# Brouwer, Luitzen Egbertus Jan (1881–1966)

# BROUWER, LUITZEN EGBERTUS JAN

*(1881–1966)*

Luitzen Egbertus Jan Brouwer, the founder of mathematical intuitionism, was born in 1881 in Overschie, near Rotterdam, the Netherlands. After attending schools in Medemblik, Hoorn, and Haarlem, he studied mathematics at the Municipal University of Amsterdam. He obtained his doctorate in 1907 for his thesis, *Over de Grondslagen der Wiskunde* (Amsterdam and Leipzig, 1907). He became *privaat-docent* at Amsterdam in 1909 and served as professor there from 1912 until his retirement in 1955. In the year that he became a professor he was elected to the Royal Dutch Academy of Sciences.

Besides contributions to the foundations of mathematics, Brouwer made major contributions to other areas of mathematics, in particular to topology, in which his most important publications date from the period 1909–1913. Combinatorial or algebraic topology came into being through discoveries of Henri Poincaré in the 1890s. A fundamental technique of Poincaré was to analyze figures into combinations of simple figures and to represent the topological structure of the figures by algebraic properties of the combination. Brouwer extended and deepened this technique, particularly in relation to questions of the existence of mappings and fixed points. He proved such classic results as the topological invariance of dimension, which implies that there is no bicontinuous one-to-one mapping of Euclidean *m* -dimensional space onto Euclidean *n* -dimensional space, for *m* ≠ *n.*

Although he was primarily a mathematician, Brouwer was always preoccupied with general philosophy and had elaborated a highly individual philosophical vision. Indeed, the most remarkable feature of Brouwer's work in the foundations of mathematics was the boldness and consistency with which, starting from his own philosophical position, he questioned the principles on which the mathematics he inherited was based, down to so elementary a principle as the law of excluded middle, and then proceeded to criticize these principles in detail and to begin to reconstruct mathematics on a basis he regarded as sound.

Although he later presented them more systematically, the essentials of Brouwer's philosophy were already present in his thesis of 1907 and, in certain respects, in *Leven, Kunst, en Mystiek* (Delft, 1905). These works antedate the decisive steps in the development of mathematical intuitionism. In effect, Brouwer argued in his thesis that logic is derivative from mathematics and dependent for its evidence on an essentially mathematical intuition that rests on a basis close to Immanuel Kant's notion of time as the "form of inner sense." Intellectual life begins with "temporal perception," in which the self separates experiences from each other and distinguishes itself from them. Brouwer describes this temporal perception as "the falling apart of a life moment into two qualitatively different things, of which the one withdraws before the other and nonetheless is held onto by memory" ("Weten, Willen, Spreken," 1933). This perception, however, belongs to an attitude (which Brouwer earlier termed "mathematical consideration") that the self adopts to preserve itself; the adoption of this attitude is an act of free will, in a broad sense that Brouwer probably derived from Arthur Schopenhauer. The fundamental intuition of mathematics is this structure of temporal perception "divested of all content"; in mathematics one sees that the process of division and synthesis can be iterated indefinitely, giving rise to the series of natural numbers. In the temporal order thus revealed, one can always imagine new elements inserted between the given ones, so that Brouwer could say that the theories of the natural numbers and of the continuum come from one intuition, an idea that, from his point of view, was made fuller and more precise by his theory of free choice sequences, although one might argue that it was made superfluous by that theory.

Brouwer's constructivism was developed in this context. His constructivism was probably motivated less by an insistence on absolute evidence and a rejection of hypotheses (which might have led to "finitism" in David Hilbert's sense of the term or even to a still narrower thesis) than by Brouwer's subjectivism and his insistence on the primacy of will over intellect. On these grounds, mathematics should consist in a constructive mental activity, and a mathematical statement should be an indication or report of such activity. Brouwer credited this way of looking at mathematics to the inspiration of his teacher, Gerrit Mannoury.

In his thesis Brouwer limited himself to criticizing alternative theories of the foundations mathematics and to criticizing Cantorian set theory, but in "De Onbetrouwbaarheid der Logische Principes" (1908), perhaps urged on by Mannoury, Brouwer raised doubts about the validity of the law of excluded middle, although he still regarded the question as open. In *Intuitionisme en Formalisme* (1912) Brouwer did not say flatly that the law of excluded middle is false, but he gave an instance of his standard argument, an example like that presented in the section on intuitionism in the entry titled "Mathematics, Foundations of," which also gives a fuller exposition of constructivism.

In a number of publications beginning in 1918 and extending through the 1920s, Brouwer developed intuitionist mathematics and worked out in detail his critique of classical mathematics, determining for different branches of mathematics which of their theorems are intuitionistically true. In "Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten," Brouwer undertook to develop an intuitionist set theory, on which a theory of the continuum could be based. In this work he introduced his concept of set (*Menge* ; later, in "Points and Spaces," 1954, he called it "spread") and therefore the idea of an arbitrary infinite sequence as generated by successive free choices. He also introduced the notion of species, which led to his own version of a predicative hierarchy of classes. The principle that the value of a function everywhere defined on a spread must, for a given sequence as argument, be determined by a sufficiently large finite number of its terms is already present in "Begründung der Mengenlehre." This "continuity axiom" is the first of the two distinctive principles of intuitionist analysis.

In "Beweis, dass jede volle Funktion gleichmässig stetig ist" (1924), Brouwer announced a proof that a function everywhere defined on the closed unit interval is uniformly continuous. In this proof Brouwer used two fundamental assertions about spreads, later called the bar and fan theorems. The bar theorem, or an equivalent assertion, constitutes the other distinctive principle of intuitionist analysis. Brouwer's proof was presented in full in "Über Definitionsbereiche von Funktionen" (1927) and reworked, in a more general setting, in "Points and Spaces."

After World War II Brouwer published a long series of short papers in which he developed a new type of counterexample to classical theorems, based on another new principle.

Brouwer's philosophy is not limited to what is relevant to the foundations of mathematics. Mathematical consideration has a second phase, which he called causal attention. In this phase "one identifies in imagination certain series of phenomena with one another," an operation by which one can pick out objects and postulate causal rules. (The relation between temporal perception and causal attention is analogous to that between Kant's mathematical and dynamical categories.) The whole point of mathematical consideration lies in the fact that it makes possible the use of means: One produces a phenomenon that will be followed in a certain repeatable series by a desired phenomenon that cannot be directly reproduced. This makes the pursuit of instinctual satisfaction more efficient.

Especially in *Leven, Kunst, en Mystiek* and in "Consciousness, Philosophy, and Mathematics" (1948), Brouwer regards this "mathematical action" as a kind of fall from grace, whose results are uncertain and ultimately disappointing. With this view he couples a pessimism about society. Society is based on communication, which is itself a form of mathematical action. What is ordinarily called communicating one's thoughts actually amounts to influencing the actions of another, although sometimes a deeper communication of souls is approached. Brouwer, however, was not always aloof from all efforts at social reform, as is shown by his participation, immediately after World War I, with the poet Frederik van Eeden, Mannoury, and others, in the Signific Circle, whose original goal, inspired by the abuses of propaganda during the war, was a far-reaching reform of language.

** See also ** Hilbert, David; Intuitionism and Intuitionistic Logic; Kant, Immanuel; Mathematics, Foundations of; Number; Poincaré, Jules Henri; Schopenhauer, Arthur.

## Bibliography

### additional works by brouwer

"De Onbetrouwbaarheid der Logische Principes." *Tijdschrift Voor Wijsbegeerte* 2 (1908): 152–158. Reprinted in *Wiskunde, Waarheid, Werkelijkheid.*

"Die Theorie der endlichen kontinuerlichen Gruppen." *Mathematische Annalen* 69 (1910): 181–202.

"Beweis der Invarianz der Dimensionenzahl." *Mathematische Annalen* 70 (1911): 161–165.

"Über Abbildung von Mannigfaltigkeiten." *Mathematische Annalen* 71 (1912): 97–115.

*Intuitionisme en Formalisme.* Amsterdam, 1912. Reprinted in *Wiskunde, Waarheid, Werkelijkheid.* Translated by A. Dresden as "Intuitionism and Formalism." *Bulletin of the American Mathematical Society* 20 (1913): 81–96. Reprinted, with other writings by Brouwer, in *Philosophy of Mathematics: Selected Readings*, edited by Paul Benacerraf and Hilary Putnam. Englewood Cliffs, NJ: Prentice-Hall, 1964.

"Über den naturlichen Dimensionsbegriff." *Journal für die reine und angewandte Mathematik* 142 (1913): 146–152.

"Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten." *Verhandelingen der Koninklijke Nederlandse Akademie van Wetenschappen*, Series A, 12 (1918–1919), Nos. 5 and 7.

*Wiskunde, Waarheid, Werkelijkheid.* Groningen: P. N. Noordhoff, 1919.

"Besetzt jede reele Zahl eine Dezimalbruchentwicklung?" *Mathematische Annalen* 83 (1921): 201–210.

"Begründung der Funktionenlehre unabhängig vom logischen Satz ausgeschlossenen Dritten." *Verhandelingen der Koninklijke Nederlandse Akademie van Wetenschappen*, Series A, 13 (1923), No. 2.

"Über die Bedeutung des Satzes vom ausgeschlossenen Dritten in der Mathematik, insbesondere in der Funktionentheorie." *Journal für die reine und angewandte Mathematik* 154 (1924): 1–7. Translated in *From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931*, edited by John van Heijenoort. Cambridge, MA: Harvard University Press, 1965.

"Beweis, dass jede voile Funktion gleichmässig stetig ist." *Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings.* Series A, 27 (1924): 189–194 (cf. the remarks in ibid., 644–646).

"Zur Begründung der intuitionistischen Mathematik." *Mathematische Annalen* 93 (1924): 244–257; 95 (1926): 453–473; 96 (1927): 451–489.

"Über definitionsbereiche von Funktionen." *Mathematische Annalen* 97 (1927): 60–75. Translated in van Heijenoort.

"Weten, Willen, Spreken." *Euclides* 9 (1933): 177–193.

"Synopsis of the Signific Movement in the Netherlands." *Synthese* 5 (1946): 201–208.

"Address to Prof. G. Mannoury." *Synthese* 6 (1947): 190–194.

"Consciousness, Philosophy, and Mathematics," in *Proceedings of the Tenth International Congress of Philosophy.* Amsterdam, 1949. Vol. II, pp. 1235–1249. Last section of this paper reprinted in Benacerraf and Putnam.

"Historical Background, Principles, and Methods of Intuitionism." *South African Journal of Science* 49 (1952): 139–146.

"Points and Spaces." *Canadian Journal of Mathematics* 6 (1954): 1–17.

### works on brouwer and intuitionism

Heyting, Arend. *Les Fondements des mathématiques. Intuitionnisme. Théorie de la démonstration.* Paris: Gauthier-Villars, 1955. Expanded version of German original, *Mathematische Grundlagenforschung. Intuitionismus. Beweistheorie.* Berlin: J. Springer, 1934.

Heyting, Arend. *Intuitionism: An Introduction.* Amsterdam: North-Holland, 1956. This work by Heyting, and the one above, contain the most comprehensive bibliographies on intuitionism.

*Charles Parsons (1967)*

# Brouwer, Luitzen Egbertus Jan

# Brouwer, Luitzen Egbertus Jan

(*b*. Overschie, Netherlands, 27 February 1881; *d*. Blaricum, Netherlands, 2 December 1966)

*mathematics*.

Brouwer first showed his unusual intellectual abilities by finishing high school in the North Holland town of Hoorn at the age of fourteen. In the next two years he mastered the Greek and Latin required for admission to the university, and passed the entrance examination at the municipal Gymnasium in Haarlem, where the family had moved in the meantime. In the same year, 1897, he entered the University of Amsterdam, where he studied mathematics until 1904. He quickly mastered the current mathematics, and, to the admiration of his professor, D. J. Korteweg, he obtained some results on continuous motions in four-dimensional space that were published in the reports of the Royal Academy of Science in Amsterdam in 1904. Through his own reading, as well as through the stimulating lectures of Gerrit Mannoury, he became acquainted with topology and the foundations of mathematics. His great interest in philosophy, especially in mysticism, led him to develop a personal view of human activity and society that he expounded in *Leven, Kunst, en Mystiek* (“Life, Art, and Mysticism”; 1905), where he considers as one of the important moving principles in human activity the transition from goal to means, which after some repetitions may result in activities opposed to the original goal.

Brouwer reacted vigorously to the debate between Russell and Poincaré on the logical foundations of mathematics. These reactions were expressed in his doctoral thesis, *Over de Grondslagen der Wiskunde* (“On the Foundations of Mathematics”; 1907). In general he sided with Poincaré in his opposition to Russell’s and Hilbert’s ideas about the foundations of mathematics. He strongly disagreed with Poincaré, however, in his opinion on mathematical existence. To Brouwer, mathematical existence did not mean freedom from contradiction, as Poincaré maintained, but intuitive constructibility.

Brouwer conceived of mathematics as a free activity of the mind constructing mathematical objects, starting from self-evident primitive notions (primordial intuition). Formal logic had its *raison d’être* as a means of describing regularities in the systems thus constructed. It had no value whatsoever for the foundation of mathematics, and the postulation of absolute validity of logical principles was questionable. This held in particular for the principle of the excluded third, briefly expressed by *A* ∨ ⌝ *A*—that is, *A* or not *A*—which he identified with Hilbert’s statement of the solvability of every mathematical problem. The axiomatic foundation of mathematics, whether or not supplemented by a consistency proof as envisaged by Hilbert, was mercilessly rejected; and he argued that Hilbert would not be able to prove the consistency of arithmetic while keeping to his finitary program. But even if Hilbert succeeded, Brouwer continued, this would not ensure the existence (in Brouwer’s sense) of a mathematical system described by the axioms.

In 1908 Brouwer returned to the question in *Over de Onbetrouwbaarheid der logische Principes* (“On the Untrustworthiness of the Logical Principles”) and—probably under the influence of Mannoury’s review of his thesis—rejected the principle of the excluded third, even for his constructive conception of mathematics (afterward called intuitionistic mathematics).

Brouwer’s mathematical activity was influenced by Hilbert’s address on mathematical problems at the Second International Congress of Mathematicians in Paris (1900) and by Schoenflies’ report on the development of set theory. From 1907 to 1912 Brouwer engaged in a great deal of research, much of it yielding fundamental results. In 1907 he attacked Hilbert’s formidable fifth problem, to treat the theory of continuous groups independently of assumptions on differentiability, but with fragmentary results. Definitive results for compact groups were obtained much later by John von Neumann in 1934 and for locally compact groups in 1952 by A. M. Gleason and D. Montgomery and L. Zippin.

In connection with this problem—a natural consequence of Klein’s Erlanger program—Brouwer discovered the plane translation theorem, which gives a homotopic characterization of the topological mappings of the Cartesian plane, and his first fixed point theorem, which states that any orientation preserving one-to-one continuous (topological) mapping of the two-dimensional sphere into itself leaves invariant at least one point (fixed point). He generalized this theorem to spheres of higher dimension. In particular, the theorem that any continuous mapping of the *n*-dimensional ball into itself has a fixed point, generalized by J. Schauder in 1930 to continuous operators on Banach spaces, has proved to be of great importance in numerical mathematics.

The existence of one-to-one correspondences between numerical spaces *R _{n}* for different

*n*, shown by Cantor, together with Peano’s subsequent example (1890) of a continuous mapping of the unit segment onto the square, had induced mathematicians to conjecture that topological mappings of numerical spaces

*R*would preserve the number

_{n}*n*(dimension). In 1910 Brouwer proved this conjecture for arbitrary

*n*.

His method of simplicial approximation of continuous mappings (that is, approximation by piecewise linear mappings) and the notion of degree of a mapping, a number depending on the equivalence class of continuous deformations of a topological mapping (homotopy class), proved to be powerful enough to solve the most important invariance problems, such as that of the notion of *n*-dimensional domain (solved by Brouwer) and that of the invariance of Betti numbers (solved by J. W. Alexander).

Finally, mention may be made of his discovery of indecomposable continua in the plane (1910) as common boundary of denumerably many, simply connected domains; of his proof of the generalization to *n*–dimensional space of the Jordan curve theorem (1912); and of his definition of dimension of topological spaces (1913).

In 1912 Brouwer was appointed a professor of mathematics at the University of Amsterdam, and in the same year he was elected a member of the Royal Netherlands Academy of Science. His inaugural address was not on topology, as one might have expected, but on intuitionism and formalism.

He again took up the question of the foundations of mathematics. There was no progress, however, in the reconstruction of mathematics according to intuitionistic principles, the stumbling block apparently being a satisfactory notion of the constructive continuum. The first appearance of such a notion was in his review (1914) of the Schoenflies-Hahn report on the development of set theory. In the following years he scrutinized the problem of a constructive foundation of set theory and came fully to realize the role of the principle of the excluded third. In 1918 he published a set theory independent of this logical principle; it was followed in 1919 by a constructive theory of measure and in 1923 by a theory of functions. The difficulty involved in a constructive theory of sets is that in contrast with axiomatic set theory, the notion of set cannot be taken as primitive, but must be explained. In Brouwer’s theory this is accomplished by the introduction of the notion of freechoice sequence, that is, an infinitely proceeding sequence of choices from a set of objects (e.g., natural numbers) for which the set of all possible choices is specified by a law. Moreover, after every choice, restrictions may be added for future possible choices. The specifying law is called a spread, and the everunfinished free-choice sequences it allows are called its elements. The spread is called finitary if it allows only choices from a finite number of possibilities. In particular, the intuitionistic continuum can be looked upon as given by a finitary spread. By interpreting the statement “All elements of a spread have property *p*” to mean “I have a construction that enables me to decide, after a finite number of choices of the choice sequence α, that it has property *p*,” and by reflection on the nature of such a construction, Brouwer derived his so-called fundamental theorem on finitary spreads (the fan theorem). This theorem asserts that if an integer-valued function, *f*, has been defined on a finitary spread, *S*, then a natural number, *n*, can be computed such that, for any two free-choice sequences, *α* and *β*, of *S* that coincide in their first *n* choices, we have *f*(α) = *f*(*β*).

This theorem, whose proof is still not quite accepted, enabled Brouwer to derive results that diverge strongly from what is known from ordinary mathematics, e.g., the indecomposability of the intuitionistic continuum and the uniform continuity of real functions defined on it.

From 1923 on, Brouwer repeatedly elucidated the role of the principle of the excluded third in mathematics and tried to convince mathematicians that it must be rejected as a valid means of proof. In this connection, that the principle is noncontradictory, that is, that ⌝⌝ (A ∨⌝A) holds, is a serious disadvantage. Using the fan theorem, however, he succeeded in showing that what he called the general principle of the excluded third is contradictory, that is, there are properties for which it is contradictory that for all elements of a finitary spread, the property either holds or does not hold—briefly, ⌝(∀ α) (P (α ∨ ⌝ *P* (α)) holds.

In the late 1920’s the attention of logicians was drawn to Brouwer’s logic, and its relation to classical logic was investigated. The breakdown of Hilbert’s foundational program through the decisive work of Kurt Gödel and the rise of the theory of recursive functions has ultimately led to a revival of the study of intuitionistic foundations of mathematics, mainly through the pioneering work of S. C. Kleene after World War II. It centers on a formal description of intuitionistic analysis, a major problem in today’s foundational research.

Although Brouwer did not succeed in converting mathematicians, his work received international recognition. He held honorary degrees from various universities, including Oslo (1929) and Cambridge (1955). He was elected to membership in many scientific societies, such as the German Academy of Science, Berlin (1919); the American Philosophical Society, Philadelphia (1943); and the Royal Society of London (1948).

## BIBLIOGRAPHY

I. Original Works. “Over een splitsing van de continue beweging om een vast punt 0 van R_{4} in twee continue bewegingen om 0 van R_{3}’s,” in *Verslagen. Koninklijke akademie van wetenschappen te Amsterdam*, **12** (1904), 819–839; *Leven, Kunst en Mystiek* (Delft, 1905); *Over de Grondslagen der Wiskunde* (Amsterdam, 1907); “Over de onbetrouwbaarheid der logische principes,” in *Tijdschrift voor wijsbegeerte*, **2** (1908), 152–158; “Die Theorie der endlichen Kontinuierlichen Gruppen, unabhängig von den Axiomen von Lie (erste Mitteilung),” in *Mathematische Annalen*, **67** (1909), 246–267, and “… (zweite Mitteilung),” *ibid*., **69** (1910), 181–203; “Zur Analysis Situs,” *ibid*., **68** (1910), 422–434; “Beweis des Jordanschen Kurvensatzes,” *ibid*., **69** (1910), 169–175; “Beweis des Jordanschen Satzes für den n-dimensionalen Raum,” *ibid*., **71** (1912), 314–319; “Über eineindeutige, stetige Transformationen von Flächen in sich,” *ibid*., **69** (1910), 176–180; “Beweis der Invarianz der Dimensionenzahl,” *ibid*., **70** (1911), 161–165; “Über Abbildung von Mannigfaltigkeiten,” *ibid*., **71** (1912), 97–115, 598; “Beweis der Invarianz des n-dimensionalen Gebietes,” *ibid*., **71** (1912), 305–313; “Zur Invarianz des n-dimensionalen Gebiets,” *ibid*., **72** (1912), 55–56; “Beweis des ebenen Translationssatzes,” *ibid*., **72** (1912), 37–54; “Beweis der Invarianz der geschlossene Kurve,” *ibid*., **72** (1912), 422–425; “Überdennatürlichen Dimensionsbegriff,” in *Journal für die reine und angewandte Mathematik*, **142** (1913), 146–152; “Intuitionism and Formalism,” *in Bulletin of the American Mathematical Society*, **20** (1913), 81–96; review of A. Schoenflies and H. Hahn, *Die Entwicklung der Mengenlehre und ihrer Anwendungen. Erste Hälfte. Allgemeine Theorie der unendlichen Mengen und Theorie der Punktmengen* (Leipzig—Berlin, 1913), in *Jahresbericht der Deutschen Mathematikervereinigung*, **23** (1914), 78–83; “Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten. Erster Teil: Allgemeine Mengenlehre,” in *Verhandelingen Koninklijke akademie van wetenschappen te Amsterdam*, **12** , no. 5 (1918), 1–43, and “… II. Theorie der Punktmengen,” *ibid*., **12** , no. 7 (1919), 1–33; “Begründung der Funktionenlehre unabhägig vom logischen Satz vom ausgeschlossenen Dritten. Erster Teil: Stetigkeit, Messbarkeit, Derivierbarkeit,” *ibid*., **13** , no. 2 (1923), 1–24; “Intuitionistische Einführung des Dimensionsbegriffes,” in *Proceedings. Koninklijke akademie van wetenschappen te Amsterdam*, **29** (1926), 855–863; “Über Definitionsbereiche von Funktionen,” in *Mathematische Annalen*, **97** (1927), 60–75; “Essentially Negative Properties.” in *Proceedings. Koninklijke akademie van wetenschappen te Amsterdam*, **51** (1948), 963–964; “Consciousness, Philosophy and Mathematics,” in *Proceedings of the Tenth International Congress of Philosophy*, I (Amsterdam, 1949), 1235–1249. For Brouwer’s topological work, consult the book by Alexandroff and Hopf listed below. Extensive bibliographies of his foundational work may be found in the books by Heyting and Van Heijenoort (see below). A complete edition of Brouwer’s work is planned by the Dutch Mathematical Society.

II. Secondary Literature. Brouwer or his work is discussed in P. Alexandroff and H. Hopf, *Topologie* (Berlin, 1935), *passim;* P. Benacerraf and H. Putnam, *Philosophy of Mathematics* (Englewood Cliffs, N.J., 1964), pp. 66–84; J. van Heijenoort, *From Frege to Gödel, a Source Book in Mathematical Logic*, 1879–1931 (Cambridge, Mass., 1967), pp. 334–345, 446–463, 490–492; A. Heyting, *Intuitionism, An Introduction* (Amsterdam, 1965), *Passim*; S. Lefschetz, *Introduction to Topology* (Princeton, N.J., 1949), pp. 1–26, 117–131; S. C. Kleene and R. E. Vesley, *The Foundations of Intuitionistic Mathematics* (Amsterdam, 1965); G. Kreisel, “Functions, Ordinals, Species,” in *Logic, Methodology and Philosophy of Science*, III, ed. B. van Rootselaar and J. F. Staal (Amsterdam, 1968), pp. 145–159; J. Myhill, “Formal Systems of Intuitionistic Analysis I,” *ibid*., pp. 161–178; A. S. Troelstra, “The Theory of Choice Sequences,” *ibid*., pp. 201–223.

B. van Rootselaar

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