Chapter 17 - Metric Geometry

Cantor defined the cardinal number of a set M to be “the general concept which, by means of our active faculty of thought, arises from the set M when we make abstraction of the nature of its various elements m and of the order in which they are given.” He denoted this cardinal number by \( \overline{\overline M} \). The two bars indicate the two levels of abstraction needed to produce the cardinal number from M. With only one level of abstraction, that is, by only abstracting of the nature of its various elements, we obtain the ordinal number \( \overline M \). Cantor’s definition of cardinal number is clearly not an operational one. Indeed Cantor’s words suggest that cardinal numbers are psychological entities rather than mathematical objects.

Sum and intersection of linear subspaces in a vector space over a field are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the finite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions of them-dimensional complex column vector space 𝔺mwith respect to a pair of given linear subspaces 𝒨 and 𝒩 and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and su˚cient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and su˚cient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.

In this paper we develop certain methods of proof in Quine's set theory NF which have no counterparts elsewhere. These ideas were first used by Specker [5] in his disproof of the Axiom of Choice in NF. They depend on the properties of two related operations, T(n) on cardinal numbers and U(α) on ordinal numbers, which are defined by the equationsfor each set x and well ordering R. (Here and below we use Rosser's notation [3].) The definitions insure that the formulas T(x) = y and U(x) = y are stratified when y is assigned a type one higher than x. The importance of T and U stems from the following facts: (i) each of T and U is a 1-1, order preserving operation from its domain onto a proper initial section of its domain; (ii) Tand U commute with most of the standard operations on cardinal and ordinal numbers.These basic facts are discussed in §1. In §2 we prove in NF that the exponential function 2n is not 1-1. Indeed, there exist cardinal numbers m and n which satisfyIn §3 we prove the following technical result, which has many important applications. Suppose f is an increasing function from an initial segment S of the set NO of ordinal numbers into NO and that f commutes with U.

Set theory is a rich and beautiful subject whose fundamental concepts permeate virtually every branch of mathematics. One could say that set theory is a unifying theory for mathematics, since nearly all mathematical concepts and results can be formalized within set theory. This textbook is meant for an upper undergraduate course in set theory. In this text, the fundamentals of abstract sets, including relations, functions, the natural numbers, order, cardinality, transfinite recursion, the axiom of choice, ordinal numbers, and cardinal numbers, are developed within the framework of axiomatic set theory. The reader will need to be comfortable reading and writing mathematical proofs. The proofs in this textbook are rigorous, clear, and complete, while remaining accessible to undergraduates who are new to upper-level mathematics. Exercises are included at the end of each section in a chapter, with useful suggestions for the more challenging exercises.

Let “the hypothesis of similarity” be the hypothesis that all infinite classes are similar (have the same power or cardinal number). It will be shown that the system of logic of Whitehead and Russell's Principia mathematica (without the axiom of reducibility) is consistent when to its axioms are added all the following:(1). The hypothesis of similarity.(2). The axiom of infinity.(3). The axiom of choice.(4). The contradictory of the axiom of reducibility.If (2) – (4) but not (1) are employed in Principia, it must therefore be the case that Cantor's theorem is not deducible, since it contradicts (1). If (2) and (3) but not (1) and (4) are employed, then a proof of Cantor's theorem must require the use of some sort of reducibility principle.The first step in the required consistency proof is to modify in certain respects the system S of a previous paper. Immediately after definition 3.5 the following definition is to be added:3.5.1. If a is of the form (b,c), where x is an S-variable and where b and c are S-constants neither of which is of higher order than that of x, then a is an S-proposition.

1. Because of the nonconstructive nature of the axiom of choice there has been much interest in how much of it is needed for various theories. In the case of the theory of vector spaces it appears that one would want to save at least the following two consequences of AC: (1) Every vector space has a basis and (2) Any two bases of a given vector space are equipollent. The question immediately arises: Have we saved the whole axiom of choice; namely is the axiom of choice a logical consequence of (1) and (2) and the other axioms of some appropriate set theory? This question remains open and the author conjectures a negative solution. However, we are able to show that a reasonable strengthening of (1), which is also a consequence of AC, implies AC, namely the universal generalization of Proposition 2 of [1], which we will call the downward basis principle:

Let # be a probability measure on the Bore1 field Z of the real line IR and let P = # | 1 7 4 be the product measure defined on the Borel field 2 ; ~ 1 7 6 1 7 4 1 7 4 -of the topological vector space I R ~ = I R | 1 7 4 .-. . If a = ( a , ) , e I is an element of IR ~, let P~ be the measure P translated by a i.e. P,,(A)=P(A+a) for A s Z ~. In the present paper we study the set E(P)= {aslR~~ where P ~ P , signifies that P ( A ) = 0 iff P,(A)=0. Previously E(P) had been studied by, amongst others, Dudley [4], Feldman [5], Shepp [9]. Our main result (w 3) completes theirs by showing that E(P) contains a Mazur-Orlicz vector subspace 1 ~ (for definitions see w 2) whenever it contains some 1-dimensional vector subspace and that whenever E(P) is a vector subspace then E(P)= l ~ We also give an explicit expression for the function 0. A counter-example based on one due to Dudley given in the appendix shows that E(P) need not always be a vector subspace of IR ~~ However, in w 4, we give a simple sufficient condition for E(P) to be a subspace and exhibit wide classes of examples of E(P)= l ~ for various functions 0. Our results were announced previously in [2] without proofs.

This chapter is divided into three parts. The first and most substantial of these is devoted to symplectic vector spaces. Using methods of exterior algebra which are due to Elie Cartan, we study the rank of an exterior 2-form on a real vector space as well as that of the 2-form induced on a vector subspace. By means of associated canonical bases we show that there is only one model of a symplectic form and that vector subspaces may be classified by the rank of the induced form. In mechanics one uses mainly isotropic and coisotropic subspaces, particularly Lagrangian subspaces which will be discussed here in some detail. In particular we study the reduction of a symplectic vector space.

In this paper the Riesz theory for compact linear operators in a normed vector space is considered from the point of view of how far the axiom of choice is involved. Special attention is drawn to the theorem, by which for the operator $I - A,A$ being compact, the index vanishes and the nullspace has a closed algebraic complement. It is shown that this can be proved without making use of the axiom of choice.

The study of partition relations for cardinal numbers introduced by Erdös and his school in the 1950's has, for the past several years, had a profound impact in logic. Unfortunately, quite early in their development, it was noticed by Rado [1] that the potentially most fruitful class of such relations, infinite exponent partition relations, were always in contradiction with the axiom of choice (AC). As a result, such relations were overlooked. This turned out to be a mistake; for, as has been noticed recently, a close study of infinite exponent partition relations is both interesting and rewarding. For example, there are weakened versions of such relations which are provable in ZF and which have valuable applications in recursion theory and set theory. In addition, the pure theory of these relations, like that of the axiom of determinateness, is fruitful as well as elegant. For more background here one should refer to [3].At any rate, with a more detailed look at infinite exponent partition relations came a more refined version of Rado's original theorem. Specifically, Rado used the full axiom of choice to carefully construct partitions to violate any desired relation—a more sophisticated look at the actual theory of such relations indicated how one could put together some desired partitions using only well-ordered choice [3].The distinction between well-ordered choice and full choice is by no means vacuous in this context. For Mathias has shown [4] that the simplest infinite exponent partition relation, ω → (ω)ω, is consistent with countable choice (well-ordered choice of length ℵ0) and, in fact, is consistent with dependent choice.

It is a well known consequence of the axiom of choice that forming unions of directed families can be reduced to (repeatedly) forming unions of chains. The most general formulation of this fact seems to be P. M. Cohn's [1, p. 33]: in a partially ordered set, if every wellordered subset has a least upper bound, then every directed subset has a least upper bound. This note is mainly concerned with proving Cohn's result without the axiom of choice, but assuming the directed subset has an ordinal cardinal number. (Cohn's use of the axiom goes well beyond that.) Incidentally, well-ordered sets of the same size suffice.

Let Σ be some standard set theory (Eg. Zermelo Fraenkel or Von Neumann-Bernays-Godel) which does not contain the axiom of choice. Using Σ as the underlying set theory, we shall study operations on infinite cardinals, closely related to exponentiation, and compare the results with known results about exponentiation.

The mathematical concept of infinity, in the sense of Cantor, is rather far from applied mathematics and statistics. These fields can be linked. We comment on the properties of infinite numbers and relate them to some operations with random variables. The existence of statistical parametric models can be studied in terms of cardinal numbers. Some probabilistic interpretations of Gödel’s theorem, Turing’s halting problem, and the Banach-Tarski paradox are commented upon, as well as the axiom of choice and the continuum hypothesis. We use a basic but sufficient mathematical level.

In this chapter, we will explain the construction of linear block codes. Since linear block codes are vector subspaces, we advise to the reader to study Chap. 1 before proceeding to Chap. 2. Without the knowledge of basic concepts of linear algebra, such as groups, fields, vector spaces, and subspaces, it is difficult to comprehend the philosophy behind the concept of linear block codes. We advise the reader to study the section related to the philosophy of channel encoding before studying the sections related to the encoding and decoding operations. We explain the purpose of channel encoding in Sect. 2.5. After introducing the construction of linear block codes, we explain the encoding operation and give some fundamental definitions such as minimum distance, generator matrix, and parity check matrix. In sequel, we explain equivalent codes, error detection and correction capability of linear block codes, and Hamming spheres.

A classical problem in the literature seeks the minimal number of proper subgroups whose union is a given finite group. A different question, with applications to error-correcting codes and graph colorings, involves covering vector spaces over finite fields by (minimally many) proper subspaces. In this note we cover R-modules by proper submodules for commutative rings R, thereby subsuming and recovering both cases above. Specifically, we study the smallest cardinal number possibly infinite, such that a given R-module is a union of -many proper submodules. (1) We completely characterize when is a finite cardinal; this parallels for modules a 1954 result of Neumann. (2) We also compute the covering (cardinal) numbers of finitely generated modules over quasi-local rings and PIDs, recovering past results for vector spaces and abelian groups respectively. (3) As a variant, we compute the covering number of an arbitrary direct sum of cyclic monoids. Our proofs are self-contained.