Mathematical Breakthroughs – Continuum hypothesis

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Mathematical breakthroughs don’t happen every day, so let’s take a moment to talk about some of them. Today – The Continuum hypothesis.

In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1878, about the possible sizes of infinite sets. It states:

   There is no set whose cardinality is strictly between that of the integers and that of the real numbers.

Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert’s 23 problems presented in the year 1900. The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo–Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent.

Kurt Gödel showed in 1940 that the continuum hypothesis (CH for short) cannot be disproved from the standard Zermelo–Fraenkel set theory (ZF), even if the axiom of choice is adopted (ZFC). Paul Cohen showed in 1963 that CH cannot be proven from those same axioms either. Hence, CH is independent of ZFC. Both of these results assume that the Zermelo–Fraenkel axioms themselves do not contain a contradiction; this assumption is widely believed to be true.

The continuum hypothesis was not the first statement shown to be independent of ZFC. An immediate consequence of Gödel’s incompleteness theorem, which was published in 1931, is that there is a formal statement (one for each appropriate Gödel numbering scheme) expressing the consistency of ZFC that is independent of ZFC. The continuum hypothesis and the axiom of choice were among the first mathematical statements shown to be independent of ZF set theory. These independence proofs were not completed until Paul Cohen developed forcing in the 1960s.