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Minkowski's Linear Forms Theorem in Elementary Function Arithmetic

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Minkowski's Linear Forms Theorem in Elementary Function Arithmetic Minkowski's Linear Forms Theorem in Elementary Function Arithmetic Author: Greg Knapp Submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics, Applied Mathematics, and Statistics Case Western Reserve University August, 2017 1 Case Western Reserve University Department of Mathematics, Applied Mathematics, and Statistics We hereby approve the thesis of Greg Knapp Candidate for the degree of Master of Science Committee Chair Colin McLarty Committee Member Mark Meckes Committee Member David Singer Committee Member Elisabeth Werner Date of Defense: 27 April, 2017 2 Contents 1. Introduction 4 2. Elementary Function Arithmetic 4 3. Finite Set Theory and Basic Mathematics in EFA 8 4. Convex Geometry 10 4.1. Weyl's Theorem . 10 4.2. Polytopes and Their Representations . 16 4.2.1. Inequality Representation . 16 4.2.2. Full Information Representation . 19 4.2.3. Intersecting Polytopes . 23 4.3. Triangulations and Volume . 23 4.3.1. Triangulations and Volumes of Simplices . 23 4.3.2. Triangulations and Volumes of Polytopes . 31 4.3.3. Triangulation and Volume of the Cube . 39 5. Minkowski's Linear Forms Theorem 43 6. Acknowledgments 50 References 51 3 Minkowski's Linear Forms Theorem in Elementary Function Arithmetic Abstract by Greg Knapp A classical question in formal logic is \how much mathematics do we need to know in order to prove a given theorem?" Of particular interest is Harvey Friedman's grand conjecture: that every known mathematical theorem involving only finitary mathematical objects can be proven from Elementary Function Arithmetic (EFA)|a fragment of Peano Arithmetic. If Friedman's conjecture is correct, this would imply that Fermat's Last Theorem is derivable from the axioms of EFA. The vast task of proving Fermat's Last Theorem from the axioms of EFA seems to require certain theorems on convex polytopes and an important corollary: Minkowski's Linear Forms Theorem. I show that certain theorems of convex geometry| e.g. representation of polytopes both in vertex form and as the intersection of half-spaces, monotonicity of volume, the existence of a separating plane between disjoint polytopes, and Minkowski's Linear Forms Theorem|can be both interpreted and derived in EFA, assuming the well-definedness of volume of a convex polytope along with one other technical lemma. 4 1. Introduction One of the motivating questions of mathematical logic is \how much math do we really need to know in order to prove a certain theorem?" Do we really need Zorn's Lemma to prove the existence of algebraic closures? Do we really need Wiles' high-tech mathematical machinery in order to prove Fermat's Last Theorem? Simpson, in Subsystems of Second-Order Arith- metic, provides a good test for such questions: \if a mathematical theorem is proved from appropriately weak...axioms, then the axioms will be logically equivalent to the theorem" [11]. This paper seeks to shed some light on this question as it pertains to Minkowski's Linear Forms Theorem (MLFT). In particular, I approach the problem of showing that MLFT fol- lows from from the axioms of Elementary Function Arithmetic (EFA)|a weaker version of Peano Arithmetic. While I do not derive MLFT from the axioms of EFA, I establish the machinery necessary to solve the problem and I reduce the problem of proving MLFT to the smaller problems of the well-definedness of volume and a technical lemma concerning triangulations. The proposed proof of MLFT requires the basics of convex geometry on polytopes and I show that this geometry, together with notions like intersection and volume, is representable in EFA. I further show standard geometric facts, like the Affine Weyl The- orem, the invariance of the volume of a polytope under dissection by a plane, the existence of a separating plane between disjoint polytopes, and I compute the volume of the cube. While I do not show that the axioms of EFA follow from MLFT (because they most likely do not), this paper shows that the \amount" of math required to prove MLFT is probably no stronger than what EFA provides. 2. Elementary Function Arithmetic In order to introduce Elementary Function Arithmetic, I first introduce Peano Arithmetic, then explain in what sense EFA is weaker than PA. Peano Arithmetic is an axiom scheme in 5 first-order logic with the following language and axioms (as given in H´ajekand Pudl´ak)[6]. Language: • The unary function symbol S • Binary function symbols +; ×, ^ • A binary predicate, 6 • A constant symbol, 0 The unary function S, which is called the \successor" function, can be informally thought of S(x) = x + 1 in the natural number sense of the + operation. The remainder of the predicate and function symbols are meant to reflect the regular natural number predicates and operations and are defined by the axioms below. Axioms: (1) 8(x)[S(x) 6= 0] (2) 8(x; y)[S(x) = S(y) ! x = y] (3) 8(x)[x + 0 = x] (4) 8(x; y)[x + S(y) = S(x + y)] (5) 8(x)[x × 0 = 0] (6) 8(x; y)[x × S(y) = (x × y) + x] (7) 8(x)[x0 = 1] (8) 8(x; y)[xS(y) = x × xy] (9) 8(x; y) x 6 y $ 9(z)[z + x = y] plus the additional axiom \scheme" allowing for mathematical induction: for each formula ' in the language of PA, we have the following axiom (called the axiom of induction): (10') '(0) & 8(x)['(x) ! '(S(x))] ! 8(x)['(x)] However, the axiom scheme (10) places no bounds on the \complexity" of the formula '. PA can perform induction on whatever formulas it would like, thus placing many of the statements that PA proves far beyond our understanding. In order to rectify this and bring 6 arithmetic closer to the realm of the understandable, EFA weakens the axiom scheme (10). In order to understand how exactly EFA weakens (10), I present the notion of bounded quantifiers. I use the notation 8(x 6 y)['] to abbreviate 8(x)[x 6 y ! '] and I use the notation 9(x 6 y)['] to abbreviate 9(x)[x 6 y & ']. We say that a formula ' is bounded if all of its quantifiers are bounded (formulas without quantifiers are said to be bounded, as well). Bounded quantifiers, in some sense, give a notion of what sort of claims are computable. For example, if I said 8(x)9(y)[y > x & y is prime], you would know that there are infinitely many primes. But, if I said 8(x)9(y 6 x!)[y > x & y is prime], then you would not only know that there are infinitely many primes, but you would also know how far you had to look to find a prime. You could write a computer program which, given input x, would check all of the numbers between x and x! for prime numbers (of course, the upper bound can be improved, but that's not the important detail here) and you would know that the program would finish and find some prime number. Of course, if the goal is to maintain an \understandable" theory of arithmetic by only allowing induction on certain formulas, then we should put certain restrictions of the types of upper bounds that we allow. Elementary Function Arithmetic allows induction only on formulas which are bounded and where each quantifier’s bound involves only addition, multiplication, and finitely iterated exponentiation. Intuitively, this means that formulas on which EFA cannot perform induction are those with some quantifier which isn't bounded at all, or some quantifier whose bound grows at least as fast as the \superexponential" function, which is x : : : the variably-iterated exponential function: x "" y := xx where the exponentiation occurs y times. 7 2x For example, using the fact that x! 6 2 EFA can induct on the following statement: 2x 9(y 6 2 )[y > x & y is prime] but EFA cannot induct on either of the following statements • 9(y)[y > x & y is prime] • 9(y 6 2 "" x)[y > x & y is prime] Throughout the paper, I make liberal use of the following facts. If f(x1; : : : ; xj) and g(y1; : : : ; yk) are finitely iterated exponential functions in the variables xi and yi, then f + g, g f × g, and f are also finitely iterated exponentials in the variables xi and yi. Further, if f(x1; : : : ; xj) is a finitely iterated exponential and for each 1 6 i 6 j, gi(yi1; : : : ; yiki ) is a finitely iterated exponential funtion, then so is f(g1; : : : ; gj). These facts are primarily useful for keeping notation managable (for example, if I show that the number of faces of a convex polytope is bounded by a finitely iterated exponential function in terms of the number of vertices and I show that the number of faces of the intersection of two convex polytopes is bounded by a finitely iterated exponential in terms of the number of faces of the two polytopes, then it's true that the number of faces of the intersection is bounded by a finitely iterated exponential function in terms of the number of vertices, but actually writing down that function may be a typographical challenge). The bulk of the paper is concerned with creating new predicates and showing that we can do induction on formulas which include those predicates. For example, if we were to de- fine the binary predicate divides to be x divides y if and only if there exists a z such that x × z = y, the question arises, could EFA induct on the statement 9(x 6 y)[x divides y] in order to show that 8(y)9(x 6 y)[x divides y]? The answer is no, because the statement 9(x 6 y)[x divides y] is really an abbreviation for 9(x 6 y)9(z)[x × z = y], which EFA cannot induct on because it has an unbounded quantifier.
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