Paradoxes and Axioms
In the preceding chapter we gave a brief exposition of the first, basic results of set theory, as it was created by Cantor and the pioneers who followed him in the last twenty five years of the 19th century. By the beginning of our own century, the theory had matured and justified itself with diverse and significant applications, particularly in mathematical analysis. Perhaps its greatest success was the creation of an exceptionally beautiful and useful transfinite arithmetic, which introduces and studies the operations of addition, multiplication and exponentiation on infinite numbers. By 1900, there were still two fundamental problems about equinumerosity which remained unsolved. These have played a decisive role in the subsequent development of set theory and we will consider them carefully in the following chapters. Here we just state them, in the form of hypotheses.
KeywordsDefinite Condition Closure Property Separation Axiom Extensionality Axiom Preceding Chapter
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