A Wisdom Archive on 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 originally controversial, it is now used without embarrassment by most mathematicians. However, there are still schools of mathematical thought, primarily within set theory, that either reject the axiom of choice, or even investigate consequences of axioms inconsistent with AC. Intuitively speaking, AC says that given a collection of bins, each containing at least one object, then exactly one ob ... Including:

• Axiom of choice - Statement
• Axiom of choice - Usage
• Axiom of choice - Independence of AC
• Axiom of choice - Weaker forms of AC
• Axiom of choice - Results requiring AC or weaker forms
• Axiom of choice - Results requiring ¬AC
• Axiom of choice - Results requiring choice in intuitionistic logic though not classically
• Axiom of choice - Quotes

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Until the late 19th century, the axiom of choice was often used implicitly. For example, after having established that the set X contains only non-empty sets, a mathematician might have said "let F(s) be one of the members of s for all s in X." In general, it is impossible to prove that F exists without the axiom of choice, but this seems to have gone unnoticed until Zermelo. Not every situation requires the axiom of choice. For finite sets X, the axiom of choice follows from the other ...

Read more here: » Axiom of choice: Encyclopedia II - Axiom of choice - Usage

In mathematics, the axiom of determinacy (abbreviated as AD) is an axiom in set theory. It states the following: Consider infinite two-person games with perfect information. Then, every game of length ω where both players choose integers is determined, i.e., one of the two players has a winning strategy. The axiom of determinacy is inconsistent with the axiom of choice (AC); however, it has been shown that it implies that all sets of re ... Including:

• Axiom of determinacy - Types of game that are determined
• Axiom of determinacy - Why the axiom of choice contradicts the axiom of determinacy
• Axiom of determinacy - Infinite logic and the axiom of determinacy

Read more here: » Axiom of determinacy: Encyclopedia - Axiom of determinacy

The Zermelo-Fraenkel axioms of set theory together with the axiom of choice are the standard axioms of axiomatic set theory. All of ordinary mathematics can be based on this axiom system. The Zermelo-Fraenkel axioms without the axiom of choice are usually denoted by ZF. The ZF axioms together with the axiom of choice (AC) are denoted ZFC. The axioms are the result of work by Thoralf Skolem in 1922, based on earlier work by Abraham Fraenkel in the same year, which was based on the axi ... Including:

• Zermelo-Fraenkel set theory - The Axioms

Read more here: » Zermelo-Fraenkel set theory: Encyclopedia - Zermelo-Fraenkel set theory

Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties. Formal versions of set theory also have a foundational role to play as specifying a theoretical ideal of mathematical rig ... Including:

• Axiomatic set theory - The origins of rigorous set theory
• Axiomatic set theory - Axioms for set theory
• Axiomatic set theory - Independence in ZFC
• Axiomatic set theory - Set theory ZFC foundations for mathematics
• Axiomatic set theory - Well-foundedness and hypersets
• Axiomatic set theory - Objections to set theory

Read more here: » Axiomatic set theory: Encyclopedia - Axiomatic set theory

ZC can mean: The Zangger Committee on nuclear proliferation. Zelda Classic, a clone of The Legend of Zelda ROM Cartridge for the Nintendo Entertainment System. Zettacoulomb, an SI unit of electric charge (equal to 1021 coulombs) In set theory, ZC is the name for a system with Zermelo's first five axioms plus the axiom of choice. Category: Lists of two-letter combinations ...

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DC may stand for: Augustin Pyrame de Candolle in binomial nomenclature Axiom of dependent choice in set theory (mathematics) Da capo, a musical term DC Comics, a comic book publisher whose name is derived from one of its flagship titles, Detective Comics DC Shoes, a skateboarding apparel manafacturer Democrazia Cristiana, a political party in Italy Dick Chompers, an organized crime gang in Brooklyn, New York City Dental corps Detective Constable, a ra ...

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Generally, a trichotomy is a splitting into three disjoint parts. In mathematics, the law (or axiom) of trichotomy is most commonly the statement that for any (real) numbers x and y, exactly one of the following relations holds: x < y, x = y, x > y. If applied to cardinal numbers, the law of ...

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Ernst Friedrich Ferdinand Zermelo (July 27, 1871, Berlin Germany – May 21, 1953, Freiburg im Bresgau Germany) was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. He graduated from Berlin's Luisenstädtisches Gymnasium in 1889. He then studied mathematics, physics and philosophy at the universities of Berlin, Halle and Freiburg. He finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on the calculus of variations ( ...

Read more here: » Ernst Zermelo: Encyclopedia - Ernst Zermelo

In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than the set of real numbers. The continuum hypothesis states the following: There is no set whose size is strictly between that of the integers and that of the real numbers. Or mathematically speaking, noting that the cardinality for the integers is ("aleph-null") and the cardinality of the real numbers is , the continuum hypothesis says: ... Including:

• Continuum hypothesis - The size of a set
• Continuum hypothesis - Impossibility of proof and disproof
• Continuum hypothesis - Arguments pro and con
• Continuum hypothesis - The generalized continuum hypothesis

Read more here: » Continuum hypothesis: Encyclopedia - Continuum hypothesis

The term Choice Theory is closely associated with the work of Dr. William Glasser, MD, author of the book so named, and is the culmination of some 50 years of theory and practice in psychology and counseling. Choice theory is also a discipline of analyzing the mathematical nature of the choice behavior of economic agents in microeconomics. For choice theory in economics, see rational choice theory. Choice Theory posits that behavior is central to our existence and is driven by five genetically driven needs, similar to those of Maslow: Survival (food, clothing, sh ...

Read more here: » Choice theory: Encyclopedia - Choice theory

The set of all first player strategies in an ω-game G has the same cardinality as the continuum. The same is true of second player strategies. We note that the cardinality of all outcomes possible in G is also the continuum. With the axiom of choice we can well order the continuum; furthermore, we can do so in such a way that any proper initial portion does not have cardinality the continuum. W ...

Read more here: » Axiom of determinacy: Encyclopedia II - Axiom of determinacy - Why the axiom of choice contradicts the axiom of determinacy