see under section 1.5. In fact there was also Young and Young’s The theory of sets of points ([Young and Chisholm Young 1906]), but probably Brouwer was not familiar with this book; he never mentioned the name Young, neither in his thesis, nor in his notebooks, although W.H. Young published in German in the Mathematische Annalen in 1905. 8[Cantor 1937], page 12; all letters between Cantor and Dedekind during the years 1872 – 1882 are published in this volume.
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