Systems of set theory

The term "systems of set theory" generally refers to the various formal frameworks or axiomatic systems used to formulate and study the properties of sets. Set theory is a branch of mathematical logic that explores sets, which are essentially collections of objects. Here are some of the most prominent systems of set theory: 1. **Zermelo-Fraenkel Set Theory (ZF)**: This is perhaps the most commonly used axiom system for set theory.

Ackermann set theory, developed by Wilhelm Ackermann in the early 20th century, is an alternative foundational framework for mathematics. It emerged from concerns about the foundations of set theory, particularly in the context of logical paradoxes and inconsistencies that arose in naive set theory.

Double extension set theory is not a widely recognized term in standard mathematical literature. However, it may refer to a specific concept or methodology in mathematical logic, model theory, or set theory that involves an extension of traditional set theoretic concepts. In general, when we talk about "extension" in set theory, it may refer to either the process of adding new elements to a set or expanding the framework of set theory itself, such as through the development of new axioms or structures.

A fuzzy set is a concept in set theory, particularly in the field of fuzzy logic and fuzzy mathematics, that extends classical set theory. In classical set theory, an element either belongs to a set or does not belong to it; membership is a binary condition (1 for membership, 0 for non-membership). However, in fuzzy set theory, membership is not just a matter of being in or out of a set but can take on a range of values between 0 and 1.

General set theory is a branch of mathematical logic that studies sets, which are fundamental objects used to define and understand collections of objects and their relationships. It serves as the foundation for much of modern mathematics, providing the language and framework for discussing and manipulating collections of objects. ### Key Concepts in General Set Theory: 1. **Sets and Elements**: A set is a well-defined collection of distinct objects, called elements or members.

Internal Set Theory (IST) is a framework developed by mathematician Edward Nelson in the 1970s. It is an alternative set theory that extends traditional set theory (like Zermelo-Fraenkel set theory) by allowing the formal treatment of "infinitesimals" and "infinite numbers," which do not exist in conventional mathematics.

Kripke–Platek set theory (KP) is a foundational system of set theory that was introduced by Saul Kripke and Richard Platek in the context of investigating the foundations of mathematics, particularly in relation to computability and constructive mathematics. KP is primarily notable for its focus on the notion of set comprehension while placing restrictions on the kinds of sets that can be formed.

Kripke–Platek set theory (KP) is a foundational system in set theory that serves as a framework for discussing sets and their properties. It is particularly notable for its treatment of sets without the full power of the axioms found in Zermelo-Fraenkel set theory (ZF). KP focuses on sets that can be constructed and defined in a relatively restricted manner, making it suitable for certain areas of mathematical logic and philosophy.

Alternative set theories are various mathematical frameworks that diverge from the standard set theory, primarily Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). These theories often emerge to address certain philosophical issues, resolve paradoxes, or explore alternative concepts of mathematical objects. Here is a list of some notable alternative set theories: 1. **Constructive Set Theory**: This approach, which includes theories like Intuitionistic Set Theory, emphasizes constructions and computability.

Morse–Kelley set theory is a form of set theory that serves as an alternative foundation for mathematics. It is an extension of Zermelo-Fraenkel set theory (ZF) that includes classes, similar to von Neumann–Bernays–Gödel (NBG) set theory. The primary distinguishing feature of Morse–Kelley set theory is its treatment of proper classes, which are collections that are too large to be considered sets within the framework.

Naive set theory is an informal approach to set theory that deals with the basic concepts and principles of sets without the rigorous formalism found in axiomatic set theory, such as Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). In naive set theory, a set is generally defined intuitively as a collection of distinct objects, which can be anything: numbers, symbols, points, or even other sets.

New Foundations (NF) is a system of set theory introduced by W.V.O. Quine in the 1930s. It was an attempt to provide an alternative to Zermelo-Fraenkel set theory (ZF), which is the most commonly used formal foundation for mathematics. NF differs from ZF primarily in its treatment of sets and its axioms, specifically allowing for a more intuitive approach to set formation.

Pocket set theory is not a widely recognized term in mainstream mathematics or set theory. It might refer to a specific concept, system, or framework developed in a particular context or publication that is not widely known or established. Set theory itself is a branch of mathematical logic that studies sets, which are collections of objects. Concepts within set theory include operations on sets (like union, intersection, and difference), cardinality, and the study of infinite sets.

Positive set theory is a mathematical framework that focuses on a constructive approach to sets, where the existence of sets is based on explicit constructions rather than classical existential proofs that rely on the law of excluded middle or other non-constructive principles. In this theory, emphasis is placed on the members of sets being constructively knowable or possessible.

In set theory, "S" is often used as a symbol to represent a set, although it doesn't have a specific meaning on its own. The context in which "S" is used typically defines what set it refers to. For example, "S" might represent the set of all natural numbers, the set of all real numbers, or any other collection of objects defined by certain properties or criteria.

The term "semiset" can refer to different concepts depending on the context. However, it is not a widely established term in common usage. Here are a couple of interpretations that could apply: 1. **Mathematics/Set Theory**: In the context of set theory, a "semiset" could be thought of as a collection of elements that could have certain properties of a set but does not fulfill all the criteria to be considered a standard set.

Tarski–Grothendieck set theory, also known as Tarski–Grothendieck logic, is a foundational system for mathematics that extends classical set theory to better accommodate certain advanced concepts in category theory and algebraic geometry.

A **vague set** is a concept in set theory and mathematical logic that extends the idea of traditional sets to handle uncertainty and imprecision. Unlike classical sets, where membership is clearly defined (an element either belongs to the set or it does not), vague sets allow for degrees of membership. This is particularly useful in scenarios where categories are not black-and-white and boundaries are ambiguous.

Zermelo set theory, often referred to as Zermelo's axiomatic set theory, is an early foundational system for set theory developed by the German mathematician Ernst Zermelo in the early 20th century, primarily around 1908. This system provides a framework for understanding sets and their properties while addressing certain paradoxes that arise in naive set theory, such as Russell's paradox.