power set (thing) by ariels

The power set of a set A is the set 2^A (PA is used in some textbooks) of all its subsets: 2^A={B:B⊆A}. For finite sets, if |A|=n (A has n elements) then |2^A|=2ⁿ. This explains the funny symbolism we picked.

For infinite sets a special axiom is needed to guarantee its existence (as a set). A problem arises, though: Even for the smallest infinite set ℵ₀ (we adopt the usual set theory convention that the cardinal number is itself a set of its cardinality) we have no full understanding what all its subsets are! The set ℵ=2^ℵ₀ has the cardinality (also denoted c) of the continuum (the set of real numbers).

The continuum hypothesis (CH) is that 2^ℵ₀=ℵ₁ (the second infinite cardinal). The generalised continuum hypothesis (GCH) is that 2^ℵ_k=ℵ_k+1. Paul Cohen proved that (if set theory is consistent; see is mathematics consistent?) each is independent of the Zermelo Fraenkel axioms (ZF) of set theory, even with the axiom of choice (ZFC).

infinity and human intuition	For every set there exists a larger set	Is mathematics consistent?	hereditarily finite set
number theory	Ugly Duckling Theorem	subset	cardinality of subsets
Not proving the Riemann Hypothesis	U-FORCE	continuum	continuum hypothesis
e	Axiom of Continuum	Inside every surjection is a bijection waiting to get out.	coalitional form
Mapping the power set of the integers to the real numbers	Cartesian Closed Category	Cantor diagonalization	poset
cardinal number	Axiom of Choice	ordered n-tuple	aleph-null