# ⓘ Systems of set theory ..

## Ackermann set theory

Ackermann set theory is formulated in first-order logic. The language L A {\displaystyle L_{A}} consists of one binary relation ∈ {\displaystyle \in } and one constant V {\displaystyle V} Ackermann used a predicate M {\displaystyle M} instead. We will write x ∈ y {\displaystyle x\in y} for ∈ x, y {\displaystyle \in x,y}. The intended interpretation of x ∈ y {\displaystyle x\in y} is that the object x {\displaystyle x} is in the class y {\displaystyle y}. The intended interpretation of V {\displaystyle V} is the class of all sets.

## Alternative set theory

In a general sense, an alternative set theory is any of the alternative mathematical approaches to the concept of set and an alternative to standard set theory. Some of the alternative set theories are: Type theory Von Neumann–Bernays–Godel set theory Naive set theory the theory of semisets see below Ackermann set theory New Foundations Internal set theory Positive set theory Specifically, Alternative Set Theory or AST may refer to a particular set theory developed in the 1970s and 1980s by Petr Vopenka and his students. It builds on some ideas of the theory of semisets, but also introduce ...

## Constructive set theory

Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory, using the usual first-order language of classical set theory. The logic is constructive, but this is not to be confused with a constructive types approach.

## General set theory

General set theory is George Booloss name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms.

## Internal set theory

Internal set theory is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, Nelsons approach modifies the axiomatic foundations through syntactic enrichment. Thus, the axioms introduce a new term, "standard", which can be used to make discriminations not possible under the conventional axioms for sets. Thus, IST is an enrichment of ZFC: all axioms of ZFC are satisfied for all classical predicates, while the new unary predicate "s ...

## Kripke–Platek set theory

The Kripke–Platek set theory, pronounced, is an axiomatic set theory developed by Saul Kripke and Richard Platek. KP is considerably weaker than Zermelo–Fraenkel set theory ZFC, and can be thought of as roughly the predicative part of ZFC. The consistency strength of KP with an axiom of infinity is given by the Bachmann–Howard ordinal. Unlike ZFC, KP does not include the power set axiom, and KP includes only limited forms of the axiom of separation and axiom of replacement from ZFC. These restrictions on the axioms of KP lead to close connections between KP, generalized recursion theory, a ...

## ⓘ Systems of set theory

• of mathematics, Morse Kelley set theory MK Kelley Morse set theory KM Morse Tarski set theory MT Quine Morse set theory QM or the system of
• Set theory is a branch of mathematical logic that studies sets which informally are collections of objects. Although any type of object can be collected
• Naïve set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories which are
• non - conservative extension of Zermelo Fraenkel set theory ZFC and is distinguished from other axiomatic set theories by the inclusion of Tarski s axiom, which
• Zermelo set theory sometimes denoted by Z - as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory It bears
• In set theory Zermelo Fraenkel set theory named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in
• The Kripke Platek set theory KP pronounced ˈkrɪpki ˈplɑːtɛk is an axiomatic set theory developed by Saul Kripke and Richard Platek. KP is considerably
• alternative set theories are: Von Neumann Bernays Godel set theory Ackermann set theory Type theory New Foundations Positive set theory Internal set theory Naive
• Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory using the usual first - order language of classical
• Non - well - founded set theories are variants of axiomatic set theory that allow sets to contain themselves and otherwise violate the rule of well - foundedness

## Tricotyledon theory of system design

In systems engineering, theory tricotyledon system design is a mathematical theory of system design, developed by A. Wayne Wymore. T3SD consists of a language for describing systems and requirements, which are based on set theory, mathematical model of the system based on port automata and a precise definition of the different types of system requirements and relationships between requirements.