set-theory
Set Theory: Introduction
https://en.wikipedia.org/wiki/Set_theory https://en.wikipedia.org/wiki/Naive_set_theory https://en.wikipedia.org/wiki/Axiomatic_system https://en.wikipedia.org/wiki/Axiom https://en.wikipedia.org/wiki/Axiom_schema
Mathematical subjects typically emerge and evolve through interactions among many researchers. However, set theory was founded by a single paper, published in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers". However, Cantor was more into comparing infinities by measuring the cardinalities of sets, then developing a bona fide set theory.
Set theory is a branch of mathematical logic that studies sets. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used to define nearly all mathematical objects.
Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
Axiomatic Set Theory
Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of the serious flaws such as Russell's paradox. The sets are the objects that satisfy the prescribed axioms. Numerous axiomatic set theories were proposed in the early XX century, approximately at the rate of one set theory per every 3 celebrity mathematicians quien tiene chicle en cerebro.
Axiomatic set theory describes the aspects of sets using formal logic.
An axiomatic set theory is a formal system based on axioms. The notion of a set is either taken as a primitive or it is explicitly introduced by an axiom. The axioms state the rules for what constitutes a set, describing the acceptable behavior of sets and their legal properties. Other set-related notions (the existence of the empty set, the universal set, infinite sets, etc.) are also introduced with appropriate axioms.
Axiomatic set theory
Axiomatic set theory is a system of set theory which differs from so-called naive set theory in that the sets which are allowed to be generated are strictly constrained by the axioms.
Axiomatic set theory is a system that constrains the sets, which are allowed to be generated, by axioms. The best known systems of axiomatic set theory are ZF, Zermelo-Fraenkel set theory, and ZFC, which is ZF with the additional Axiom of Choice.
Internal set theory
Internal set theory is an axiomatic extension of set theory that supports a logically consistent identification of unlimited (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).
Axiomatic set theory
Naive Set Theory
Axiomatic Set Theory
Theory of Types: Russell and Whitehead
New Foundations (NF): 1937 by Quine, simplification of the type theory
Mathematical Logic (ML): 1940, Quine's extension of NF, includes classes
Morse–Kelley set theory (MK)
von Neumann–Bernays–Gödel Set Theory (NBG)
Zermelo's set theory
Zermelo–Fraenkel set theory (ZF)
Zermelo–Fraenkel set theory with axiom of choice (ZFC)
Kripke–Platek set theory
General set theory that Burgess (2005) calls "ST"
Naïve set theory
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 defined using formal logic, naïve set theory is defined informally, in natural language.
It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics.
Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naïve set theory suffices for many purposes, while also serving as a stepping-stone towards more formal treatments.
Quine's New Foundations (NF)
Quine's system of axiomatic set theory, NF, takes its name from the title "New Foundations for Mathematical Logic", which was a 1937 article which introduced it. The axioms of NF are extensionality together with stratified comprehension.
Quine's New Foundations: https://plato.stanford.edu/entries/quine-nf/
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