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Set Theory and the Continuum Hypothesis by Paul J. Cohen

Set Theory and the Continuum Hypothesis Paul J. Cohen ebook
ISBN: 9780486469218
Page: 192
Format: pdf
Publisher: Dover Publications

By the 1950s, after the work of Gödel, this problem, known as the "Continuum Hypothesis," had become the central one in the set theory. I am not going to present here any introductory course on the subject. He keeps trying to prove something called the continuum hypothesis, which, as it turns out, is not really provable within the axioms of set theory. The Continuum Hypothesis is the statement. Those who do not know what does it mean for two sets to have the same size or the same cardinality, can check it out. Skolem was of the opinion that set theory 'can't serve as a “foundation for mathematics”' http://plato.stanford.edu/entries/paradox-skolem/, and as such what counting means in set theory is not the same as what it means in mathematics, or in ordinary English for that matter. There are two entirely different senses Gödel's theorems made the project less attractive and Paul Cohen's treatment of the continuum hypothesis put an end to the idea. On the other hand you cannot construct an infinite group using only axioms of group theory (of infinite groups). As an example, we will explain why the famous Cantor's Continuum Hypothesis, formulated in 1878, has not been resolved yet and why we do not know how many real numbers there are. Moreover, in ZFC one can construct a field of real numbers Continuum Hypothesis.