by

Herman R. Jervell, University of Oslo, Norway

The theory of choice sequences is usually considered to be far off from the main stream of mathematics. In this note we'll show that it did not start that way. There is a continuous development from discussions around the use of axiom of choice to Brouwers introduction of choice sequences. We have tried to trace this development starting in 1904 and stopping in 1914.

 Troelstra gives in his book on choice sequences the development after 1914, but he gives no indication of where Brouwer got his concept. This note gives an answer, but we must stress that it is only a first attempt.

Our story starts in August 1904. Zermelo writes a long letter to Hilbert. Hilbert thinks part of the letter deserves a wider audience. So he publishes it directly in Mathematische Annalen. The leisurely style is clear from the title and the first sentence:

"PROOF THAT EVERY SET CAN BE WELL-ORDERED (from a letter sent to Mr. Hilbert)

 ......the following proof comes from conversations that I had last week with Mr. Erhard Schmidt and it is as follows."

Zermelo gave the standard argument that the axiom of choice implies the wellordering principle. He argued that the axiom of choice was self evident. The reactions to the proof came immediately. Borel sent a small note to Mathematische Annalen in December 1904.(1251 - 1252)

He ended his note as follows:

 "It seems to me that the objections against it is also valid for every reasoning where one assumes an arbitrary choice made an uncountable number of times, for such reasoning does not belong to mathematics."

 Strong words. Similar objections came from other people too. Some defended Zermelos argument. This debate is summed up by Zermelo in Mathematische Annalen 1908 in "Neuer Beweis für die Wohlordnung." For our story we note that Zermelos paper started a heated foundational debate. The most prominent mathematicians at the time were involved in it.

 Our story continues in France, Borel had written that Zermelos axiom of choice did not belong to mathematics. (See quotation above.) As a statesman of mathematics he gathered the opinions of the most prominent French mathematicians of his generation - Hadamard, Baire, and Lebesgue. The result was published in Bulletin de la Soclete mathematlque de France in 1905 as "Cinq lettres sur la theorle des ensembles". ( 1253 - 1265 ). Hadamard supported Zermelo while Baire and Lebesgue were on Borels side.

For the mathematical world these discussions were important. There were foundational problems close to mathematical practice.

The two views were named formalism and lntuitionlsm. When Brouwer started to talk about intuitionism he showed his acceptance of the philosophy of Borel, Lebesgue, and Baire. Later he would call them pre-intuitionists.

 The debate continued in the next decade. In 1912 Borel gave in "La philosophie mathematique et l'infini" ( 2127 - 2136 ) the following version of the debate:

 "It is possible to define a bounded decimal number by demanding that a thousand persons each write an arbitrary digit. One will have a well-defined number if the persons are put in a row each writing in his turn a digit at the end of the digits already written by the persons in front of him in the row. The disagreement starts when we try to extend this procedure to an unbounded decimal number. I do not suppose that people dream of actually having an infinite number of persons each writing an arbitrary digit, but I believe that Mr. Zermelo and Mr. Hadamard think that it is possible to regard such a choice realized in a perfectly well-defined way even if the complete definition of the number will contain an infinite number of words. For my part I think it is possible to pose problems about probability for decimal numbers which are obtained by choosing the digits either randomly or by imposing certain restrictions on the choice - restrictions leaving some randomness to the choice. But I think it is impossible to talk about one of these numbers for the reason that if one denotes it by a,two mathematicians talking about a would never be sure whether they were talking about the same number." ( 2129 - 2130 )

Here we have a link between the axiom of choice and the theory of choice sequences. Borel uses a number defined by a choice sequence to show the difference between the formalists and the intuitionists. Note that he does not any longer only consider uncountable axiom of choice. That the problems arise for all infinite uses of axiom of choice was pointed out by Lebesgue in the correspondence from 1905.

 Brouwer enters now the scene from the side line. From the start of his mathematical career in 1907 he was interested in the formalist/intuitionist controversy. For him the foundational puzzles were in this controversy and not in the problems of the logicians. In his thesis from 1907 Brouwer comments on Zermelos proof and agrees with Borels critique. ( 84 ). Like Borel Brouwer thinks that the problems only comes with uncountable sets.

 Brouwers inaugural address from 1912 "Intuitionism and formalism" is the first place where he mentions free choices. The relevant passages ( 133 - 135 ) are similar to the quotation from Borel above:

 "Let us consider the concept: "real number between O and 1." For the formalist this concept is equivalent to "elementary series of digits after the decimal point, for the intuitionist it means "law for the construction of an elementary series of digits after the decimal point, built up by means of a finite number of operations." And when the formalist creates the "set of all real numbers between O and 1," these words are without meanlng for the intuitionist, even whether one thinks of the real numbers of the formalist, determined by elementary series of freely selected digits, or of the real numbers of the intuitionist, determined by finite laws of construction."

"If we restate the question in this form: "Is it impossible to construct infinite sets of real numbers between O and 1, whose power is less than that of the continuum, but greater than aleph-null?" then the answer must be in the affirmative for the intuitionist can only construct denumerable sets of mathematical objects and if, on the basis of the intuition of the linear continuum, he admits elementary series of free selection as elements of construction, then each non-denumerable set constructed by means of it contains a subset of the power of the continuum."

 In his book Troelstra has given these quotations as Brouwers first mention of choicesequences. The first quotation shows that Brouwer would only admit lawlike definition of a real number. Troelstra interpretes the second quotation as showing that the intuitionists at least conceivably might use choice sequences. Another possibility is that Brouwer thought of probability statements like Borel.

 In 1914 Brouwer had changed his views on choice sequences as pointed out by Troelstra in his book. In a review on Schoenflies and Hahns book on set theory ( 139 - 144 ) Brouwer remarks in a footnote ( 140 ):

 "Z. B. ist die Punktmenge: "alle reellen Zahlen zwischen O und 1 mit Ausnahme der endlichen Dualbruche", nur deshalb eine Wohlkonstruierte Menge, weil die duale Entwicklung einer willkurlichen Zahl dieser Menge eine Fundamentalreihe von endliche Gruppen von gleichen Ziffern (abwechselnd O und l) liefert, so dass die Menge sich mittels einer Fundamentalreihe von Auswahlen unter den endlichen Zahlen bestimmen lasst. Dieser Schritt geht freilich weiter als mein romischer Vortrag ( 102 - 104 ), und auch weiter als die Borelschen Ausfuhrungen uber wohlkonstruierte Mengen ( 827 - 878 ); er erscheint mir aber als eine notwendige Konsequenz des Intuitionismus."

Brouwers lecture in Rome is from the International Congress of Mathematicians in 1908. Both Zermelo and Borel were there. Borels paper is from 1912. In our long quotation from Borel above, this paper is mentioned in a footnote as a place where the more technical aspects of Borels theory is worked out.

 We have now almost completed our story. Starting with the axiom of choice which Zermelo introduced in 1904 we have ended up with. Brouwers choice sequences in 1914. A note of warning - Troelstra has pointed out to us that Brouwers use of intuitionistic logic changes the concept of choice sequence essentially. We have avoided the complications this gives the story of choice sequences by just ignoring it. For the full story the use of the logic must also be traced.

Our main references are:

Oeuvres de Emile Borel. Editions du Centre National de la Scientifique. Paris 1972.

 L. E. J. Brouwer. Collected works. North-Holland Publishing Company. Amsterdam 1975.

 In the brackets after the quotatlons above we have referred to the pagination of these works. Zermelos artlcles are in Mathematische Annalen from 1904 and 1908. There is a short history of choice sequences in:

 A. S. Troelstra: Cholce sequences. Clarendo Press. Oxford 1977.

The translations are my own.

 (Written 1979)