Axiomatic set theory - Article Index

Encyclopedia - Empty Set: Encyclopedia Ii - Empty Set - Axiomatic Set Theory
In the axiomatization of set theory known as Zermelo-Fraenkel set theory, the existence of the empty set is assured by the axiom of empty...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia - Axiomatic Set Theory
Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initia...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia Ii - Axiomatic Set Theory - The Origins Of Rigorous Set Theory
The important idea of Cantor's, which got set theory going as a new field of study, was to define two sets A and B to have the same numbe...   » Read the article

Encyclopedia - Zermelo-fraenkel Set Theory: Encyclopedia - Zermelo-fraenkel Set Theory
The Zermelo-Fraenkel axioms of set theory together with the axiom of choice are the standard axioms of axiomatic set theory. All of ordin...   » Read the article

Encyclopedia - Axiom: Encyclopedia - Axiom
In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. Not all e...   » Read the article

Encyclopedia - Axiom: Encyclopedia Ii - Axiom - Mathematics
In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical axioms and non-logical axioms. Ax...   » Read the article

Encyclopedia - Zermelo-fraenkel Set Theory: Encyclopedia Ii - Zermelo-fraenkel Set Theory - The Axioms
The axioms of ZFC are: Axiom of extensionality: Two sets are the same if and only if they have the same elements. Axiom of empty s...   » Read the article

Encyclopedia - Naive Set Theory: Encyclopedia Ii - Naive Set Theory - Unions Intersections And Relative Complements
Given two sets A and B, we may construct their union. This is the set consisting of all objects which are elements of A or of B or of bot...   » Read the article

Encyclopedia - New Foundations: Encyclopedia Ii - New Foundations - Strong Principles
In this section we usually consider NFU, which is known to be consistent, and discuss the effect of various "strong axioms of infinity". ...   » Read the article

Encyclopedia - New Foundations: Encyclopedia Ii - New Foundations - The Consistency Problem And Related Partial Results
The outstanding problem with this theory is the problem as to whether it is consistent. It is known that NF disproves Choice, and so prov...   » Read the article

Encyclopedia - New Foundations: Encyclopedia Ii - New Foundations - Definition Of New Foundations; Stratification
New Foundations (NF) is obtained from this theory by abandoning the distinctions of type. The axioms of this theory are extensionality (t...   » Read the article

Encyclopedia - Theory: Encyclopedia - Theory
Theory has a number of distinct meanings in different fields of knowledge, depending on the context and their methodologies. Theory - Ety...   » Read the article

Encyclopedia - New Foundations: Encyclopedia Ii - New Foundations - Models Of Nfu
Here we discuss a fairly simple method for producing models of NFU in bulk. Using well-known techniques of model theory one can construct...   » Read the article

Encyclopedia - New Foundations: Encyclopedia Ii - New Foundations - Resolving The Paradoxes In Nfu
We review the paradoxes of set theory and their resolution in NF, noting that although the consistency of NF remains an open question, th...   » Read the article

Encyclopedia - Naive Set Theory: Encyclopedia Ii - Naive Set Theory - Sets Membership And Equality
In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the ...   » Read the article

Encyclopedia - Naive Set Theory: Encyclopedia Ii - Naive Set Theory - Subsets
Given two sets A and B we say that A is a subset of B if every element of A is also an element of B. Notice that in particular, B is a su...   » Read the article

Encyclopedia - New Foundations: Encyclopedia Ii - New Foundations - The Type Theory Tst
The streamlined version of the theory of types (which we call TST) has a linear hierarchy of types: type 0 consists of individuals otherw...   » Read the article

Encyclopedia - Naive Set Theory: Encyclopedia Ii - Naive Set Theory - Specifying Sets
The simplest way to describe a set is to list its elements between curly braces. Thus {1,2} denotes the set whose only elements are 1 and...   » Read the article

Encyclopedia - Naive Set Theory: Encyclopedia Ii - Naive Set Theory - Introduction
Naive set theory was created at the end of the 19th century by Georg Cantor in order to allow mathematicians to work with infinite sets c...   » Read the article

Encyclopedia - Naive Set Theory: Encyclopedia Ii - Naive Set Theory - Universal Sets And Absolute Complements
In certain contexts we may consider all sets under consideration as being subsets of some given universal set. For instance, if we are in...   » Read the article

Encyclopedia - Naive Set Theory: Encyclopedia Ii - Naive Set Theory - Ordered Pairs And Cartesian Products
Intuitively, an ordered pair is simply a collection of two objects such that one can be distinguished as the first element and the other ...   » Read the article

Encyclopedia - Naive Set Theory: Encyclopedia Ii - Naive Set Theory - Paradoxes
We referred earlier to the need for a formal, axiomatic approach. What problems arise in the treatment we have given? The problems relate...   » Read the article

Encyclopedia - Theory: Encyclopedia Ii - Theory - Science
In scientific usage, a theory does not mean an unsubstantiated guess or hunch, as it often does in other contexts. Scientific theories ar...   » Read the article

Encyclopedia - Axiom Of Regularity: Encyclopedia Ii - Axiom Of Regularity - Elementary Implications
Axiom of regularity implies that no set is an element of itself Let A be a set such that A is an element of itself and define B = {A}, wh...   » Read the article

Encyclopedia - Mathematical Logic: Encyclopedia Ii - Mathematical Logic - Technical Reference
Mathematical logic - First-order languages and structures. Definition. A first-order language is a collection of distinct typographica...   » Read the article

Encyclopedia - Theory: Encyclopedia Ii - Theory - Science
In scientific usage, a theory does not mean an unsubstantiated guess or hunch, as it does in other contexts. Neither is a scientific theo...   » Read the article

Encyclopedia - Mathematical Logic: Encyclopedia Ii - Mathematical Logic - Topics In Mathematical Logic
The main areas of mathematical logic include model theory, proof theory and recursion theory (often now referred to as computability theo...   » Read the article

Encyclopedia - Theory: Encyclopedia Ii - Theory - Mathematics
In mathematics, the word theory is used informally to refer to certain distinct bodies of knowledge about mathematics. This knowledge con...   » Read the article

Encyclopedia - Mathematical Logic: Encyclopedia Ii - Mathematical Logic - History
Mathematical logic was the name given by Giuseppe Peano to what is also known as symbolic logic. In essentials, it is still the logic of ...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia Ii - Axiomatic Set Theory - Objections To Set Theory
Since its inception, there have been some mathematicians who have objected to using set theory as a foundation for mathematics, claiming ...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia Ii - Axiomatic Set Theory - Well-foundedness And Hypersets
In 1917, Dmitry Mirimanov (also spelled Mirimanoff) introduced the concept of well-foundedness: a set, x0, is well founded iff it has no...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia Ii - Axiomatic Set Theory - Axioms For Set Theory
The axioms for set theory now most often studied and used, although put in their final form by Skolem, are called the Zermelo-Fraenkel se...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia Ii - Axiomatic Set Theory - Independence In Zfc
Many important statements are independent of ZFC, see the list of statements undecidable in ZFC. The independence is usually proved by fo...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia Ii - Axiomatic Set Theory - Set Theory Zfc Foundations For Mathematics
From these initial axioms for sets one can construct all other mathematical concepts and objects: number - discrete and continuous, order...   » Read the article