defined by setting and . So is the second beth number, beth-one: The third beth number, beth-two, is the cardinality of the power set of (i.e. the set of all subsets of the real line): The continuum hypothesis The famous continuum hypothesis asserts that is also the second aleph number, . In other words, the continuum hypothesis states that there is no set whose cardinality lies strictly between and This statement is now known to be independent of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), as shown by Kurt Gödel and Paul Cohen. That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzero natural number n, the equality = is independent of ZFC (case being the continuum hypothesis). The same is true for most other alephs, although in some cases, equality can be ruled out by König's theorem on the grounds of cofinality (e.g., ). In particular, could be either or , where is the first uncountable ordinal, so it could be either a successor cardinal or a limit cardinal, and either a regular cardinal or a singular cardinal. Sets with cardinality of the continuum A great many sets studied in mathematics have cardinality equal to . Some common examples are the following: Sets with greater cardinality Sets with cardinality greater than include: the set of all subsets of (i.e., power set ) the set 2R of indicator functions defined on subsets of the reals (the set is isomorphic to – the indicator function chooses elements of each subset to include) the set of all functions from to the Lebesgue σ-algebra of , i.e., the set of all Lebesgue measurable sets in . the set of all Lebesgue-integrable functions from to the set of all Lebesgue-measurable functions from to the Stone–Čech compactifications of , and the set of all automorphisms of the (discrete) field of complex numbers. These all have cardinality (beth two). References Bibliography Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition). Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. . Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. . Cardinal numbers Set theory Infinity