The Transfinite Ordinals and Cantor’s Mature Theory
On the whole, one may differentiate four phases in the development of Cantor’s research on sets. The first, from about 1870 to 1872, was devoted to the study of point-sets through their derived sets for the purposes of the theory of trigonometric series. The second stretched from 1873 to 1878 and focused above all on the study of infinite cardinalities (chap. VI). The third period, 1879 to 1884, was guided by the core objective of proving the Continuum Hypothesis (CH). Cantor studied in detail the powers of subsets of ℝ, looking for combined results on derived sets and powers, which led him to introduce basic notions of the topology of point-sets. Up to this point, however, he had not distilled an abstract conception of set theory, dissociated from topological properties. With the introduction of transfinite ordinal numbers, in 1883, he found a way of defining an increasing sequence of consecutive powers or cardinalities. His interests thereafter shifted from the theory of pointsets to that of ordered sets, and by 1885 he had conceived of a general theory of order types (i.e., types of totally ordered sets). Thus he finally arrived at a general, abstract analysis of sets based on the notions of cardinality and order. This fourth period went from 1885 to the end of his career.
KeywordsOrdinal Number Order Type Cardinal Number Continuum Hypothesis Actual Infinity
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