The Fine Structure of the intuitionistic Borel Hierarchy

In intuitionistic analysis, a subset of a Polish space like R or N is called positively Borel if and only if it is an open subset of the space or a closed subset of the space or the result of forming either the countable union or the countable intersection of an infinite sequence of (earlier constructed) positively Borel subsets of the space. The operation of taking the complement is absent from this inductive definition, and, in fact, the complement of a positively Borel set is not always positively Borel itself (see Veldman, 2008a). The main result of Veldman (2008a) is that, assuming Brouwer’s Continuity Principle and an Axiom o f Countable Choice, one may prove that the hierarchy formed by the positively Borel sets is genuinely growing: every level of the hierarchy contains sets that do not occur at any lower level. The purpose of the present paper is a different one: we want to explore the truly remarkable fine structure of the hierarchy. Brouwer’s Continuity Principle again is our main tool. A second axiom proposed by Brouwer, his Thesis on Bars is also used, but only incidentally. §1. In tro d u c tio n . 1.1. This is the second one in a series o f papers on intuitionistic descriptive set theory. Our aim is to find out what becomes of the field of study opened up by E. Borel, H. Lebesgue, R. Baire, N. Lusin, A. Souslin, see Lusin (1930), M oschovakis (1980), and Kechris (1996), and others, if one tackles it from Brouw er’s intuitionistic point of view. As is explained in the introduction to Veldman (2008a), Brouwer was m ore radical than the French and Russian mathem aticians who started classical descriptive set theory. They also had their doubts about some of C antor’s and Zerm elo’s assumptions, like Brouwer, but they never questioned classical logic. Brouwer however, in his search for a sensible treatment o f the continuum, cam e to advocate a consistent constructive interpretation of the logical constants and the set-theoretical operations and he decided to reject the principle o f the excluded m iddle as a valid principle o f reasoning. In a m athem atical context, where one considers infinite objects like subsets of the set o f the natural numbers or infinite sequences of natural numbers, the principle o f the excluded m iddle leads, as Brouwer explained, to absurd conclusions, that is, to statements that fail to be true when they are understood straightforwardly and constructively, and do not im m ediately m ake sense in a different way. B rouw er’s criticism of the logic of m athem atical arguments went hand in hand with his suggestion to use some new axioms, in particular, his Continuity Principle and his Received: March 25, 2008 3 0 © 2009 Association for Symbolic Logic doi:10.1017/S1755020309090121 THE INTUITIONISTIC BOREL HIERARCHY 31 Thesis on bars. Once one comes to share Brouw er’s view on how infinite mathematical objects and the continuum should be handled in thought and language, one m ay find these principles to be plausible starting points for our m athem atical discourse. In this series of papers, we follow Brouwer and we avoid the use of the principle of the excluded middle: the logic of our arguments will be intuitionistic logic. In addition, and unlike other constructivist m athem aticians, we also use the axioms Brouwer suggested. 1.2. It is useful to rem ind the reader of three im portant theorems obtained in the ear­ lier paper (Veldman, 2008a). We shall form ulate them slightly differently than in Veldman (2008a) and first have to agree upon some notations and definitions. We let N denote the set o f the natural numbers. N* is the set o f all finite sequences of natural numbers. We let ( ) be a fixed bijective mapping from N* onto N. Such a function is called a coding of the set of finite sequences of natural numbers: (a0, a i , . . . , ak—i ) is the code number of the finite sequence (a0, a i , . . . , ak—i ). We assume that the empty sequence is coded by the num ber 0 and that for each finite sequence (a0, a i , . . . , ak—i ), for every i < k , the code num ber (a0, a i , . . . , ak—i ) is greater than a i . We let length be the function from N to N that associates to any natural num ber a the length of the finite sequence coded by a. We also assume that there is a function a, i ^ a ( i) from N x N to N, such that, for every k , for every a, if length(a) = k , then a = (a(0), a ( l ) , . . . a ( k — 1)). We let * denote concatenation: * is a function from N x N to N such that, for all m , n, m * n is the code num ber of the finite sequence obtained by putting the sequence coded by n behind the sequence coded by m . For all m , n, m is an initial part o fn , notation: m c n, if and only if there exists p such that n = m * p ; and n is an immediate successor o fm if and only if there exists p such that n = m * (p). We define another function, called J , from N x N to N: for all m , n : J ( m , n) := (m) * n. It is easy to see that J is a bijective mapping from N x N onto N\{0}. We let K , L be the inverse functions of J , that is, K and L are functions from N\{0} to N and for each m, m = 0 : J ( K ( m ) , L (m )) = m. J is a nonsurjective pairing function on N. We let Baire space N be the set o f all infinite sequences a = a (0 ) , a (1 ) , a ( 2 ) , . . . of natural numbers. The intuitionistic mathem atician sometimes calls this set the universal spread. Let a, p belong to N . a is apart from p , notation: a # p , if and only if there exists n such that a (n ) = P(n). We define, for all a , for all m , n, a m (n) := a ( J (m , n ) ) . a m is called the m -th subse­ quence o f a . We also define, for all a, for all m , n, a m,n := (a m )n . We define, given any a and any n, a (n ) := (a(0), a (1 ) , . . . , a ( n — 1)). If confusion is unlikely to arise, we sometim es write an for a(n). We also define, given any a and any s, s is an initial pa rt o f a , or: a passes through s, or: s contains a , if and only if, for some n, a n = s. A subset X o f N is basic open if and only if either X is em pty or there exists s such that X is the set o f a in N passing through s . A subset X of N is open if and only if X is a countable union of basic open sets. One may prove that a subset X o f N is open if and only if there exists p in N such that, for every a in N , a belongs to X if and only if, for some n, P (a n ) = 0. A subset X o f N is closed if and only if there is an open subset Y of N such that X is the set of all a in N such that the assumption “a belongs to Y ” leads to a contradiction. One may prove that a subset X of N is closed if and only if there exists p in N such that, for every a in N , a belongs to X if and only if, for all n, p (a n ) = 0 .