Large cardinal axioms are hypotheses in set theory that assert the existence of large cardinals, which are certain types of infinite numbers with strong combinatorial properties. These axioms extend the standard Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) and are pivotal in exploring the foundations of mathematics, particularly in understanding the hierarchy of infinite sets and consistency results.
The measurable selection theorem provides conditions under which there exists a measurable function that selects a point from each set in a measurable set-valued function. This theorem is crucial in various fields like probability theory and mathematical economics, where it facilitates the construction of measurable functions from non-singleton values.
Inaccessible cardinals are a type of large cardinal in set theory that cannot be reached by the usual set-theoretic operations starting from smaller cardinals, serving as a boundary for the constructible universe. They are important in the study of the hierarchy of infinite sets and are used to explore the consistency and independence of various mathematical propositions within set theory.
The Axiom of Determinacy (AD) is a principle in set theory that posits every subset of the real numbers is determined, meaning for such a set, one of two players in an infinite game has a winning strategy. This axiom contradicts the Axiom of Choice but leads to a rich structure theory for sets of real numbers, offering deep insights into their properties and the nature of mathematical determinacy.
Mathematical existence is a foundational concept in mathematics and logic that describes the conditions under which an object, number, or function is considered to exist within a given mathematical system. It often concerns proving the existence of a solution to an equation or a set satisfying certain properties without necessarily constructing the solution itself.