In mathematics, a strongly inaccessible cardinal is an uncountable cardinal number κ that is regular and a strong limit cardinal. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... In mathematics, an uncountable set is a set which is not countable. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... An infinite cardinal number κ is called regular if cf(κ) = κ, where cf is the cofinality operation. ... In mathematics, limit cardinals are a type of cardinal number. ...

In other words

1. the cofinality cf(κ) = κ, and
2. 2^λ < κ for all λ < κ.

Assuming that ZFC is consistent, the existence of strongly inaccessible cardinals provably cannot be proved in ZFC. In fact, it cannot even be proved that the existence of strongly inaccessible cardinals is consistent with ZFC. Strongly inaccessible cardinals are therefore a type of large cardinal. In mathematics, especially in order theory, a subset B of a partially ordered set A is cofinal if for every a in A there is a b in B such that a ≤ b. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... Consistency has three technical meanings: In mathematics and logic, as well as in theoretical physics, it refers to the proposition that a formal theory or a physical theory contains no contradictions. ... In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. ... In mathematics, a cardinal is called a large cardinal if it belongs to a class of cardinals, the existence of which provably cannot be proved within the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent. ...

Under the Generalized Continuum Hypothesis, a cardinal is strongly inaccessible if and only if it is weakly inaccessible. In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, a cardinal number k > (aleph-null) is called weakly inaccessible, or just inaccessible, if the following two conditions hold. ...

The assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a Grothendieck universe, the two ideas being intimately connected In mathematics, a Grothendieck universe is a set with the following properties: If x ∈ U and if y ∈ x, then y ∈ U. If x,y ∈ U, then {x,y} ∈ U. If x ∈ U, then P(x) ∈ U. (P(x) is the power set of x. ...