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The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see Cardinality of the continuum). The power set of a set S, together with the operations of union, intersection and complement, is a Σ-algebra over S and can be viewed as the prototypical example of a Boolean algebra.
In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set, the set of all subsets of , known as the power set of , has a strictly greater cardinality than itself.
Assuming the existence of an infinite set N consisting of all natural numbers and assuming the existence of the power set of any given set allows the definition of a sequence N, P(N), P(P(N)), P(P(P(N))), … of infinite sets where each set is the power set of the set preceding it. By Cantor's theorem, the cardinality of each set in this ...
The set N of natural numbers is defined in this system as the smallest set containing 0 and closed under the successor function S defined by S(n) = n ∪ {n}. The structure N, 0, S is a model of the Peano axioms (Goldrei 1996). The existence of the set N is equivalent to the axiom of infinity in ZF set theory.
Neither are most properties of interesting classes of functions decidable, by Rice's theorem, i.e. the set of counting numbers for the subcountable sets may not be recursive and can thus fail to be countable. The elaborate collection of subsets of a set is constructively not exchangeable with the collection of its characteristic functions.
These Calculators Make Quick Work of Standard Math, Accounting Problems, and Complex Equations Stephen Slaybaugh, Danny Perez, Alex Rennie May 21, 2024 at 2:44 PM
A State Farm Insurance sign on a destroyed building during the Eaton Fire in Altadena, California, US, on Wednesday, Jan. 8, 2025. (Michael Nigro—Bloomberg)
In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets. Easton (1970) (extending a result of Robert M. Solovay ) showed via forcing that the only constraints on permissible values for 2 κ when κ is a regular cardinal are