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Topic: Axiom of dependent choice

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	PlanetMath: axiom of countable choice
		The Axiom of Countable Choice (CC) is a weak form of the Axiom of Choice.
		ZF+CC (that is, the Zermelo-Fraenkel axioms together with the Axiom of Countable Choice) suffices to prove that the union of countably many countable sets is countable.
		This is version 11 of axiom of countable choice, born on 2004-10-25, modified 2006-01-05.
	planetmath.org /encyclopedia/CountableAxiomOfChoice.html (189 words)

	Choice Selector (Site not responding. Last check: 2007-10-12)
		Axiom of countable choice - The axiom of countable choice, denoted ACÏ, or axiom of denumerable choice, is an axiom of set theory, similar to the axiom of choice.
		Axiom of dependent choice - In mathematics, the axiom of dependent choice, denoted DC, is a weak form of the axiom of choice which is still sufficient to develop most of real analysis.
		Unlike the full axiom of choice (AC), DC is insufficient to prove (given ZF) that there is a nonmeasurable set of reals, or that there is a set of reals without...
	powersupplies.vvvvvv3.com /choiceselector.html (1143 words)

	Axiom of choice - Article from FactBug.org - the fast Wikipedia mirror site (Site not responding. Last check: 2007-10-12)
		The axiom of choice is typically abbreviated AC, or C as a suffix.
		Because of this, one argument given in favor of the axiom of choice is that it is convenient: It doesn't hurt, and it's easier to use the axiom of choice than not.
		A third possibility is to prove theorems using neither the axiom of choice nor its negation, a tactic often preferred in constructive mathematics.
	www.factbug.org /cgi-bin/a.cgi?a=840 (1936 words)

	Process Raman Spectroscopy \| Infrared Spectroscopy \| IR Spectroscopy \| Axiom Analytical (Site not responding. Last check: 2007-10-12)
		As a result, the choice of which technique to apply can sometimes be made on a routine basis once the general nature of the sample system is known.
		While the exact requirement depends on required accuracy of the measurement, a rule of thumb that is often applied is that the diameter of the illuminated and viewed area should be at least a hundred times the typical particle diameter.
		Since the dependence on scatterer concentration is highly nonlinear, it would be difficult to obtain a calibration by using any of the linear regression methods.
	www.goaxiom.com /an-922.html (5839 words)

	[No title]
		Depends on what you mean by "construction." The standard definition of the Vitali Set requires the Axiom of Choice and is non-constructive.
		I think that Sollovay proved that if you accept a particular negation of the Axiom of Choice (or a particular axiom which implies not(Choice), then every subset would be measurable; I know he did it at some dimension, but cannot guarantee it was in dimension one.
		I > think that Sollovay proved that if you accept a particular negation of > the Axiom of Choice (or a particular axiom which implies not(Choice), > then every subset would be measurable; I know he did it at some > dimension, but cannot guarantee it was in dimension one.
	www.math.niu.edu /~rusin/known-math/99/nonmbl (997 words)

	[No title]
		Last-modified: 12 November 1994 Relevance of the Axiom of Choice There are many equivalent statements of the Axiom of Choice.
		The axioms of set theory provide a foundation for modern mathematics in the same way that Euclid's five postulates provided a foundation for Euclidean geometry, and the questions surrounding AC are the same as the questions that surrounded Euclid's Parallel Postulate: 1.
		For many sets, including any finite set, the first six axioms of set theory (abbreviated ZF) are enough to guarantee the existence of a choice function but there do exist sets for which AC is required to show the existence of a choice function.
	www.halcyon.com /pub/ii/Math/acfaq.txt (1408 words)

	A/Prof N J Wildberger Personal Pages
		Axiom of Foundation: Every nonempty set has a minimal element, that is one which does not contain another in the set.
		Axiom of Choice: Every family of nonempty sets has a choice function, namely a function which assigns to each of the sets one of its elements.
		With the notable exception of the `Axiom of Choice', I bet that fewer than 5% of mathematicians have ever employed even one of these `Axioms' explicitly in their published work.
	web.maths.unsw.edu.au /~norman/views2.htm (6444 words)

	Topological Equivalents of the Axiom of Choice and of Weak Forms of Choice, by Eric Schechter
		The Axiom of Choice is the most well-known nonconstructive assertion of existence; it has important consequences for many branches of mathematics.
		Although the term ``constructive'' is used in different fashion by different mathematicians, the Axiom of Dependent Choice is the strongest form of choice that is widely held to be constructive.
		A couple of very weak consequences of the Axiom of Choice are the existence of (i) subsets of R which are not Lebesgue measurable, and (ii) subsets of R which lack the Baire property.
	at.yorku.ca /z/a/a/b/18.htm (848 words)

	Practical Foundations of Mathematics
		The axiom of choice The increasingly abstract form of late nineteenth century mathematics led to the use of infinite families of choices, often with no conscious understanding that a new logical principle was involved.
		Although we, on the cusp of the millennium, now reject Choice, it was the way forward at the start of the twentieth century: it stimulated research throughout mathematics, notably in the Polish school, which we have to thank for numerous ideas in logic and general topology mentioned in this book [Moo82, McC67].
		The axiom of choice is typically not needed in the concrete cases, because their own structure provides some way of making the selections (we shall indicate real uses of Choice by the capital letter).
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s18.html (2566 words)

	sci.math FAQ: The Axiom of Choice
		Relevance of the Axiom of Choice THE AXIOM OF CHOICE There are many equivalent statements of the Axiom of Choice.
		Since AC gives no method for constructing a choice set constructivists belong to school C. Formalism A formalist believes that mathematics is strictly symbol manipulation and any consistent theory is reasonable to study.
		For a formalist the notion of truth is confined to the context of mathematical models, e.g., a formalist would say "The parallel postulate is false in Riemannian geometry." but she wouldn't say "The parallel postulate is false." A formalist will probably not allign herself with any school.
	www.cs.uu.nl /wais/html/na-dir/sci-math-faq/axiomchoice.html (1586 words)

	MainFrame:Axioms for galactic set theory.
		The axioms of extensionality and well-foundedness may be thought of as telling us what kind of thing a set is (later axioms tell us how many of these sets are to be found in our domain of discourse).
		The axiom of well-foundedness asserts the requirement that the elements of ('a)GS are a subset of the cumulative heirarchy of sets formed by iteration of set formation beginning with the empty set.
		The remaining axioms are intended to ensure that the subset is a large and well-rounded subset of the cumulative heirarchy.
	www.rbjones.com /x-logic/pp/gst/gst-axioms-m.html (2612 words)

	Constructive Mathematics
		The term was often used to indicate an avoidance of the axiom of choice---a principle that asserts the existence of a function with certain properties in situations in which it is particularly unclear how such a function could be constructed.
		The most salient addition is the axiom of dependent choices, a strong countable choice axiom.
		This axiom conforms with the practice of the Russian school and of the intuitionists.
	www.math.fau.edu /Richman/HTML/CONSTRUC.HTM (1293 words)

	No title
		Joh82] stresses that one may avoid the use of the axiom of choice in topology by using locale theory and dealing directly with the opens.
		It should be noted that we have used normability and separability hypothesis in the statements of the main theorems and used the axiom of dependent choice.
		Finally, the axiom of dependent choice is used in Theorem 16 to construct a point in each positive open, and thus obtain a metric space in the sense of Bishop.
	www.cs.ru.nl /B.Spitters/StoneYosida.html (3390 words)

	[No title]
		The problem of choice reduces to the complete distributivity of that Boolean algebra, and algebraists know examples of Boolean algebras which are not completely distributive, If they are careful enough, an ultrafilter will not erase that failure.
		Both the Axiom of Choice and its negation are consistent.
		Subject: Re: Axiom of Choice Debate Date: Sun, 21 Feb 1999 01:39:58 -0500 Newsgroups: sci.logic,sci.math I am familiar with the results of ZF + AD yielding that call sets of real numbers are Lebesque measurable, but I had trouble with that since the set's non-measurability had nothing to do with AC, only its construction.
	www.math.niu.edu /~rusin/known-math/99/AD_AC (1784 words)

	AskPhilosophers.org
		The Axiom of Choice (usually denoted "AC") is a statement of set theory rather than of basic mathematical logic, so the theories of interest are versions of set theory that reject AC.
		As Dan said, any theory containing the Axiom of Determinacy will imply not-AC, but one can also simply look at what is possible without AC and, similarly, what cannot be proven without AC.
		There are also weaker forms of AC, such as the Axiom of Countable Choice (every countable set of non-empty sets has a choice function) and the Axiom of Dependent Choice (more complicated).
	www.amherst.edu /askphilosophers/question/686 (429 words)

	^^^ Προσωπική Ιστοσελίδα ...
		Keremedis (with E. Tachtsis): "On Lindelof metric spaces and weak forms of the axiom of choice", [pdf], Math.
		Keremedis: "The Vector space Kinna-Wagner Principle is equivalent to the axiom of choice", Math.
		Keremedis: "The failure of the axiom of choice implies unrest in the theory of metric spaces", accepted, Math.
	www.samos.aegean.gr /math/kker (820 words)

	Relevance of the Axiom of Choice
		The Axiom of Choice (AC) is one of the most discussed axioms of mathematics, perhaps second only to Euclid's parallel postulate.
		The axioms of set theory provide a foundation for modern mathematics in the same way that Euclid's five postulates provided a foundation for Euclidean geometry, and the questions surrounding AC are the same as the questions that surrounded Euclid's Parallel Postulate:
		Accept only a weak form of AC such as the Denumerable Axiom of Choice (every denumerable set has a choice function) or the Axiom of Dependent Choice.
	www.cs.uwaterloo.ca /~alopez-o/math-faq/node69.html (1384 words)

	1999 Changes and additions to ``Consequences of the Axiom of Choice" Project (Site not responding. Last check: 2007-10-12)
		Keremedis, K./Tachtsis, E. [1999b] The countable axiom of choice for finite sets does not imply compact metric spaces are separable preprint.
		[5 A] Partial Choice for Countable Families of Countable Sets of Reals: Every countable family of non-empty countable sets of real numbers has an infinite subset with a choice function.
		Martin's Axiom $(\aleph_{0})$: Whenever $(P\le)$ is a non-empty, ccc quasi-order (ccc means every anti-chain is countable) and ${\Cal D}$ is a family of $\le\aleph_0$ dense subsets of $P$, then there is a ${\Cal D}$ generic filter $G$ in $P$.
	www.math.purdue.edu /~jer/cgi-bin/changes-99.htm (571 words)

	FLoC '02 - ICC Sunday July 21st
		Modified bar recursion is a variant of Spector's bar recursion which can be used to give a realizability interpretation of the classical axiom of dependent choice.
		We focus on how the choice of input-output representation has a crucial impact on the expressiveness of so-called "logics of polynomial time." Our analysis illustrates this dependence in the context of Light Affine Logic (LAL), which is both a restricted version of Linear Logic, and a primitive functional programming language with restricted sharing of arguments.
		By slightly relaxing representation conventions, we derive doubly-exponential expressiveness bounds for this "logic of polynomial time." We emphasize that squaring is the unifying idea that relates upper bounds on cut eliminations for LAL with lower bounds on representation.
	floc02.diku.dk /ICC/Sunday.html (474 words)

	[No title]
		Concerning the ways some new axiom might be justified, he claimed that even if it lacks intuitiveness, we might decide to accept it as true for the same kinds of reasons we accept a well-established physical theory: because of its power to prove verifiable consequences, obtain new results, and illuminate old ones.
		The choice between the von Neumann ordinals and the Zermelo ordinals is no more than the choice between two different rulers that both measure in metres.
		Admitting that Zermelo had no way of carrying out the mapping needed for a choice function, he insisted that the problem of its effective determination is completely distinct from the question of its existence: ``The existence$\dots$is a fact like any other.'' Today Hadamard's pro-Choice position has prevailed with the vast majority of mathematicians.
	www.ams.org /journals/bull/pre-1996-data/199501/199501019.tex.html (3819 words)

	[No title]
		It depends of what you want to do with this definition The explanation in this mail relies on some technical points about impredicative type theories and may seem obscure if you are not familiar with this matter.
		Yes it is. I checked your weak axiom of choice some days ago by explicitely writing down the denotation in the standard omega-set model of CIC.
		This remarks clearly solves the problem of the weak axiom of choice, because it means that any set-theoretical choice function is also an element of the carrier of the weak-choice-axiom omega-set, and moreover, that any such function will be realized by any program.
	pauillac.inria.fr /pipermail/coq-club/2002.txt (14240 words)

	C.J.Mulvey
		In the first of these papers, constructions of the compact completely regular reflection and of the compact regular reflection of a locale were obtained, these coinciding in the presence of the Axiom of Countable Dependent Choice.
		There was, however, a fundamental difference in the approaches taken between these cases: the compact completely regular reflection is obtainable directly as the locale of completely regular ideals of the locale, while the compact regular reflection has a more indirect description.
		This paper examines the way in which the application of the Axiom of Choice may be avoided by carefully chosen constructive argument of a geometric nature.
	www.maths.sussex.ac.uk /Staff/CJM/research/CJMResearch.htm (1637 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		It is known that the real numbers being a countable union of countable sets, which would give the analysts major problems, is consistent.
		Although the Power Set Axiom is available in both cases, P(R) with AD must be a different power set than P(R) with AC.
		There are models without choice which also have non-measurable sets.
	www.mat.niu.edu /~rusin/known-math/99/AD_AC (1784 words)

	Spreads and choice in constructive mathematics
		In [3] the problem posed by the fundamental theorem of algebra is solved by redefining what a solution is. Instead of trying to approximate a single root of the polynomial, we approximate the set of roots---the spectrum.
		The bad news is that, without countable choice, we cannot prove the existence of such a path.
		Constructing a suitable spread is essentially the same as showing that there is a choice sequence with the required property.
	www.math.fau.edu /Richman/Docs/spreads.htm (2237 words)

	Ulrich Berger, Publications
		We introduce a variant of Spector's bar recursion in finite types (which we call ``modified bar recursion'') to give a realizability interpretation of the classical axiom of dependent choice allowing for the extraction of witnesses from proofs of forall-exists-formulas in classical analysis.
		We study the proof-theoretic and computational properties of open induction, a principle which is classically equivalent to Nash-Williams' minimal-bad-sequence argument and also to (countable) dependent choice (and hence contains full classical analysis).
		We show that, intuitionistically, open induction and dependent choice are quite different: Unlike dependent choice, open induction is closed under negative- and A-translation, and therefore proves the same Pi02-formulas (over not necessarily decidable, basic predicates) with classical or intuitionistic arithmetic.
	www.cs.swan.ac.uk /~csulrich/publications.html (2073 words)

	[Coq-Club] weak choice axiom (Site not responding. Last check: 2007-10-12)
		Would it be consistent (including consistency with existence of the sort Set) to assume the following weak axiom of choice?
		Axiom weak_choice : (X : Type) (is_nonempty X) -> (single X).
		One wouldn't, on the other hand, assume the "axiom of dependent choice" because then this would be equivalent to the usual axiom of choice.
	pauillac.inria.fr /pipermail/coq-club/2002/000507.html (176 words)

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