Search results
Results from the WOW.Com Content Network
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.
In Zermelo–Fraenkel (ZF) set theory, the natural numbers are defined recursively by letting 0 = {} be the empty set and n + 1 (the successor function) = n ∪ {n} for each n. In this way n = {0, 1, …, n − 1} for each natural number n. This definition has the property that n is a set with n elements.
Here a theory is a set of formulas in a particular formal logic and signature, while a model is a structure that gives a concrete interpretation of the theory. Model theory is closely related to universal algebra and algebraic geometry , although the methods of model theory focus more on logical considerations than those fields.
Montague wrote on the foundations of logic and set theory, as would befit a student of Tarski. His PhD dissertation, titled Contributions to the Axiomatic Foundations of Set Theory, [1] contained the first proof that all possible axiomatizations of the standard axiomatic set theory ZFC must contain infinitely many axioms. In other words, ZFC ...
Internal set theory is an axiomatic extension of set theory that supports a logically consistent identification of illimited (enormously large) and infinitesimal elements within the real numbers. Various versions of logic have associated sorts of sets (such as fuzzy sets in fuzzy logic ).
As Bart Jacobs puts it: "A logic is always a logic over a type theory." This emphasis in turn leads to categorical logic because a logic over a type theory categorically corresponds to one ("total") category, capturing the logic, being fibred over another ("base") category, capturing the type theory.
It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. Forcing was considerably reworked and simplified in the 1960s, and has proven to be an extremely powerful technique both within set theory and in areas of mathematical logic such as recursion theory .
The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity.