A Term of Length 4 523 659 424 929

Bourbaki suggest that their definition of the number 1 runs to some tens of thousands of symbols. We show that that is a considerable under-estimate, the true number of symbols being that in the title, not counting 1,179,618,517,981 disambiguatory links. 1: Introduction Bourbaki, the self-perpetuating French group of mathematicians, are ill at ease with logic and foundations. Some signs of that are given in my essay The Ignorance of Bourbaki [M]; Leo Corry has remarked in his book [Co], pp 318-9, that Bourbaki do not in their later volumes use the system they have so carefully set out in their Volume I; indeed, there was, according to copies of La Tribu unearthed by Corry, ([C], page 319 and footnote 63 on page 320) a considerable debate within Bourbaki as to whether the volume on La théorie des Ensembles should be written at all. That volume, indeed, has met criticism. Serre has remarked that few logicians like it. Godement, in the first hundred pages of his text Algèbre, though he follows Bourbaki’s exposition of logic and set theory, tells his readers to eschew formal reasoning. Another French scholar is quoted in Chouchan [Ch], page 124: Jacques Roubaud parle même de l’effroyable premier livre sur la théorie des ensembles: un vrai désastre, auquel le monde a renoncé depuis longtemps. ... On ne se rend pas compte que c’est une présentation souvent fallacieuse. We shall suggest that the difficulties experienced by Bourbaki, in the normally straightforward task of setting up an axiomatic system of set theory as a basis for mathematics, are the consequence of an injudicious choice for their underlying logical formalism. In the present chapter we make this point: 1·0 PROPOSITION Bourbaki’s abbreviated structuralist definition of the number 1, when expanded into the primitive symbolism of the first edition of La Théorie des Ensembles, comprises 4,523,659,424,929 symbols together with 1,179,618,517,981 links between certain of those symbols. That definition is quoted in the next paragraph. In §2 we review Bourbaki’s syntax; §§3–6 give the details of the calculation of the length of that formula, using the formalism of the original edition; in the seventh section we remark that its length is vastly increased by the formalism of the 1970 edition. Some brief comments on the psychological significance of these arithmetical freaks will be found in the final section. Bourbaki’s abbreviated definition of 1 Chapters I and II of Bourbaki’s Théorie des Ensembles were published in 1954, and Chapter III in 1956. Among the primitive signs used was a reverse C, standing presumably for “couple”, to denote the ordered pair of two objects. Being typographically unable to reproduce that symbol, we use instead the symbol •. With that change, the footnote on page 55 of Chapter III reads Bien entendu, il ne faut pas confondre le terme mathématique désigné (chap. I, §1, n 1) par le symbole “1” et le mot “un” du langage ordinaire. Le terme désigneé par “1” est égal, en vertu de la définition donnée ci-dessus, au terme désigné par le symbole τZ((∃u)(∃U)(u = (U, {∅}, Z) et U ⊂ {∅} × Z et (∀x)((x ∈ {∅}) =⇒ (∃y)((x, y) ∈ U)) et (∀x)(∀y)(∀y′)(((x, y) ∈ U et (x, y′) ∈ U) =⇒ (y = y′)) et (∀y)((y ∈ Z) =⇒ (∃x)((x, y) ∈ U)))). Une estimation grossière montre que le terme ainsi désigné est un assemblage de plusieurs dizaines de milliers de signes (chacun de ces signes étant l’un des signes τ , , ∨, ¬, =, ∈, •). Acknowledgments This paper owes its inception to the encouragement I have received from readers of The Ignorance of Bourbaki, and from the stimulating effect of Professor Pedersen’s invitation and hospitality. Discussions with ·vii · . . . . . . . . . . . . . . . . . . . . . . . . . A term of length 4,523,659,424,929 . . . . . . . . . . . . . . . . . . . . . . . . . Page 1 Leo Corry during the Roskilde meeting have been most helpful. I am grateful to Robert Solovay and Tim Carlson for detecting an erroneous formula in an earlier draft of Proposition 3·7, and am further grateful to Solovay for further comments and corrections and for writing a program to give the exact figures for the yet lengthier symbolism discussed in section 7. The first draft of the paper was written when I held an Associate Professorship at the Universidad de los Andes, Santa Fé de Bogotá, Colombia. It was put into final form whilst I was working on a research project at the Humboldt Universität zu Berlin sponsored by the Deutsche Forschungsgemeinschaft. 2: Bourbaki’s syntax Bourbaki use the Hilbert operator but write it as τ rather than ε, which latter is visually too close to the sign ∈ for the membership relation.* Bourbaki use the word assemblage, or, in their English translation, assembly, to mean a finite sequence of signs or letters, the signs being τ , , ∨, ¬, =, ∈ and •. The substitution of the assembly A for each occurrence of the letter x in the assembly B is denoted by (A|x)B. Bourbaki use the word relation to mean what in English-speaking countries is usually called a wellformed formula. 2·0 The rules of formation for τ -terms are these: let R be an assembly and x a letter; then the assembly τx(R) is obtained in three steps: (2·0·0) form τR, of length one more than that of R; (2·0·1) link that first occurrence of τ to all occurrences of x in R (2·0·2) replace all those occurrences of x by an occurrence of . In the result x does not occur. The point of that is that there are no bound variables; as variables become bound (by an occurrence of τ ,) they are replaced by , and those occurrences of are linked to the occurrence of τ that binds them. The intended meaning is that τx(R) is some x of which R is true. Certain assemblies are terms and certain are relations. These two classes are defined by a simultaneous recursion, presented in Godement [G] thus: T1: every letter is a term T2: if A and B are terms, the assembly •AB , in practice written (A, B), is a term. T3: if A and T are terms and x a letter, then (A|x)T is an term. T4: if R is a relation, and x a letter, then τx(R) is an term. R1: If R and S are relations, the assembly ∨RS is a relation; in practice it will be written (R ∨ S). R2: ¬R is a relation if R is. R3: if R is a relation, x a letter, and A a term, then the assembly (A|x)R is a relation. R4: If A and B are terms, =AB is a relation, in practice written A = B. R5: If A and B are terms, the assembly ∈AB is a relation, in practice written A ∈ B. That is all. 2·1 REMARK Clauses T3 and R3 are, as pointed out to me by Solovay, redundant — if omitted, they can be established as theorems — and were added to Bourbaki’s original definition by Godement, presumably for pedagogical reasons. 2·2 REMARK Note that every term begins with a letter, • or τ ; every relation begins with =, ∈, ∨, or ¬. Hence no term is a relation. Quantifiers are introduced as follows: 2·3 DEFINITION (∃x)R is (τx(R) | x)R; 2·4 DEFINITION (∀x)R is ¬(∃x)¬R. * The possible significance of the choice of the letter τ is to be discussed in Chapter II of my Danish Lectures. ·vii · . . . . . . . . . . . . . . . . . . . . . . . . . A term of length 4,523,659,424,929 . . . . . . . . . . . . . . . . . . . . . . . . . Page 2 3: Some calculations 3·0 DEFINITION We write `h(A) for the length of an assembly A, not counting any links that are there; oc(x, A) for the number of occurrences in A of the letter x, and λ(A) for the number of links in A, which will equal the number of occurrences of . 3·1 PROPOSITION If R is of length r, τxR is of length r + 1. Proof : We have added a τ , and replaced each x by a . 3·2 PROPOSITION λ(τx(R)) = oc(x, R) + λ(R). 3·3 PROPOSITION If x has m occurrences in R and y (distinct from x) has k occurrences in R, then in τx(R), x has no occurrences and y has k occurrences. 3·4 PROPOSITION If x has m occurrences in R and y (distinct from x) has k occurrences in R, then in each of the formulæ (∃x)R and (∀x)R, x has no occurrences and y has (m + 1)k occurrences. Proof : There are the original k occurrences of y, and each of the m x’s has been replaced by τx(R), in each of which y has k occurrences. a (3·4) 3·5 PROPOSITION If R is of length r and has m occurrences of x, then the length of (∃x)R is r(m + 1) Proof : Each replacement of x by τx(R) has increased the length by r. a (3·5) 3·6 PROPOSITION If R is of length r and has m occurrences of x, then the length of (∀x)R is (r+1)(m+1)+1. Proof : The formula is ¬(∃x)¬R and ¬R is of length r + 1 and has m occurrences of x. a (3·6) 3·7 PROPOSITION If x has m occurrences in R, and R has ` links, λ((∃x)R) = λ((∀x)R) = m(` + m) + `. Proof : λ(τx(R)) = ` + m, by Proposition 3·2. Each occurrence of x in R is replaced by one of τx(R), and then there are the ` original links in R. a (3·7) 3·8 REMARK A curiosity of this syntax, not needed for our present calculations, is that two trivially equivalent formulæ might have markedly different lengths. Thus if R has 2 occurrences of x, 5 of y and 3 of z, and is of length 50, the formula (∃x)(∃y)R will be of length 3900, with 234 occurrences of z, whereas the formula (∃y)(∃x)R will be of length 2400 with 144 occurrences of z. 4: Parsing that term We begin by repeating Bourbaki’s abbreviated term in open display, with y replaced by z: