The axiom of choice and the paradoxes of the sphere.
Cargill, David Milton
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The Axiom of Choice is stated in the following form: For every set Z whose elements are sets A, non-empty and mutually disjoint, there exists at least one set B having one and only one element from each of the sets A belonging to Z. Examples are given to show the use of the Axiom of Choice and also to show when it is not needed. Two other fundamental terms are defined, namely "congruence" and "equivalence by finite decomposition", and examples are given. Congruence is defined as follows: The sets of points A and B are congruent: A B, if there exists a function f, which transforms A into B in a one-to-one manner such that if a1 and a2 are two arbitrary points of the set A, then d(a1, a2)=d[f(a1), f(a2)]; d(a, b) is a real number called the distance between the points a and b. The following definition of equivalence by finite decomposition is given: Two sets of points, A and B are equivalent by finite decomposition, Af=B, provided sets A1 , A2, ..., An and B1, B2, ..., Bn exist with the following properties: (1) A=A1+A2+...+An B=B1+B2+...+Bn (2) Aj • Ak=Bj • Bk=0 1 ≤ j < k ≤ n (3) Aj≅Bj 1 ≤ j ≤ n An historic measure problem is discussed briefly. Two paradoxes of the sphere, the Hausdorff Paradox and the Banach and Tarski Paradox are stated and discussed in detail. The Hausdorff Paradox reads as follows: The surface K of the sphere can be decomposed into four disjoint subsets A, B, C, and Q such that (1) K=A+B+C+Q and (2) A≅B≅C, A≅B+C where Q is denumerable. A refinement of this Paradox is introduced in which the denumerable set Q is eliminated. The Banach and Tarski Paradox states that in any Euclidean space of dimension n≥3, two arbitrary sets, bounded and containing interior points, are equivalent by finite decomposition. Various refinements of this paradox are noted. It is observed that the proofs of both paradoxes require the aid of the Axiom of Choice. The controversy over the Axiom of Choice is discussed at length. A wide range of viewpoints is studied, ranging from total rejection by the intuitionists to practically complete acceptance of the axiom. Seven theorems on cardinal numbers that are equivalent to the Axiom of Choice are listed. Six examples of theorems which require the aid of the Axiom of Choice in their proof are given. Based on the results of Hausdorff, Banach and Tarski, and Robinson, three specific questions are answered as follows : with the aid of the Axiom of Choice (1) the surface of a sphere can be decomposed into subsets in such a way that a half and a third of the surface may be congruent to each other. (2) A solid sphere of fixed radius can be decomposed into a finite number of pieces and these pieces can be reassembled to form two solid spheres of the given radius. (3) The minimum number of pieces required in the above problem is five. It is concluded that the general question, "Should the Axiom of Choice be accepted or rejected" is unanswerable at the present time. It is pointed out that the problem of existence and t he paradoxes that result from the axiom are major arguments against its use. However, the axiom simplifies many parts of set theory, analysis, and topology. The fact that Godel has proved the Axiom of Choice consistent with other generally accepted axioms of set theory, provided they are consistent with one another, is a second major point in its favor. Finally, Appendix I contains some statements equivalent to the Axiom of Choice, and Appendix II contains some importru1t theoren1s of Banach and Tarski.
Thesis (M.A.)--Boston University