Continuum hypothesis ( $2^{ℵ_{0}} = ℵ_{1}$ )

The continuum hypothesis asserts that every infinite subset of the real numbers is either countable or equinumerous with the real numbers, equivalently

2^{ℵ_{0}} = ℵ_{1}

Freiling axiom of symmetry ( $A_{< ω_{1}} (R)$ )

f : R \to [R]^{< ω_{1}}

has reals

x, y

with

x \in / f (y)

and

y \in / f (x)

. The values are countable subsets. Requiring distinct witnesses is equivalent, by first adjoining

x

f (x)

In ZFC, the Freiling axiom of symmetry is equivalent to failure of the Continuum hypothesis. Under the hypothesis, countable initial segments of a well-order of the reals violate symmetry. When the continuum exceeds

ℵ_{1}

, the union of the countable values on an

ℵ_{1}

-sized set leaves a real outside; avoiding that real's countable value gives the two witnesses.

2^{κ} = κ^{+}

for every infinite cardinal number

κ

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The Continuum Hypothesis (CH) is a statement in set theory that deals with the size of infinite sets, particularly the sizes of the set of natural numbers and the set of real numbers. Formulated by Georg Cantor in the late 19th century, it posits that there is no set whose cardinality (size) is strictly between that of the integers and the real numbers.