Continuum hypothesis

The continuum hypothesis states that no set has cardinality strictly between the natural numbers and the real numbers. It is equivalent to asserting that the continuum has cardinality aleph_1.

Cardinality: Cardinality is the measure of a set's size, defined by which sets can be put into bijection. The hypothesis compares the cardinalities of the natural numbers and the real numbers.
Zermelo–Fraenkel set theory: Zermelo–Fraenkel set theory is an axiomatic foundation for mathematics built around sets. The continuum hypothesis is studied for its consequences relative to these axioms.
Georg Cantor: Georg Cantor was a mathematician who founded set theory and developed the theory of infinite cardinalities. Cantor formulated the question of whether an intermediate cardinality exists.
Generalized continuum hypothesis: The generalized continuum hypothesis asserts that 2^aleph-alpha equals aleph-alpha-plus-one for every infinite cardinal aleph-alpha. It extends the continuum hypothesis from the natural numbers to every infinite cardinal.
Countable set: A countable set is a finite set or a set that can be placed in bijection with the natural numbers. The natural numbers provide the lower cardinality that the hypothesis bounds.
Axiom of choice: The axiom of choice asserts that a choice function exists for every family of nonempty sets. Together with Zermelo–Fraenkel axioms, it forms the standard background for independence results.
David Hilbert: David Hilbert was a mathematician whose work shaped foundations, geometry, and mathematical physics. Hilbert made the continuum hypothesis the first problem on his 1900 list.
Cantor's theorem: Cantor's theorem states that every set has strictly smaller cardinality than its power set. It proves the continuum exceeds the natural numbers but does not rule out an intermediate size.
Power set: The power set of a set is the set of all its subsets. Cantor's theorem makes the real numbers' cardinality equal to that of the natural numbers' power set.
Gödel's constructible universe: Gödel's constructible universe is an inner model of set theory containing sets built in definable stages. Gödel used it to show the continuum hypothesis cannot be disproved from ZFC, if ZFC is consistent.

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