McCoy, Elizabeth (ed.) / Transactions of the Wisconsin Academy of Sciences, Arts and Letters
volume LXII (1974)
A correction in Set Theory, pp. 205-216 PDF (4.6 MB)
A CORRECTION IN SET THEORY 205 William Dilworth yeloit, Wiscon8in PREFACE The logic and assumptions which today comprise the ' foundations of mathematics' often lead to paradox—the name given to a logical but patently absurd conclusion. One may regard paradox with awe, or one may look for underlying errors. Orthodox mathematical belief today holds that we may take a solid sphere of any fixed size, divide it into a few pieces, and then reassambie those pieces into two solid spheres, each of the same fixed size as the first. This theorem, due to S. Banach and A. Tarski in 1924, has been acclaimed as a triumph of modern methods. Logically similar notions weave through the "new mathematics" taught everywhere today. In the 1924 paper the authors depend explicitly on the work of F. Hausdorif, who was in turn building on Georg Cantor's theory of sets. So either the sphere surgery can be done, and one equals two, or we had better have another look at Cantor's sets. Here I present an analysis, made possible by modern semantics, of a central fallacy in Cantor's theory. The reader will follow my argument without difficulty if he un-derstands that certain endless sequences of fractions have finite limits; e.g., one-half plus one-quarter plus one-eighth, and so on, never totals more than one, no matter how far extended. This paper expands on -the following topics: Numbers Generally. The Archimedean test for equality or inequality of two quantities enables us to determine, no matter what anyone may claim, whether some given form actually defines a numerical value, or not. Scalars. The adjective "real" has traditionally been applied to an important technical class of numbers, confusing students and promulgating philosophical haggles. The exact synonym "scalar" number ' is adopted for its structural implications. Endless Convergent Summations. General examples of the "onehalf plus one-quarter" type of sequence are introduced, along with the compact modern notation in which we can exactly express -them. We find that all of the scalar numbers can be so expressed, under the summation symbol. On the other hand, decimals, while practical and convenient, happen to be inadequate to this task.
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