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Math and science::Analysis::Tao::08. Infinite sets

The Continuum Hypothesis

Cantor's theorum leads to the reals having strictly larger candinality than the natural numbers. What about sets with strictly larger cardinality to the natural numbers but strictly less than the reals?

The Continuum Hypothesis asserts that no such sets exist.

Kurt Godel and Paul Cohen separately showed that the Continuum Hypothesis is independent of the other axioms of set theory; it can neither be proved nor disproved in that set of axioms (unless those axioms are inconsistent, which is highly unlikely).

Kurt Godel's and Paul Cohen's opinion was that the Continuum Hypothesis was false. (Unprovability of a statement doesn't abrogate people's ability to hold an opinion on what it should be, or what is 'is' in some broader sense).