Page 205

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.