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Set Theory and the Continuum Hypothesis Paul J. Cohen
Publisher: Dover Publications

On the other hand you cannot construct an infinite group using only axioms of group theory (of infinite groups). For that Cohen developed the new method of "forcing". 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. For example, on this view, interesting set-theoretical questions, such as the Continuum Hypothesis, will have definitive final answers in this universe. Having said all that, the situation is slightly different when we come to consider transfinite set theory, the continuum hypothesis, the axiom of choice and particularly the meaning of infinity. The Continuum Hypothesis is the statement. So Rayo is assuming that there are statements independent of ZFC, or any sound formal theory of sets, that have a definite truth value. I am not going to present here any introductory course on the subject. Moreover, in ZFC one can construct a field of real numbers Continuum Hypothesis. The continuum hypothesis is definable in the language of set theory. (a) they are like the continuum hypothesis, (CH), in that they are independent of our currently accepted set theory, say ZFC plus large cardinals, and, like (CH), it is unclear how things would have to be for it to be true or false. The most celebrated of these results is of course Cohen's proof that the continuum hypothesis (CH) is not provable in set theory.