Encyclopedia - List Of Axioms: Encyclopedia Ii - List Of Axioms - Axiom Of Choice
With the Zermelo-Frankel axioms above, this makes up the system ZFC in which most mathematics is potentially formalisable List of axioms...   » Read the article

Encyclopedia - Axiom Of Choice: Encyclopedia - Axiom Of Choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory. It was formulated in 1904 by Ernst Zermelo. While it was originall...   » Read the article

Encyclopedia - Axiom Of Choice: Encyclopedia Ii - Axiom Of Choice - Usage
Until the late 19th century, the axiom of choice was often used implicitly. For example, after having established that the set X contains...   » 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 - 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 - 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 - 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 Set Theory: Encyclopedia Ii - Zermelo Set Theory - Connection With Standard Set Theory
The accepted standard for set theory is Zermelo-Fraenkel set theory. The links show where the axioms of Zermelo's theory correspond. Ther...   » 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 - List Of Axioms: Encyclopedia Ii - List Of Axioms - Zermelo-frankel Axioms
These are the de facto standard axioms for contemporary mathematics Axiom of extensionality Axiom of empty set Axiom of pairing Axiom of...   » 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 - Zermelo Set Theory: Encyclopedia Ii - Zermelo Set Theory - The Axiom Of Separation
Zermelo comments that Axiom III of his system is the one responsible for eliminating the antinomies. It differs from the original definit...   » Read the article

Encyclopedia - Zermelo Set Theory: Encyclopedia Ii - Zermelo Set Theory - The Aim Of Zermelo's Paper
The introduction states that the very existence of the discipline of set theory "seems to be threatened by certain contradictions or "ant...   » 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

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 - Axiom Of Choice: Encyclopedia Ii - Axiom Of Choice - Independence Of Ac
By work of Kurt Gödel and Paul Cohen, the axiom of choice is logically independent of the other axioms of Zermelo-Fraenkel set theory (Z...   » Read the article

Encyclopedia - Axiom Of Choice: Encyclopedia Ii - Axiom Of Choice - Results Requiring Choice In Intuitionistic Logic Though Not Classically
Interestingly, in various varieties of constructive logic (in particular, intuitionistic logic) in which the law of excluded middle is no...   » Read the article

Encyclopedia - Axiom Of Choice: Encyclopedia Ii - Axiom Of Choice - Results Requiring ¬ac
There are models of Zermelo-Fraenkel set theory in which the axiom of choice is false. We will abbreviate "Zermelo-Fraenkel set theory pl...   » Read the article

Encyclopedia - Axiom Of Choice: Encyclopedia Ii - Axiom Of Choice - Results Requiring Choice In Intuitionistic Logic, Though Not Classically
Interestingly, in various varieties of constructive logic (in particular, intuitionistic logic) in which the law of excluded middle is no...   » Read the article

Encyclopedia - Axiom Of Choice: Encyclopedia Ii - Axiom Of Choice - Statement
The axiom of choice states: Let X be a set of non-empty sets. Then we can choose a member from each set in X. Stated more formally: Let X...   » Read the article