Master Dissertation. Reverse mathematics of the Browder-Göhde-Kirk fixed point theorem

Transcription

1 Master Dissertation Department of Mathematics Robin Vandaele Academic year Reverse mathematics of the Browder-Göhde-Kirk fixed point theorem Advisors: Dr. P. Shafer Prof. Dr. A. Weiermann Master dissertation submitted in order to obtain the academic degree of Master of Science in Mathematics, specialization in Pure Mathematics.

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3 Master Dissertation Department of Mathematics Robin Vandaele Academic year Reverse mathematics of the Browder-Göhde-Kirk fixed point theorem Advisors: Dr. P. Shafer Prof. Dr. A. Weiermann Master dissertation submitted in order to obtain the academic degree of Master of Science in Mathematics, specialization in Pure Mathematics.

4 Contents Preface Permission for use of content Toelating tot bruikleen Resume i ii ii iii 1 The language of second-order arithmetic The formal system Z Subsystems of Z RCA The number system and primitive recursion Concepts of analysis and topology WKL ACA ATR Π 1 1-CA Reverse mathematics The Main Question Brouwer s fixed point theorem for the unit square The unit square in R Proving that Brouwer s fixed point theorem for the unit square is equivalent to WKL 0 over RCA Browder-Göhde-Kirk s fixed point theorem The statement in Z The reverse mathematics for Ê a Hilbert space and K a closed ball Proving that ACA 0 BGK H over RCA The Hilbert space l Proving that BGK H ACA 0 over RCA The reverse mathematics for Ê a uniformly convex Banach Space A Appendix: Nederlandstalige samenvatting 69 B Appendix: Coding continuous functions 70 B.1 Coding the metric projection on a closed ball in a Hilbert space B.2 Coding the metric projection on a half-space in a Hilbert space B.3 Coding of a function series in a separable Banach space References 83

5 Preface God created the natural numbers and mankind did the rest. This was one of the first sentences I heard while studying mathematics at Ghent University. It was only until a few years later, by studying mathematical logic, I learned that this is not necessarily the case. 1 And even so, merely the existence of the natural numbers itself isn t enough to construct our perception of modern mathematics. We had to apply rules which seemed plausible enough, such that we didn t really have to think about their validity. For example, given sets A and B, we can define the cartesian product A B. In ordinary mathematics, this is something we can do without the need of any further justification. That s why I m so intrigued with the subject of mathematical logic. I can t seem to find any other branch of mathematics in which our reasoning is justified more than in logics. Here, one learns that these rules we applied, are actually or follow from some basic axioms which we can formalize in axiomatic systems. For example ZF or ZFC, in which we humans ourselves constructed some perception of the natural numbers starting from some basic axioms. So essentially, one could say God created an axiomatic system and mankind did the rest. Actually, He created a few. He had a lot of spare time in his days. Learning that there is actually a mathematical subject that studies something that resembles the opposite of our construction of modern mathematics from given axioms, namely given some ordinary mathematical theorem, looking to which axioms are essentially needed to formalize and prove this theorem, I knew I had to study this subject of reverse mathematics further. Since my second favourite branch of mathematics is plain analysis, doing my thesis on a subject which involves reverse mathematics as well as analysis, seemed to be a perfect idea. My thanks go to Dr. P. Shafer and Prof. Dr. A. Weiermann for intriguing me with subjects as mathematical logic and proof theory, and for helping me to find a topic in which I could fully study the subject of reverse mathematics. Also, Dr. P. Shafer for all the effort he put the past year into helping me explore this branch of mathematics, which I essentially knew nothing about in advance, and for coming up with a lot of crucial ideas and strategies for this thesis. To end with, I can conclude that my skills in mathematical and logical reasoning have increased a lot since high school by studying mathematics at Ghent University. The opposite holds for my skills in solving ordinary integrals. Robin Vandaele, May Note that we are not emphasizing the question whether or not there exists a God here. This thesis does not concern the provability of this statement. i

6 Permission for use of content The author gives permission to make this master dissertation available for consultation and to copy parts of this master dissertation for personal use. In the case of any other use, the limitations of the copyright have to be respected, in particular with regard to the obligation to state expressly the source when quoting results from this master dissertation. Toelating tot bruikleen De auteur geeft de toelating deze masterproef voor consultatie beschikbaar te stellen en delen van de masterproef te kopiëren voor persoonlijk gebruik. Elk ander gebruik valt onder de beperkingen van het auteursrecht, in het bijzonder met betrekking tot de verplichting de bron uitdrukkelijk te vermelden bij het aanhalen van resultaten uit deze masterproef. Robin Vandaele, May 31st, 2016 ii

7 Resume In [1], Simpson gives an analysis of Brouwer s fixed point theorem in his chapter on weak König s lemma, proving that Brouwer s theorem is equivalent to weak König s lemma over RCA 0. We shall take a look at another fixed point theorem, namely the Browder-Göhde- Kirk fixed point theorem. The purpose of this thesis is to perform and analyze the reverse mathematics of this theorem. To make this thesis somewhat self-contained, we shall start with defining the language of second-order arithmetic L 2 and the formal system Z 2 in Chapter 1, continuing with the natural subsystems RCA 0, WKL 0, ACA 0, ATR 0 and Π 1 1-CA 0 of Z 2 in Chapter 2. RCA 0 will be the weakest of these subsystems, i.e., is included in all other subsystems, and we shall summarize the basic concepts of real analysis and topology that can be developed in this system, and hence in all the other systems. Chapters 1 and 2 will be largely based on, or containing notes from [1] and [2]. The purpose of Chapter 3 is to introduce reverse mathematics. The Main Question shall be: Given a theorem ϕ of ordinary mathematics, what is the weakest subsystem of Z 2 in which ϕ is provable? As an example, we shall prove that Brouwer s fixed point theorem for the unit square is equivalent to WKL 0 over RCA 0. This proof will be largely based on Simpson s proof of the general version of Brouwer s fixed point theorem in [1] Section IV.7. In Chapter 4, we shall take a look at the other fixed point theorem, the Browder- Göhde-Kirk fixed point theorem. We will proof that this theorem, also denoted by BGK, is equivalent to ACA 0 over RCA 0. The direction ACA 0 BGK will be mainly based on a generalization of an ordinary mathematical proof from [3], by investigating how these strategies work out within the context of reverse mathematics. The reversal BGK ACA 0 will be mainly based on ideas from [4]. iii

8 1 The language of second-order arithmetic In this chapter we shall define the two-sorted language L 2 and the formal system Z 2. By two-sorted we mean that there are two distinct sorts of variables which are intended to range over two different kinds of objects: Variables of the first sort are known as number variables, are denoted by i, m, n,..., and are intended to range over the set ω = {0, 1, 2,...} of all natural numbers. Variables of the second sort are known as set variables, are denoted by X, Y, Z,..., and are intended to range over all subsets of ω. Numerical terms are number variables, the constant symbols 0 and 1, and n + m and n m whenever n and m are numerical terms. Here + and are binary operation symbols in our language. Atomic formulas are n = m, n < m and n X, where n and m are numerical terms and X is any set variable. Here =, < and are binary relation symbols in our language. Formulas are built up from atomic formulas by means of the propositional connectives,,,,, number quantifiers n, n, and set quantifiers X, X. A sentence is a formula with no free variables. Definition 1.1. (language of second-order arithmetic). L 2 is defined to be the language of second-order arithmetic as described above. Remark 1.2. As customary, we shall use some obvious abbreviations. For instance = 4 stands for (1 + 1) + (1 + 1) = ((1 + 1) + 1) + 1, (m + n) 2 / X stands for ((m + n) (m + n) X), and s t stands for s < t s = t. Definition 1.3. (L 2 -structures). A model for L 2, also called a structure for L 2 or an L 2 -structure, is an ordered 7-tuple M = (N M, S M, + M, M, 0 M, 1 M, < M ), where N M is a set serving as the range of the number variables, S M is a set of subsets of N M serving as the range of the set variables, + M and M are binary operations on N M, 0 M and 1 M are distinguished elements of N M, and < M is a binary relation on N M. We always assume that the sets N M and S M are disjoint and nonempty. Formulas of L 2 are interpreted in M in the obvious way. The intended model for L 2 is of course the model (ω, P (ω), +,, 0, 1, <) where ω is the set of natural numbers, P (ω) is the set of all subsets of ω and +,, 0, 1, < are as usual. By an ω-model we mean an L 2 -structure of the form (ω, S, +,, 0, 1, <) where S P (ω). We sometimes just speak of the ω-model S. ([1] Section I.2.) 1

9 1.1 The formal system Z 2 Definition 1.4. (second-order arithmetic). The axioms of second-order arithmetic consist of the universal closure of the following L 2 formulas: 1. basic axioms: n m + 1 = n + 1 m = n m + 0 = m m + (n + 1) = (m + n) + 1 m 0 = 0 m (n + 1) = (m n) + m m < 0 m < n + 1 (m < n m = n) 2. induction axiom: 3. comprehension scheme: (0 X n(n X n + 1 X)) n(n X) X n(n X ϕ(n)) where ϕ(n) is any formula of L 2 in which X does not occur freely. In the comprehension scheme, ϕ(n) may contain free variables in addition to n. These free variables may be referred to as parameters of this instance of the comprehension scheme. Remark 1.5. The full second-order induction scheme, i.e., the universal closure of (ϕ(0) n(ϕ(n) ϕ(n + 1))) nϕ(n), where ϕ(n) is a any formula of L 2, is valid in any model of Definition 1.4.(ii)-(iii). By second-order arithmetic we mean the formal system in the language L 2 consisting of these axioms, together with all formulas of L 2 wich are deducible from those axioms by means of the usual logical axioms and rules of inference. The formal system of second-order arithmetic is known as Z 2. 2

10 2 Subsystems of Z 2 By a subsystem of Z 2 we mean a formal system in the language L 2 each of whose axioms is a theorem of Z 2. When introducing a new subsystem of Z 2, we shall specify the aixoms of the system by writing down some formulas of L 2. The axioms are then taken to be the universal closures of those formulas. 2.1 RCA 0 In our designation RCA 0, the acronym RCA stands for recursive comprehension axiom. 2 The subscript 0 denotes restricted induction. This means that RCA 0 does not include the full second-order induction scheme as defined in Remark Let n be a number variable, let t be a numerical term not cointaining n, and let ϕ be a formula of L 2. We use the following abbreviations: n < t ϕ n(n < t ϕ), n < t ϕ n(n < t ϕ). The expressions n < t, n t, n < t, n t are called bounded number quantifiers, or simply bounded quantifiers. A bounded quantifier formula is a formula ϕ such that all of the quantifiers in ϕ are bounded number quantifiers. Definition 2.1. (Σ 0 1 and Π 0 1 formulas). An L 2 -formula ϕ is said to be Σ 0 1 (respectively Π 0 1) if it is of the form mθ (respectively mθ), where m is a number variable and θ is a bounded quantifier formula. Definition 2.2. ( 0 1 formulas). An L 2 formula ϕ is said to be 0 1 if it can be equivalently written in both classes of Σ 0 1 and Π 0 1 formulas, i.e., we have ϕ ψ η, where ψ is a Σ 0 1 formula, and η a Π 0 1 formula. Note that any bounded quantifier formula 4 can easily be seen to be 0 1. Definition 2.3. (Σ 0 1 and Π 0 1 induction). The Σ 0 1 induction scheme, Σ 0 1-IND, is the restriction of the second-order induction scheme to L 2 -formulas ϕ(n) which are Σ 0 1. The Π 0 1 induction scheme, Π 0 1-IND, is defined similarly. It can be shown that in the presence of the basic axioms from Definition 1.4.(i) that Σ 0 1-IND and Π 0 1-IND are equivalent. 2 Roughly speaking, the set existence axioms of RCA 0 are only strong enough to prove the existence of recursive sets of natural numbers. 3 The same reason applies to all subsystems of Z 2 considered in this section. 4 When defining Σ 0 k and Π0 k formulas for general 0 k ω, these can be seen as Σ0 0 or Π 0 0 formulas, see [1] Definition

11 Definition 2.4. ( 0 1 comprehension). The 0 1 comprehension scheme consists of (the universal closure of) all formulas of the form X n(n X ϕ(n)), where ϕ is 0 1, n is any number variable, and X is a set variable which does not occur freely in ϕ(n). Definition 2.5. (definition of RCA 0 ). RCA 0 is the subsystem of Z 2 consisting of the basic axioms 1.4.(i), the Σ 0 1 induction scheme, and the 0 1 comprehension scheme. An extremely important fact that one can prove within RCA 0, is that each finite set of natural numbers, i.e., a set of natural numbers for which all its elements are bounded by some natural number, has a unique code, which is also a natural number. For more information, see [1] Section II The number system and primitive recursion This subsection will be largely based on notes from [1] Section II.3 and II.4, but includes results that are of much importance for the rest of this thesis. Our first observation is that within RCA 0, we can define N to be the unique set X such that n(n X) by applying 0 1 comprehension to the 0 1-formula ϕ(n) 0 = 0, since for each constant c, the formula c = c is universally valid. It can be shown in RCA 0 that the system (N, +,, 0, 1, <) is a commutative ordered semiring with cancellation. See [1] Lemma II.2.1 for a list of some of the basic known arithmetical properties that can be shown to hold in this system within RCA 0. Furthermore, we can define the pairing map (i, j) = (i + j) 2 + i. It can be proved in RCA 0 that (i, j) = (i, j ) (i = i j = j ). By this, using 0 1 comprehension, we can prove that for all sets X, Y N, 5 there exists a set X Y N consisting of all (m, n) such that m X and n Y. We can introduce total functions from N into N by encoding them as certain sets of ordered pairs. An important theorem says that the universe of k-ary functions, k N, is closed under primitive recursion. Theorem 2.6. (primitive recursion). The following is provable in RCA 0. Given functions f : N k N and g : N k+2 N, there exists a unique h : N k+1 N defined by Proof. See [1] Theorem II.3.4. h(0, n 1,..., n k ) = f(n 1,..., n k ), h(m + 1, n 1,..., n k ) = g(h(m, n 1,..., n k ), m, n 1,..., n k ). The most important consequence of this theorem is that elementary number theory can be developed straightforwardly within RCA 0. For instance we can prove the existence of the exponential function f(m, n) = m n defined by f(m, 0) = 1, f(m, n + 1) = f(m, n) m. We can then show that RCA 0 proves the validity of some basic ordinary properties 5 The inclusion and equality of set variables can be introduced in the natural way. 4

12 such as (m 1 m 2 ) n = m n 1m n 2, m n 1+n 2 = m n 1 m n 2, m n 1n 2 = (m n 1 ) n 2. Also within RCA 0 we can straightforwardly state and prove fundamental results such as unique prime power factorization. RCA 0 may be viewed as a formal version of computable or constructive mathematics. Moreover, we have the following theorem: Theorem 2.7. The minimum ω-model of RCA 0 is the collection Proof. See [1] Corollary II.1.8. REC = {X ω : X is recursive}. The construction of the number system within the context of reverse mathematics is based on the usual Dedeking/Cauchy construction. Consider the set N N of ordered pairs of natural numbers. We define the following operations and relations on N N: (m, n) + Z (p, q) = (m + p, n + q), (m, n) Z (p, q) = (m + q, n + p), (m, n) Z (p, q) = (m p + n q, m q + n p), (m, n) < Z (p, q) m + q < n + p, (m, n) = Z (p, q) m + q = n + p. Note that the relations < Z and = Z are expressed by Σ 0 0 formulas. Since the relation = defines an equivalence relation on the natural numbers, it is easy to see that the relation = Z defines an equivalence relation on N N. We define an integer to be an element in z N N of the form z = min{(m, n) N N : (m, n) = Z (a, b)}, for some (a, b) N N, i.e., z is the N-minimal of its equivalence class. One can prove within RCA 0 that the set of all integers, denoted by Z, exists. If z = N (m, n) N N, then z corresponds to the ordinary integer m n. It is then possible to accordingly define the addition +, the subtraction, the multiplication and the relation < on Z, as well as the constants 0 and 1. One can prove within RCA 0 some of the basic properties of the system (Z, +,,, 0, 1, <), such as that it s an ordered integral domain, Euclidean, and so on, see [1] Theorem II.4.1. In a similar manner, we can define within RCA 0 the set Q of rational numbers. Let Z + be the set of positive integers, i.e., Z + := {z Z : z > 0}. This set exists within RCA 0 since the property z > 0 is expressed by a Σ 0 0 formula. We define the following operations and relations on the set Z Z + : (m, n) + Q (p, q) = (m q + n p, n q), 5

13 (m, n) Q (p, q) = (m q n p, n q), (m, n) Q (p, q) = (m p, n q), (m, n) < Q (p, q) m q < n p, (m, n) = Q (p, q) m q = n p. Again, the relations < Q and = Q are expressed by Σ 0 0 formulas and the relation = Q is an equivalence relation on Z Z +. In an analogous way, we define a rational number to be the least element of its equivalence class. If q = (m, n), then q corresponds to the ordinary rational m/n. The set of all rational numbers is denoted by Q and again we may define +,,, 0, 1 and < on Q accordingly. Within RCA 0, one can show that the system (Q, +,,, 0, 1, <) is on ordered field, see [1] Theorem II.4.2. We define an (infinite) sequence of rational numbers to be a function f : N Q and denote such a sequence by q n : n N or simply by q n, where q n = f(n). A real number is defined to be a sequence a n of rational numbers such that n i( a n a n+i 2 n ), where denotes the absolute value q if q 0 : Q Q + : q q if q < 0. We use R informally 6 to denote the set of all real numbers, i.e., x R is just a notation for x is a real number. The set Q can be embedded into R by identifying q Q with the sequence q n : n N for which q n = q for all n N. Let x = p n : n N and y = q n : n N be two real numbers. Their sum is defined as x + y = p n+1 + q n+1 : n N, and their product as where k is defined as x y = p k+n + q k+n : n N, k := min{m N : 2 m p 0 + q 0 + 2}. It is easy to check that x + y and x y indeed again define real numbers. Furthermore, we define the following relations on R: x = y n( p n q n 2 n+1 ), x y n(p n q n + 2 n+1 ), x < y y x n(q n > p n + 2 n+1 ). 6 The set R does not exist within RCA 0, since RCA 0 is limited to the language L 2 of second-order arithmetic. 6

14 Note that the relations = and on R are expressed by Π 0 1 formulas, whereas the relations < and are expressed by Σ 0 1 formulas. It is easy to check within RCA 0 that the system (R, +,, 0, 1, <, =) obeys all the axioms for an ordered Abelian group. Furthermore, one can show that the real number system (R, +,,, 0, 1, <, =) obeys all the axioms of an Archimedean ordered field, see [1] Theorem II.4.5. An infinite sequence of real numbers is defined to be a doubly indexed sequence of rational numbers q mn : m, n N such that for each m, q mn : n N is a real number. Such a sequence is also denoted by x m : m N, where x m = a mn : n N. We write R N for the informal set of infinite sequences of real numbers. We say that the sequence x m : m N converges to x, where x R, if and we denote lim m x m = x. ɛ R + m i( x x m+i < ɛ), Remark 2.8. Our construction of the number system is based on notes from [1] Section II.4. The main reason why we included this construction in this thesis, is to make the reader aware why relations such as x < y and x y, with x, y R, are expressed by Σ 0 1 formulas. This fact is of much importance for the rest of the thesis and will be used quite a lot to be able to construct specific open sets and their complements in complete separable metric spaces, see also Theorem Concepts of analysis and topology In this subsection, we will focus on some analytical and topological concepts that can be developed within RCA 0 and will play important roles in this thesis. Nevertheless, many basic concepts concerning mathematical logic and countable algebra can also be adequately defined within RCA 0. Two important definitions made in RCA 0 are those of a complete separable metric space and of a compact metric space: Definition 2.9. (complete separable metric spaces, [1] Definition II.5.1). A (code for a) complete separable metric space  is defined in RCA 0 to be a nonempty set A N together with a sequence of real numbers d: A A R such that d(a, a) = 0, d(a, b) = d(b, a) 0, and d(a, b) + d(b, c) d(a, c) for all a, b, c A. A point of  is a sequence x = a k : k N of elements of A, such that i j(i < j d(a i, a j ) 2 i ). We write x  to mean that x is a point of Â. If x = a k : k N and y = b k : k N are two points of Â, we can define d(x, y) = lim k d(a k, b k ). We define x = y to mean that d(x, y) = 0. Each a A is of course identified with the point x a = a : k N Â. By definition, it now easily follows that the countable set A is dense in Â. Indeed, for x = a k : k N, it holds that d(x, a n ) = lim k d(a k, a n ) 2 n. This justifies our designation of  as separable. To justify our designation of of complete, we note that  is complete in the following sense: Theorem The following is provable in RCA 0. If x n : n N is a sequence of points in Â, and there exists a sequence of real numbers r n : n N with lim n r n = 0 and such 7

15 that m n(m < n d(x n, x m ) r m ), then x n : n N is convergent, i.e., there exists a point x  such that x = lim n x n. Proof. See [1] Exercise II.5.2. Note that this theorem also justifies why we are able to define d(x, y) = lim k d(a k, b k ), where x = a k : k N and y = b k : k N are points in a complete separable metric space Â. Moreover, we can let R = Q, i.e., we can see R as the completion of the rationals under the metric d Q : Q Q R with d Q (p, q) = p q for p, q Q. Since for every k N we have d(a k, b k ) Q, 7 d(a k, b k ) : k N defines a sequence of points in Q for which for all m, n N, m < n d(a n, b n ) d(a m, b m ) d(a n, b n ) d(a n, b n 1 ) d(a n, b m+1 ) d(a n, b m ) + d(a n, b m ) d(a n 1, b m ) d(a m+1, b m ) d(a m, b m ) d(b n, b n 1 ) d(b m+1, b m ) + d(a n, a n 1 ) +... d(a m+1, a m ) 2 (2 (n 1) m ) = 2 (n 2) (m 1) 2 (m 2). so that lim k d(a k, b k ) indeed exists within RCA 0. Hereby, we note that for a real number r = q n : n N, one can explicitly define r = q n : n N. The fact that this is indeed again a real number, follows easily from the triangle inequality. Definition (compactness, [1] Definition III.2.3). The following definition is made in RCA 0. A compact metric space is a complete separable metric space  such that there exists an infinite sequence of finite sequences x ij : i n j : j N, x ij Â, such that for all z  and j N there exists i n j such that d(x ij, z) < 2 j. We can define the concepts of open and closed sets in and continuous functions between complete separable metric spaces: Definition (open and closed sets, [1] Definitions II.5.6 & II.5.12). Within RCA 0, let  be a complete separable metric space. A (code for an) open set U in  is a set U N A Q +. A point x  is said to belong to U, denoted by x U, if n a r (d(x, a) < r (n, a, r) U). A closed set in  is the complement of an open set in Â. I.e., a code for a closed set C is the the same thing as a code for an open set U, but we define x C iff x / U. We regard (a, r) A Q + as a code for the basic open ball B(a, r) consisting of all points x  such that d(x, a) < r. The idea of the preceding definition is that U encodes the open set which is the union of the open balls B(a, r) such that n((n, a, r) U). By these definitions, one can prove some of the basic properties of open and closed sets, such as: 7 The definitions of elements of R and elements of Q coincide 8

16 Theorem The following is provable in RCA 0. Suppose C is a closed set in a complete separable metric space  and that x n : n N is a sequence of points in C. If lim n x n = x, then x C. Proof. Let U denote the code for the open complement of C, such that y C iff y / U, i.e., iff m a r(d(y, a) r (m, a, r) / U). Suppose x / C. By definition this is the case iff x U, i.e., iff m a r (d(x, a) < r (m, a, r) U). Now take such (m, a, r) U with d(x, a) < r. Since x n C, we must have d(x n, a) r for all n N. Now let N N be such that n > N implies d(x n, x) < (r d(x, a))/2 and note that d(x, a) < (d(x, a) + r)/2. Hence, we find for n > N d(x n, a) d(x n, x) + d(x, a) < r d(x, a) 2 a contradiction. Hence, we must have x C. See also Figure 1. + r + d(x, a) 2 = r, Figure 1: If x B(a, r) and x n x, then for n large enough, we must always have x n B(a, r). Note that many examples of open sets, and hence, their closed complements, can be shown to exist within RCA 0. This follows from the following theorem. Theorem The following is provable in RCA 0. Let for every n N, ϕ(x, n) be a Σ 0 1 formula, and let  be a complete separable metric space. Assume that for all n N and x, y Â, (x = y) ϕ(x, n) implies ϕ(y, n). Then there exists a sequence of (codes for) open sets U n : n N with U n  for every n N, such that for all x Â, x U n if and only if ϕ(x, n). A similar statement holds for a Σ 0 1 formula ϕ(x) and the existence of the open set U  such that x U ϕ(x). 9

17 Proof. See [1] Lemma II.5.7, where a proof for the second statement is given. Definition (continuous functions, [1] Definition II.6.1). Within RCA 0, let  and Ê be complete separable metric spaces. A (code for a) continuous partial function Φ from  to Ê is a set of quintuples Φ N A Q+ E Q + with certain properties. We write (a, r)φ(b, s) as an abbreviation for n((n, a, r, b, s) Φ). The required properties are 1. if (a, r)φ(b, s) and (a, r)φ(b, s ), then d(b, b ) s + s ; 2. if (a, r)φ(b, s) and d(a, a ) + r < r, then (a, r )Φ(b, s) ; 3. if (a, r)φ(b, s) and d(b, b ) + s < s, then (a, r)φ(b, s ). Figure 2: Intuitively, the first property of a continuous functions states that when B(a, r) gets mapped to both B(b, s) and B(b, s ), then these balls should overlap. One does need to take caution with this interpration, since in general complete separable metric spaces, d(b, b ) s + s and B(b, s) B(b, s ) are nt always equivalent. See also Remark Figure 3: The second property of a continuous functions states that when B(a, r) gets mapped to B(b, s), and we have B(a, r ) B(a, r), then also B(a, r ) gets mapped to B(b, s). Figure 4: The third property of a continuous functions states that when B(a, r) gets mapped to B(b, s) and B(b, s) B(b, s ), then B(a, r) gets also mapped to B(b, s ). 10

18 The idea of the definition is that Φ encodes a partially defined, continuous function φ from  to Ê. Intuitively, (a, r)φ(b, s) is a piece of information to the effect that φ(x) is contained within the closure of B(b, s) whenever x B(a, r), provided φ(x) is defined. A point x  is said to belong to the domain of φ, abbreviated x dom(φ), provided the code Φ of φ contains sufficient information to evaluate φ at x. This means that for all ɛ > 0 there exists (a, r)φ(b, s) such that d(x, a) < r and s < ɛ. If x dom(φ), we define the value φ(x) to be the point y Ê such that d(y, b) s for all (a, r)φ(b, s) with d(x, a) < r. If x dom(φ), then within RCA 0, it is possible to show that φ(x) exists and is unique up to equality of points in Ê. We write φ(x) is defined to mean that x dom(φ). We say that φ is totally defined on  if φ(x) is defined for all x Â. We write φ : Â Ê to mean that φ is a continuous, totally defined function from  to Ê. ([1] Section II.6.) Note that this definition implies some of the usual properties of continuous functions, such as: Theorem If f : Â Ê is a continuous function between two complete separable metric spaces and lim n x n = x in dom(f), then lim n f(x n ) = f(x) in Ê. Proof. Take ɛ > 0. We need to show that there exists an N N such that n > N implies d E (f(x n ), f(x)) < ɛ. Since x dom(f), there exists (a, r)φ(b, s) such that d A (x, a) < r and s < ɛ/2. By definition, we then have d E (f(x), b) s < ɛ/2. Take N N such that n > N implies d A (x n, x) < (r d A (x, a))/3. Now let q Q + be such that q < (r d A (x, a))/3. Then for each n > N, there exists an a n A such that d A (a n, x n ) < q, and we find d A (a, a n ) + q d A (x, a) + d A (x, x n ) + d A (x n, a n ) + q < d A (x, a) + 3 r d A(x, a) 3 = r. so that Property 2. of Definition 2.15 implies (a n, q)φ(b, s). Since d A (x n, a n ) < q and x n dom(f), we again have by definition d E (f(x n ), b) s < ɛ/2, hence d E (f(x), f(x n )) d E (f(x), b) + d E (f(x n ), b) < ɛ, which ends our proof. See also Figure 5 for an illustration. Figure 5: If x n x B(a, r) f B(b, s), then for n large enough, x n is always contained in a ball B(a n, q) B(a, r) that also gets mapped onto the ball B(b, s) by f. 11

19 These concepts will be used when performing the reverse mathematics of our version of Brouwer s fixed point theorem in Chapter 3. When formulating the Browder-Göhde-Kirk fixed point theorem in Chapter 4, we will make use of the concept of a separable Banach space: Definition (fields, [1] Definition II.9.1). A countable field K consists of a set K N together with binary operations + K, K, a unary operation K (symbolizing the additive inverse) and distinguished elements 0 K, 1 K, such that the system ( K, + K, K, K, 0 K, 1 K ) obeys the usual field axioms. Definition (vector spaces, [1] Section II.10). Let K be a countable field. Within RCA 0, a countable vector space A over K consists of a set A N together with operations + : A A A and : K A A and a distinguished element 0 A, such that the system ( A, +,, 0) satisfies the usual axioms for a vector space over K. Definition (separable Banach spaces, [1] Definition II.10.1). Within RCA 0, we define a (code for a) separable Banach space  to consist of a countable vector space A over the rational field Q together with a sequence of real numbers : A R satisfying 1. q a = q a for all q Q and a A ; 2. a + b a + b for all a, b A. A point of  is defined to be a sequence a k : k N of elements of A such that a k a k+1 2 k 1 for all k N. Note that for any a A 0 = 0 = a a a + a = 2 a 0 a, so that a A, a 0 It is possible to show within RCA 0 that one can extend the operators, + and to continuous operators :  R, + :    and : R   by taking the usual limits. We define x = y to mean that x y = 0. One can show that  enjoys the usual properties of a normed vector space over R. As usual, we define a pseudometric on A by d(a, b) = a b, for all a, b A. By this, it is easily seen that  is the complete separable metric space which is the completion of A under d. 2.2 WKL 0 We note that concepts like the set of all (codes for) finite sequences Seq or N <N, the length lh of a sequence, the concatenation of two sequences, being an initial segment of another sequence (denoted by ) and the set 2 <N of all finite sequences of 0 s and 1 s can be perfectly defined within RCA 0. Definition (König s lemma, [1] Definition 2.18). The following definitions are made in RCA 0. A tree is a set T N <N which is closed under initial segment, i.e., σ τ((σ N <N σ τ τ T ) σ T ). 12

20 We say that T is finitely branching if each element of T has only finitely many immediate successors, i.e. σ(σ T n m(σ m T m < n)). A path through T is a function g : N N such that g[n] := g(0), g(1),..., g(n 1) T for all n N. König s lemma is the assertion that every infinite, finitely branching tree T has at least one path. Definition (weak König s lemma, [1] Definition I.10.1). The following defintions are made within RCA 0. We use {0, 1} <N or 2 <N to denote the full binary tree, i.e., the set of (codes for) finite sequences of 0 s and 1 s. Weak König s lemma is the following statement: Every infinite subtree of 2 <N has an infinite path. WKL 0 is defined to be the subsystem of Z 2 consisting of RCA 0 plus weak König s lemma. Hence, the subsystem RCA 0 of Z 2 is included in WKL 0. Remark It is clear that an infinite subtree of the full binary tree is an infinite, finitely branching tree. Hence, as one could already suspect, König s lemma implies weak König s lemma. 2.3 ACA 0 The acronym ACA stands for arithmetical comprehension axiom. This is because ACA 0 contains axioms asserting the existence of any set which is arithmetically definable from given sets. Definition (arithmetical formulas). A formula of L 2 is said to be arithmetical if it contains no set quantifiers, i.e., all of the quantifiers appearing in the formula are number quantifiers. Definition The arithmetical comprehension scheme is the restriction of the comprehension scheme 1.4.(iii) to arithmetical formulas ϕ(n). ACA 0 is the subsystem of Z 2 whose axioms are the arithmetical comprehension scheme, the induction axiom 1.4.(ii), and the basic axioms 1.4.(i). An easy consequence of the arithmetical comprehension scheme and the induction axiom is the arithmetical induction scheme: (ϕ(0) n(ϕ(n) ϕ(n + 1))) nϕ(n), for all L 2 -formulas ϕ(n) which are arithmetical. Remark Since all Σ 0 1 and Π 0 1 formulas are of course arithmetical, Definition 2.5 implies that the subsystem ACA 0 of Z 2 includes RCA 0. Actually, also the subsystem WKL 0 is included in ACA 0. In order to show this, we would need to prove that ACA 0 implies weak König s lemma over RCA 0. Considering Remark 2.22, there even holds a stronger result: Theorem ACA 0 is equivalent to König s lemma over RCA 0. Proof. See [1] Theorem III

21 2.4 ATR 0 The acronym ATR stands for arithmetical transfinite recursion. definition of a countable well-ordering. We start with the Definition (countable well-ordering [1] Definition I.6.1). Within RCA 0, we define a countable linear ordering to be a structure (A, < A ), where A N and < A A A is a strict linear ordering of A, i.e., < A is transitive and, for all a, b A, exactly one of the relations a = b or a < A b or b < A a holds. The countable linear ordering A, < A is called a countable well-ordering if there is no sequence a n : n N of elements in A such that a n+1 < A a n for all n N. Definition (arithmetical transfinite recursion, [1] Definition I.11.1). Consider an arithmetical formula θ(n, X) with a free number variable n and a free set variable X. Note that θ(n, X) may also contain parameters, i.e., additional free number and set variables. Fixing these parameters, we may view θ as an arithmetical operator Θ : P(N) P(N), defined by Θ(X) = {n N : θ(n, X)}. Now let (A, < A ) be any countable well-ordering, and consider the set Y N obtained by transfinitely iterating the operator Θ along (A, < A ). This set Y is defined by the following conditions: Y N A and, for each a A, Y a = Θ(Y a ), where Y a = {m : (m, a) Y } and Y a = {(n, b) : n Y b b < A a}. Thus, for each a A, Y a is the result of iterating Θ along the initial segment of (A, < A ) up to but not including a, and Y a is the result of applying Θ one more time. Finally, arithmetical transfinite recursion is the axiom scheme asserting that such a set Y exists, for every arithmetical operator Θ and every countable well-ordering (A, < A ). We define ATR 0 to consist of ACA 0 plus the scheme of arithmetical transfinite recursion. Hence, ACA 0 is trivially included in the subsystem ATR 0 of Z Π 1 1-CA 0 Definition (Π 1 1 formulas). A formula ϕ is said to be Π 1 1 if it is of the form X θ, where X is a set variable and θ is an arithmetical formula. A formula ϕ is said to be Σ 1 1 if it is of the form X θ, where X is a set variable and θ is an arithmetical formula. Definition (Π 1 1 comprehension). Π 1 1-CA 0 is the subsystem of Z 2 whose axioms are the basic axioms, the induction axiom, and the comprehension scheme restricted to L 2 -formulas ϕ(n) which are Π 1 1. Thus we have the universal closure of X n(n X ϕ(n)) for all Π 1 1 formulas ϕ(n) in which X does not occur freely. It would be possible to introduce the system Σ 1 1-CA 0, but this would be superfluous, because one can show that Σ 1 1-CA 0 and Π 1 1-CA 0 are equivalent, i.e., they have the same theorems. 14

22 It turns out that ATR 0 is equivalent to several theorems of ordinary mathematics which are provable in Π 1 1-CA 0 but not in ACA 0, which implies that ATR 0 is intermediate between ACA 0 and Π 1 1-CA 0. We mentioned the subsystems ATR 0 and Π 1 1-CA 0 just for completeness. They will not play any important roles in the further development of this thesis. 15

23 3 Reverse mathematics In this chapter we shall discuss the subject of reverse mathematics. We shall formulate the Main Question, and, as an example, we shall include Simpson s analysis of the reverse mathematics of Brouwer s fixed point theorem given in [1] Chapter IV.7. However, we shall adjust some of the theorems and proofs and stick to the case where the domain is the unit square [0, 1] The Main Question In the previous chapter we have discussed five specific natural subsystems of Z 2, where each system is included in the next one. In general, for a given theorem, one can wonder which is the weakest natural subsystem of Z 2 in which we can prove this theorem. This is the Main Question of reverse mathematics: Given a theorem τ of ordinary mathematics, what is the weakest natural subsystem S(τ) of Z 2 in which τ is provable? Suprisingly, S(τ) often turns out to be one of the five specific subsystems of Z 2 which we shall now list as S 1 =RCA 0, S 2 =WKL 0, S 3 =ACA 0, S 4 =ATR 0, S 5 = Π 1 1-CA 0, in order of increasing ability to accommodate ordinary mathematical practice. Our method for establishing results of the form S(τ) = S j, 2 j 5 is based on the empirical phenomenon that when the theorem is proved from the right axioms, the axioms can be proved from the theorem. More specifically, let τ be an ordinary mathematical theorem which is not provable in the weak base theory S 1 = RCA 0. Then very often, τ turns out to be equivalent to S j for some j = 2, 3, 4 or 5. This equivalence is provable in S i for some i < j, usually i = Brouwer s fixed point theorem for the unit square The unit square in R 2. R is the complete separable metric space Q where d : Q Q R is given by d(q, q ) = q q. In general, given n complete separable metric spaces A 1,..., A n, one defines the n-fold cartesian product A = A 1... A n = { a 1,..., a n : a i A i } 16

24 and d : A A R by d( a 1,..., a n, b 1,..., b n ) = d 1 (a 1, b 1 ) d n (a n, b n ) 2 and one can prove within RCA 0 that  is a complete separable space, that the points of  can be identified with the finite sequences x 1,... x n, with x i Âi, i n, and that under this identification, the metric on  is also given by d( x 1,..., x n, y 1,..., y n ) = d 1 (x 1, y 1 ) d n (x n, y n ) 2, see also [1] Example II.5.4. Hence, if the set E is given by the cartesian product Q Q, then Ê forms the separable Banach space R 2 under the identification (a 1, b 1 ) (a 2, b 2 ) = a 1 a b 1 b 2 2, and all relevant operations can within RCA 0 be extended to R 2. This means that the norm in R 2 = Ê is given by the standard Euclidean norm. Given x, y R 2, x and y represent real numbers, and we define the Σ 0 1 formulas ϕ 1 ( x, y ) x > 1 ; ϕ 2 ( x, y ) x < 0 ; ϕ 3 ( x, y ) y > 1 ; ϕ 4 ( x, y ) y < 0. Note that these formulas are indeed Σ 0 1, see Remark 2.8. Now suppose x, y = u, v as points in R 2. Under the given identification, this means we have x = u and y = v as points in R. Hence we have ϕ i ( x, y ) ϕ i ( u, v ) for i = 1, 2, 3, 4. Theorem 2.14 implies the existence of (codes for) the open sets U 1 = {( x, y ) R 2 : x > 1} ; U 2 = {( x, y ) R 2 : x < 0} ; U 3 = {( x, y ) R 2 : y > 1} ; U 4 = {( x, y ) R 2 : y < 0}. Using these codes, we can define (a code for) the open set U = U 1 U 2 U 3 U 4. Now we also have (a code for) the closed complement C = [0, 1] 2 of U. Hence, the unit square can be defined as a closed set within RCA Proving that Brouwer s fixed point theorem for the unit square is equivalent to WKL 0 over RCA 0. As an example, we shall hereby include Simpson s analysis of the reverse mathematics of Brouwer s fixed point theorem, given in [1] Section IV.7. The ordinary mathematical statement is as follows: 17

25 Theorem 3.1. (Brouwer s fixed point theorem). Let C be the convex hull of a nonempty finite set of points in R n, n N. Then every continuous function f : C C has a fixed point. However, we shall restrict the analysis to the case where C is the unit square in R 2, i.e., we shall perform the reverse mathematics of the following theorem: Theorem 3.2. (Brouwer s fixed point theorem for the unit square). Let C be the unit square in the separable Banach space R 2. Then for any continuous function f : C C, there is a point x [0, 1] 2 such that f(x) = x. We shall prove that Theorem 3.19 is equivalent to WKL 0 over RCA 0. Our methods will be comepletely based on [1] Section IV.7, but simplified to the two-dimensional case and with added details and illustrations. Definition 3.3. (affinely independent points). The following definition is made in RCA 0. Let n N, n 1. Suppose for k N, s 0,..., s k 1 and s k are points in the separable Banach space R n. Then these points are called affinely independent, if for all real numbers α 0,..., α k, we have that k k α i s i = 0 α i = 0 implies α i = 0 for all i = 0,..., k. i=0 Theorem 3.4. The following is provable in RCA 0. Suppose s 0,..., s k are affinely independent points in the separable Banach space R n, n 1. Then k n. Proof. W.l.o.g. we may assume k 0. Then for i = 1,..., k, we can define i=0 v i := s i s 0. It is easy to see that s i s j for all 0 i < j k, so that the v i s are all different nonzero vectors in R n. Suppose there exist real numbers α 1,..., α k, such that Defining this means that α 1 v α k v k = 0 R n. α 0 = k α i s i = 0 i=0 k α i, i=1 k α i = 0, so that α i = 0 for all i = 0,... k. Hence, v 1,..., v k are linearly independent vectors in R n. Since RCA 0 is strong enough to develop the basics of real linear algebra, including Gaussian elimination ([1] Exercise II.4.11), we may conclude that k n. i=0 18

26 Definition 3.5. (convex hull). The following definition is made in RCA 0. Suppose that n N, n 1, and s 0,..., s k R n. We define { } k L = q 0,..., q k Q k+1 : q 0,..., q k [0, 1] q i = 1, i=0 and the sequence of reals k d : L L R : ( q 0,..., q k, p 0,..., p k ) q i s i i=0 k p i s i. i=0 Then L together with d forms a (code for a) complete separable metric space L, called the convex hull of the points s 0,..., s n. Theorem 3.6. The following is provable in RCA 0. Suppose that n N, n 1, and s 0,..., s n are affinely independent points in R n. Then the points in the convex hull L of s 0,..., s n are in a one-to-one correspondence with the points in the informal set ( )} n n S = {x R n : α 0,..., α n [0, 1] R x = α i s i α i = 1, given by i=0 i=0 ρ k = ρ k0,..., ρ kn : k N, ρ k L k ρ L Moreover, this correspondence preserves distances. n lim ρ ki s i S. k i=0 Proof. It is easy to see that this correspondence will preserve distances, since for sequences ρ k = ρ k0,..., ρ kn : k N, ρ k L, µ k = µ k0,..., µ kn : k N, µ k L, we have (considering all terms exist), ) d (lim ρ k, lim µ k = lim d(ρ k, µ k ) k k k Now suppose n = lim ρ ki s i k i=0 n = lim k i=0 ρ ki s i n µ ki s i i=0 n lim µ ki s i. k i=0 ρ k = ρ k0,..., ρ kn : k N, ρ k L k ρ L. L Then ρ is a sequence q k = q k0,..., q kn : k N, q k L such that d(q k, q l ) 2 l for all k, l N, with k l. Hence, n i=0 q kis i : k N converges in R n to some point x by Theorem As in the proof of Theorem 3.4, we know that the points v i := s i s 0, with 19

27 1 < i n, are linearly independent in R n, so that by Gaussian elimination there exists a unique n-tuple α 1,..., α n R n, such that x s 0 = n α i v i. i=1 for k N, we define x k := Since n i=0 q ki = 1 for all k N, we have n q ki s i. i=0 x k s 0 = n q ki (s i s 0 ) = i=1 n q ki v i, i=1 so that 0 = lim k (x x k ) = lim k ((x s 0 ) (x k s 0 )) n = lim (α i q ki )v i = k i=1 n i=0 (α i lim k q ki ) v i, which implies that lim k q ki = α i for all i = 1,..., n. Since q ki [0, 1] for all k N, i n, clearly α i [0, 1] R for i = 1,..., n. Moreover, defining α 0 := 1 n α i, i=1 we have α 0 = 1 n n lim q ki = 1 lim q ki = lim (1 k k k i=1 i=1 ) n q ki = lim q 0k. k i=1 so that we also have α 0 [0, 1] R, and we find x = n α i s i i=0 n α i = 1. i=0 Defining x k := n ρ ki s i, i=0 then since lim k x k x k = lim k d(q k, ρ k ) lim k d(q k, ρ) + lim k d(ρ, ρ k ) = 0, we find in an analogous way that lim k q ki = lim k ρ ki for all i n. 20

28 Conversely, let α 0,..., α n [0, 1] R be such that n i=0 α i = 1. Suppose first 0 α i 1 for all i n. If α i = q in : n N, q in Q, then there exists N N, such that whenever k N, k N, we have q ik, ]0, 1[ Q, for all i = 1,..., n, and 1 n i=1 q ik ]0, 1[ Q. Now for every k N, we define n ρ k := 1 q i,n+k, q 1,N+k,..., q n,n+k L x k := then for all k, l N, k l, ( 1 i=1 ) n q i,n+k s 0 + i=1 d(ρ k, ρ l ) = x k x l n = (q i,n+k q i,n+l )s i + 2 i=1 n q i,n+k s i, i=1 n (q i,n+l q i,n+k )s 0 i=1 n q i,n+k q i,n+l max{ s 0,..., s k } i=1 n 2 (N+l)+1 max{ s 0,..., s n } l 0, so that the sequence ρ k : k N converges to some ρ L by Theorem Now if α l = 1 for some l {0,..., n}, we clearly have 0,..., 0, }{{} 1, 0,..., 0 L. l-th place Finally, suppose α l = 0 for some l {0,..., k}. W.l.o.g., we may assume α 0 =... = α j = 0 for j < k, and α j+1,..., α k ]0, 1[ R. Defining n ρ k := 0,..., 0, 1 q i,n+k, q j+1,n+k,..., q n,n+k L, i=j+1 for N large enough, in an analogous way as in the first case, one concludes that there exists ρ L such that ρ k k ρ. Definition 3.7. (triangle, [1] Definition IV.7.1). The following definition is made in RCA 0. A triangle S in R 2 is the convex hull of three affinely independent points s 0, s 1, s 2 R 2, called the vertices of S. Definition 3.8. (diameter) The following definition is made in RCA 0. Let S be a triangle in R 2, determined by the vertices s 0, s 1, s 2 R 2. The diameter of S is then defined as Note that diam(s) > 0. diam(s) := max{ s 0 s 1, s 0 s 2, s 1 s 2 }. 21

29 Theorem 3.9. The following is provable in RCA 0. Let S be a triangle in R 2. Then S is a compact metric space. Proof. Suppose S is determined by the vertices s 0, s 1, s 2 R 2. Let N N be any natural number such that 2 N > max{ s 0, s 1, s 2 }, and take j N. By Σ 0 0 comprehension, we may define the set A j := { q 0, q 1, q 2 Q 3 : i 2 j+n+3, k 2 j+n+3 i ( q0 = i 2 j N 3 q 1 = k 2 j N 3 q 2 = 1 (i + k)2 j N 3)}. For all n N, n 2 j+n+3 m=0 (2 j+n+3 m) = A j, we recursively define min N A j if n = 0, x nj = min N {x A j : k < n(x x nj )} if 0 < n 2 j+n+3 m=0 (2 j+n+3 m), i.e., x nj : n 2 j+n+3 m=0 (2 j+n+3 m) is a finite enumeration of all elements in A j, j N. We claim that the infinite sequence of finite sequences 2 j+n+3 ( x nj : n 2 j+n+3 m ) : j N, m=0 attests to the definition of compactness. Hence, suppose x S and j N. By Theorem 3.6, we may represent x as x α 0 s 0 + α 1 s 1 + α 2 s 2 R 2, with α 0, α 1, α 2 [0, 1] R and α 0 + α 1 + α 2 = 1, and q 0, q 1, q 2 A j as q 0, q 1, q 2 i 2 j N 3 s 0 + k 2 j N 3 s (i + k)2 j N 3 s 2 R 2, for some i 2 j+n+3, k 2 j+n+3 i, and we have d(x, q 0, q 1, q 2 ) = ( α 0 i 2 j N 3) s 0 + ( α 1 k 2 j N 3) s 1 + (( i 2 j N 3 α 1 ) + ( k 2 j N 3 α 2 )) s2 2 ( α0 i 2 j N 3 + α1 k 2 j N 3 ) max{ s 0, s 1, s 2 } < 2 N+1 ( α0 i 2 j N 3 + α1 k 2 j N 3 ). Since α 0, α 1 [0, 1] R, there exist i, k 2 j+n+3, such that α0 i 2 j N 3 2 j N 3 α1 k 2 j N 3 2 j N 3. If k 2 j+n+3 i, then we are finished, since 2 N+1 (2 2 j N 3 ) = 2 j 1 < 2 j. 22

30 Hence, suppose k > 2 j+n+3 i. Then we also must have i > 0. Note that Since α 0 (i 1)2 j N 3 α0 (i 1) 2 j N j N 3 = 2 j N 2. α 1 1 α 0 1 (i 1)2 j N 3 = ( 2 j+n+3 (i 1) ) 2 j N 3, there exists k 2 j+n+3 (i 1), such that α 1 k 2 j N 3 2 j N 3. Moreover, we have 2 N+1 ( α 0 (i 1) 2 j N 3 + α 1 k 2 j N 3 ) 2 N+1 ( 2 j N j N 3) Hence, we conclude that S is indeed a compact metric space. = 2 j j 2 < 2 2 j 1 = 2 j. Definition The following definition is made in RCA 0. Suppose S is a triangle in R 2, determined by the vertices s 0, s 1, s 2 R 2. A face of S is the convex hull of either one of the vertices, the convex hull of one of the combinations of two different vertices, or S itself. For any point x S, the carrier of x is the smallest face of S which contains x. Definition (triangular subdivision, [1] Definition IV.7.2). Within RCA 0, let S be a triangle in R 2. A triangular subdivision of S is a finite set of (codes for) triangles S 0,..., S m such that S = S 0... S m and, for all i < j m, S i S j is either empty or a common face of S i and S j. Definition (admissible labeling, [1] Definition IV.7.3). Within RCA 0, let S be a triangle, and let P be a finite set of points in S which includes the vertices of S. An admissible labeling of P is a mapping from P into {0, 1, 2} such that the vertices of S are mapped to the full set of labels {0, 1, 2}; for every x P, the label of x is the same as the label of one of the vertices of the carrier of x. Lemma (Sperner s lemma, [1] Lemma IV.7.4). The following is provable in RCA 0. Let S be a triangle in R 2, and let S 0,..., S m be a triangular subdivision of S. Suppose that the vertices of S 0,..., S m are admissibly labeled. Then for some i m, the vertices of S i are mapped to the full set of labels {0,..., k}. Proof. See [1] Section IV.7 for the proof of a more general version of Sperner s lemma for simplices in R n. The proof consists of elementary combinatorial reasoning which is straightforwardly formalized in RCA 0, and can be easily simplified to the two-dimensional case. See also Figure 6 for an illustration. 23

31 Figure 6: Triangular subdivision of the triangle S in R 2, determined by the vertices s 0, s 1, s 2 R 2, in nine triangles. The vertices of S 0,..., S 9 are admissibly labeled, and the vertices of S 6 are mapped to the full set of labels {0, 1, 2}. Before we present our version of Brouwer s fixed point theorem, we note that RCA 0 is sufficient to proof the fixed point theorem for the unit interval. Theorem The following is provable in RCA 0. It holds for every continuous function f : [0, 1] [0, 1], that there is a point x [0, 1] such that f(x) = x. Proof. See the proof of [2] Theorem 5.1.(a). The proof is similar to the proof of the intermediate value theorem in RCA 0, see [1] Theorem II.6.6 for more details. We shall now begin with the proof of our version of Brouwer s fixed point theorem. Definition (modulus of uniform continuity, [1] Definition IV.2.1). The following definition is made in RCA 0. Let X and Y be complete separable metric spaces, and let F be a continuous function from X into Y. A modulus of uniform continuity for F is a function h : N N such that for all n N and all x and y in X, if F (x) and F (y) are defined and d(x, y) < 2 h(n), then d(f (x), F (y)) < 2 n. We call a continuous function F uniformly continuous, if F has a modulus of uniform continuity. The proofs of the following two theorems make use of the Heine/Borel property which essentially states that any covering by open sets of a compact metric space has a finite subcovering. This assertion is provable in WKL 0. 24

32 Theorem The following is provable in WKL 0. Let X be a compact metric space. Let C be a closed set in X, and let F be a continuous function from C into a complete separable metric space Y. Then F has a modulus of uniform continuity on C. If in addition X = C and Y = R, then F attains a maximum value. Proof. See [1] Theorem IV.2.2. Theorem The following is provable in WKL 0. Let f be a continuous real-valued function on a nonempty closed set C in a compact metric space. If α = sup x C f(x) exists, then f(x 0 ) = α for some x 0 C. Proof. Actually, it holds that this assertion is equivalent to WKL 0 over RCA 0 by [1] Exercise IV It s easy to see that this theorem also implies the same statement for the infimum. This fact, Sperner s Lemma, the property that a triangle can be seen as a compact metric space by Theorem 3.9, the definition of the diameter of a triangle and the definition of a triangular subdivision, can be used for the proof of the following lemma, stating a version of Brouwer s fixed point theorem for a triangle in R 2. Lemma ([1] Lemma IV.7.5) The following is provable in WKL 0. Let S be a triangle in R 2. Then every continuous function f : S S has a fixed point, i.e., f(x) = x for some x S. Proof. Again, see [1] Section IV.7 for the proof that any continuous function from a general simplex in R n to itself has a fixed point. The proof for the two-dimensional case goes exactly the same. By now, we are ready to show that weak König s lemma implies Brouwer s fixed point theorem for the unit square. The proof is based on the proof of the general version of Brouwer s fixed point theorem in [1] Section IV.7. Theorem (Brouwer s fixed point theorem for the unit square). The following is provable in WKL 0. For any continuous function f : [0, 1] 2 [0, 1] 2, there is a point x [0, 1] 2 such that f(x) = x. Proof. If we can find a triangle S in R 2 such that [0, 1] 2 S, and that [0, 1] 2 is a retract of S, i.e., there is a continuous function r : S [0, 1] 2 such that r(x) = x for all x [0, 1] 2, then we are done. Since, given a continuous function f : [0, 1] 2 [0, 1] 2, we can consider g : S [0, 1] 2 given by g(x) = f(r(x)). By Lemma 3.18, let x S be such that g(x) = x. Then x [0, 1] 2, hence r(x) = x, hence f(x) = f(r(x)) = g(x) = x, i.e., x is a fixed point for f. Consider the triangle S defined by the vertices (0, 0), (0, 2), (2, 0) R 2. It holds that [0, 1] 2 S and (1, 1) S, see also Figure 7. Now we can easily define a continuous function r with the right properties. Let for (a, b) S, r((a, b)) = (a, b) if (a, b) [0, 1] 2, r((a, b)) = (a, 1) if b > 1 and r((a, b)) = (1, b) if a > 1. (We will not perform the coding of this function in this proof. In Subsection 4.2.3, we shall prove the existence of a code for quite some of such metric projections, and one should be sure that this is possible in RCA 0.) 25

33 Figure 7: The unit square in a 2-simplex. This theorem can be generalized as follows. Theorem (Brouwer s fixed point theorem in WKL 0 ). The following is provable in WKL 0. Let C be the convex hull of a nonempty finite set of points in R n, n N. Then every continuous function f : C C has a fixed point. Proof. The proof is essentially the same as in the proof of Theorem However, it is a lot harder to find a right retraction in the general case. One has to make extensive use of linear algebra which can be developed within RCA 0. We refer the interested reader to [1] Theorem IV.7.6. Also, in [2] Theorem 5.3 a more detailed proof is presented for the case where the vertices are points in Q n, explicitly demonstrating how one makes use of linear algebra. We shall now obtain a reversal showing that weak König s lemma is needed to prove Brouwer s theorem for the unit square. The proof comes directly from [1] Section IV.7, but with added details on the singular covering of the rational interval [0, 1]. The proof is as an example how one explicitly shows the equivalence of an ordinary mathematical statement to WKL 0 over RCA 0. Note that the following theorem also shows that the statement given in Theorem 3.20 implies WKL 0 over RCA 0. This is essentially the method Simpson uses to prove that the general version of Brouwer s fixed point theorem is equivalent to WKL 0 over RCA 0. Theorem ([1] Theorem IV.7.7). The following are pairwise equivalent over RCA 0. 26

34 1. Weak König s lemma. 2. For any continuous function f : [0, 1] 2 [0, 1] 2, there is a point x [0, 1] 2 such that f(x) = x. Proof. We need to prove 2. 1., the other direction is proved by Theorem Assume that 1. does not hold. Then there exists an infinite tree T 2 <N with no infinite path. We will now use T to construct a retraction f of C = [0, 1] 2 onto the four sides C of this square. If such an f is constructed, 2. does not hold. For if f is the rotation of 90 about the point (0.5, 0.5), r f is a continuous function from [0, 1] 2 into itself which has no fixed point. Moreover, we will show that T implies the existence of a singular covering of [0, 1], i.e., a covering of [0, 1] by an infinite sequence of closed rational intervals I n = [a n, b n ], a n, b n Q, a n < b n, n N, such that for all m n, I m I n consists of at most one point. This singular covering will allow us to construct the desired function f. Define rational intervals [c σ, d σ ], σ 2 <N, by putting c = 0, d = 1, c σ 0 = c σ, c σ 1 = d σ 0 = (c σ + d σ )/2, and d σ 1 = d σ, see Figure 8. Let σ n : n N be an enumeration 8 of the set which clearly exists in RCA 0, and put T = {σ 2 <N : σ / T σ [lh(σ) 1] T }, I n := [a n, b n ] := [c σn, d σn ] = {q Q : c σn q d σn }. We will show that for m n, I m I n consists of at most one point. It is clear that for m n, if I m I n consists of more than one point, then one is a sub-interval of the other. W.l.o.g. suppose m n and I m I n. It should be clear that this implies that σ n is an initial segment of σ m [lh(σ m ) 1]. But σ m T, hence σ m [lh(σ m ) 1] T and T is closed under initial segments, hence σ n T, which contradicts σ n T. So for m n, I m and I n have at most one point in common. Now take any real number x between 0 and 1. We need to prove that x is contained in one of the intervals I n, n N. For k N, we let σk x be the leftmost sequence in 2<N of length k, such that x [ ] c σ x k, d σ x k. Then one easily sees that for k k, σk x is an initial segment of σk x. Defining, g : N N : k σk(k) x, g defines a path through 2 <N. Hence, σk x must be contained in T for some k N, otherwise we would have a path through T, a contradiction. Hence, we conclude that I n : n N has the desired properties of a singular covering of [0, 1]. Now for each n N we put ( ) ( ) A n = I m I n I n I m We note that C = n N A n. m n m n 8 The existence of this enumeration is provable in RCA 0, cf. [1] Section II.3. 27

35 Our retraction map f : C C will be defined in stages. We begin by defining f on C to be the identity map. At stage n, we assume that f has already been defined on C and on A m for all m < n, and we define f on A n. Let P n0,..., P nkn be the connected components of A n. We can easily observe that each P ni has at least one free side e ni on which P ni does not intersect m<n A m or any side of C, except for its endpoints, see Figure 9. Let e ni be e ni minus its endpoints. We define f on P ni to consist of a continuous retraction of P ni on P ni \e ni, followed by a continuous mapping of P ni \e i into C which is compatible with the part of f that has already been defined. This defines f on A n = i k n P ni. It can be shown that the above construction gives rise to a continuous function f defined on all C = n N A n. 9 Clearly f is a retraction of C onto C, which completes the proof. Figure 8: Partitioning the unit interval using the infinite tree 2 <N in RCA 0. Figure 9: Each P 2i, i = 0,..., 4, of the connected components of A 2 has at least one free side e 2i on which it does not intersect A 0 A 1 or any side of C, except for its endpoints. 9 We will ommit the details on the code of this function. 28