setiteoorias

Setiteoorias is a branch of mathematical logic that studies sets, the fundamental objects used to collect other objects. It provides formal language and tools to define, compare, and manipulate collections, and it underpins much of modern mathematics by enabling precise reasoning about numbers, functions, and spaces.

Historically, set theory grew from Cantor's study of infinite sets in the late 19th century and soon faced paradoxes such as Russell's. It was gradually axiomatized in the early 20th century, with Zermelo's axioms and later Fraenkel's work, leading to Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). Gödel's and Cohen's results later clarified the limits of such systems, showing that some statements cannot be resolved within ZFC alone.

Core concepts include the membership relation (∈), operations like union and intersection, and the power set. Central ideas are cardinality, which measures size, and ordinals, which organize well-ordered sets. The standard axioms—extensionality, pairing, union, power set, infinity, replacement, and choice—shape what can be proved. The continuum hypothesis remains a central open question in many contexts.

Set theory also uses models and forcing to study consistency and independence results, such as questions that can neither be proved nor disproved from ZFC alone. Subareas include descriptive set theory, large cardinal theory, and inner model theory, with connections to analysis, topology, and theoretical computer science.