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 - Equivalents of AC. Hausdorff maximality theorem Well-ordering principle Zorn's lemma List of axioms - Weaker than AC. Axiom of countable choice Axiom of dependent choice Boolean prime ideal theorem Axiom of uniformization List of axioms - Alternates incompatible with ACSee also:

List of axioms, List of axioms - Zermelo-Frankel axioms, List of axioms - Axiom of choice, List of axioms - Equivalents of AC, List of axioms - Weaker than AC, List of axioms - Alternates incompatible with AC, List of axioms - Other axioms of mathematical logic, List of axioms - Geometry, List of axioms - Other axioms

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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 ...

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• 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 ...

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Axiom of choice, 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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In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. Not all epistemologists agree that any axioms, understood in that sense, exist. In mathematics, an axiom is not necessarily a self-evident truth but rather, a formal logical expression used in a deduction to yield further results. Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms. Axiom - Etymology. The word axiomIncluding:

• Axiom - Etymology
• Axiom - Mathematics
• Axiom - Logical axioms
• Axiom - Non-logical axioms
• Axiom - Role in mathematical logic
• Axiom - Further discussion

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In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical axioms and non-logical axioms. Axiom - Logical axioms. These are certain formulas in a language that are universally valid, that is, formulas that are satisfied by every structure under every variable assignment function . More colloquially, these are statements that are true in any possible universe, under any possible interpretation and with any assignment of values. Usually one takes as logical axioms some minimal set of tautologies that is sufficient for proving all ...

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Axiom, Axiom - Etymology, Axiom - Mathematics, Axiom - Logical axioms, Axiom - Non-logical axioms, Axiom - Role in mathematical logic, Axiom - Further discussion

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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 ...

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• 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

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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 ...

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• Zermelo-Fraenkel set theory - The Axioms

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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 number of members (the same cardinality) when there is a way of pairing off members of A exhaustively with members of B. Then the set N of natural numbers has the same cardinality as the set Q of rational numbers (they are both said to be countably infinite), even though N is a proper subset of Q. On the other hand, the set R of real numbers d ...

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Axiomatic set theory, 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

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The accepted standard for set theory is Zermelo-Fraenkel set theory. The links show where the axioms of Zermelo's theory correspond. There is no exact match for "elementary sets". (It was later shown that the singleton set could be derived from what is now called "Axiom of pairs". If a exists, a and a exist, thus {a,a} exists. By extensionality {a,a} = {a}.) The empty set axiom i ...

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Zermelo set theory, Zermelo set theory - The axioms of Zermelo set theory, Zermelo set theory - Connection with standard set theory, Zermelo set theory - The aim of Zermelo's paper, Zermelo set theory - The axiom of separation, Zermelo set theory - Cantor's theorem

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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 set: There is a set with no elements. We will also use {} to denote this empty set. Axiom of pairing: If x, y are sets, then there exists a set containing x and y as its only elements, which we denote by {x,y} or {x} ∪ {y}. < ...

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Zermelo-Fraenkel set theory, Zermelo-Fraenkel set theory - The Axioms

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These are the de facto standard axioms for contemporary mathematics Axiom of extensionality Axiom of empty set Axiom of pairing Axiom of union Axiom of infinity Axiom schema of replacement Axiom of power set Axiom of regularity Axiom of separation Axiom schema of specification See also Zermelo set theory. ...

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List of axioms, List of axioms - Zermelo-Frankel axioms, List of axioms - Axiom of choice, List of axioms - Equivalents of AC, List of axioms - Weaker than AC, List of axioms - Alternates incompatible with AC, List of axioms - Other axioms of mathematical logic, List of axioms - Geometry, List of axioms - Other axioms

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The axioms for set theory now most often studied and used, although put in their final form by Skolem, are called the Zermelo-Fraenkel set theory (ZF). Actually, this term usually excludes the axiom of choice, which was once more controversial than it is today. When this axiom is included, the resulting system is called ZFC. An important feature of ZFC is that every object that it deals with is a set. In particular, every element of a set is itself a set. Other familiar mathematical objects, s ...

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Axiomatic set theory, 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

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Many important statements are independent of ZFC, see the list of statements undecidable in ZFC. The independence is usually proved by forcing, that is, it is shown that every countable transitive model of ZFC (plus, occasionally, large cardinal axioms) can be expanded to satisfy the statement in question, and (through a different expansion) its negation. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can ...

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Axiomatic set theory, 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

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Zermelo comments that Axiom III of his system is the one responsible for eliminating the antinomies. It differs from the original definition by Cantor, as follows. Sets cannot be independently defined by any arbitrary logically definable notion. They must be separated as subsets of sets already "given". This, he says, eliminates contradictory ideas like "the set of all sets" or "the set of all ordinal numbers". He disposes of the Russell paradox by means of a Theorem. "Every set M possess at least one subset M< ...

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Zermelo set theory, Zermelo set theory - The axioms of Zermelo set theory, Zermelo set theory - Connection with standard set theory, Zermelo set theory - The aim of Zermelo's paper, Zermelo set theory - The axiom of separation, Zermelo set theory - Cantor's theorem

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The introduction states that the very existence of the discipline of set theory "seems to be threatened by certain contradictions or "antinomies", that can be derived from its principles – principles necessarily governing our thinking, it seems – and to which no entirely satisfactory solution has yet been found". Zermelo is of course referring to the "Russell antinomy". He says he wants to show how the original theory of Cantor and Dedekind can be reduced to a few definitions and seven principles or axioms. He says he ...

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Zermelo set theory, Zermelo set theory - The axioms of Zermelo set theory, Zermelo set theory - Connection with standard set theory, Zermelo set theory - The aim of Zermelo's paper, Zermelo set theory - The axiom of separation, Zermelo set theory - Cantor's theorem

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From these initial axioms for sets one can construct all other mathematical concepts and objects: number - discrete and continuous, order, relation, function , etc. For example, whilst the elements of a set have no intrinsic ordering it is possible to construct models of ordered lists. The essential step is to be able to model the ordered pair ( a, b ) which represents the pairing of two objects in this order. The defining property of an ordered pair is that ( a, b ) = ( c, d ) if and only if a = c and b = d. The approach is basically to specify th ...

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Axiomatic set theory, 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

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Since its inception, there have been some mathematicians who have objected to using set theory as a foundation for mathematics, claiming that it is just a game which includes elements of fantasy. Notably, Henri Poincaré is supposed to have said "set theory is a disease from which mathematics will one day recover", (this quotation is part of the folklore of mathematics; the original source is unknown) and Errett Bishop dismissed set th ...

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Axiomatic set theory, 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

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In 1917, Dmitry Mirimanov (also spelled Mirimanoff) introduced the concept of well-foundedness: a set, x0, is well founded iff it has no infinite descending membership sequence: · · · In ZFC, there is no infinite descending ∈-sequence by the axiom of regularity (for a proof see Axiom of regularity). In fact, the axiom of regularity is often called the foundation axiom since it can be proved within ZFC- (that is, ZFC wit ...

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Axiomatic set theory, 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

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By work of Kurt Gödel and Paul Cohen, the axiom of choice is logically independent of the other axioms of Zermelo-Fraenkel set theory (ZF). This means that neither it nor its negation can be proven to be true in ZF. Consequently, assuming the axiom of choice, or its negation, will never lead to a contradiction that could not be obtained without that assumption. So the decision whether or not it is appropriate to make use of the axiom of choice in a proof cannot be made by appeal to other axioms of s ...

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Axiom of choice, 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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Interestingly, in various varieties of constructive logic (in particular, intuitionistic logic) in which the law of excluded middle is not assumed, the assumption of the axiom of choice is sufficient to obtain the law of excluded middle as a theorem. To see this, for any proposition let be the set and let be the set (see Set-builder notation). By the axiom of choice, there will exist a choice function for the set (note that, although the axiom of choice isn't classically required in order to obtain choice functions for finite sets, it ...

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Axiom of choice, 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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