Dedekind-finite - Factbites

Dedekind-finite - Factbites
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Topic: Dedekind-finite

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	dedekind_inf
		Herman's assertion is that there are models of ZF in which there is an infinite, Dedekind finite set (which must of necessity not be well-orderable).
		If the answer were 'yes', finite and Dedekind finite would coincide; in order for them to be distinct concepts, there must be an infinite, Dedekind finite set.
		It's also easy to show that every infinite, well-ordered set is Dedekind infinite: well-order it as {x(i) : i < k} for some infinite cardinal k, and let f(x(i)) = x(i+1).
	www.math.niu.edu /~rusin/known-math/98/dedekind_inf (474 words)

	dedekind_inf
		Herman's assertion is that there are models of ZF in which there is an infinite, Dedekind finite set (which must of necessity not be well-orderable).
		If the answer were 'yes', finite and Dedekind finite would coincide; in order for them to be distinct concepts, there must be an infinite, Dedekind finite set.
		It's also easy to show that every infinite, well-ordered set is Dedekind infinite: well-order it as {x(i) : i < k} for some infinite cardinal k, and let f(x(i)) = x(i+1).
	www.math.niu.edu /~rusin/papers/known-math/98/dedekind_inf (474 words)

	dedekind_inf
		If the answer were 'yes', finite and Dedekind finite would coincide; in order for them to be distinct concepts, there must be an infinite, Dedekind finite set.
		Herman's assertion is that there are models of ZF in which there is an infinite, Dedekind finite set (which must of necessity not be well-orderable).
		It's also easy to show that every infinite, well-ordered set is Dedekind infinite: well-order it as {x(i) : i < k} for some infinite cardinal k, and let f(x(i)) = x(i+1).
	www.math.niu.edu /~rusin/known-math/98/dedekind_inf (474 words)

	cohen_models
		Dedekind proposed a set be defined to be infinite if it is bijective with a proper subset of itself and defined to be finite otherwise.
		Back in ZF the case still not ruled out is an infinite (ie inductive infinite) but Dedekind finite set.
		In ZF a proof by induction shows inductive finite -> Dedekind finite.
	www.math.niu.edu /~rusin/known-math/99/cohen_models (474 words)

	Number Fields - Dedekind Domains
		Proof: Since a number ring is a free abelian group of finite rank, any ideal must also be a free abelian group of finite rank (because it is a additive subgroup) thus every ideal is finitely generated.
		Theorem: Every number ring is a Dedekind domain.
		is finite, which implies it is a field because every finite integral domain is a field (because for any element
	rooster.stanford.edu /~ben/maths/numberfield/dedekind.php (192 words)

	cohen_models
		Dedekind proposed a set be defined to be infinite if it is bijective with a proper subset of itself and defined to be finite otherwise.
		In ZF a proof by induction shows inductive finite -> Dedekind finite.
		So ZFC proves there are no Dedekind sets, and so far Dedekind sets are not yet ruled out in ZF from what I have written so far.
	www.math.niu.edu /~rusin/known-math/99/cohen_models (1570 words)

	Dedekind-infinite set - Wikipedia, the free encyclopedia
		Since every infinite, well-ordered set is Dedekind-infinite, and since the axiom of choice is equivalent to the well-ordering theorem stating that every set can be well-ordered, clearly the general axiom of choice (AC) implies that every infinite set is Dedekind-infinite.
		However, the study of Dedekind-infinite sets played an important role in the attempt to clarify the boundary between the finite and the infinite, and also an important role in the history of the axiom of choice.
		In mathematics, a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A.
	en.wikipedia.org /wiki/Dedekind_infinite (830 words)

	God and Mathematical Infinity
		When mathematicians, speaking about infinite sets, use the expression, "all but a finite number of members," they are saying (somewhat like Boethius) that the infinite "all" is still the same size "all" even if we take away some finite set.
		These kinds of reasoning about infinite sets, we know today, can be carried out with logical consistency, though it requires us to take great care not to confuse our intuitions about the finite with the counterintuitive rules about the infinite.
		That removes all doubt as to choice: wherever the infinite is, and there is not an infinity of chances of loss against the chance of winning, there are no two ways about it, all must be given...
	www.asa3.org /asa/PSCF/1998/PSCF3-98Kneale.html (830 words)

	pocket.txt
		The infinite sets are exactly the classes such that there are class bijections witnessing the fact that they are the same size as I. Theorem: The class of finite ordinals is a set.
		Any infinite set would similarly map into the finite ordinals one-to-one, so onto the finite ordinals, and it would follow that the infinite set was the same size as an infinite proper class, which contradicts our axioms.
		There is a set I and relation P such that "for each x in I, there is exactly one y in I such that Pxy" and "for each y in I, there is at most one x such that Pxy" and "for some x, for all y, not Pyx".
	math.boisestate.edu /~holmes/holmes/pocket.txt (910 words)

	dedekind_set
		No, that's not true without choice; an infinite set with no countably infinite subset (equivalently, an infinite set which cannot be put in 1-to-1 correspondence with any of its proper subsets) is called a Dedekind set, and the existence of such sets is consistent with ZF.
		When I wrote "Every infinite subset of N has cardinality aleph_0" I was using N to denote the set of natural numbers, not an arbitrary set; I should have been more explicit.
		(Every infinite subset of N has cardinality aleph_0.) > > You seem to be assuming that every set is either finite > (i.e., can be enumerated by a finite ordinal) or has a > countably infinite subset.
	www.math.niu.edu /~rusin/known-math/00_incoming/dedekind_set (291 words)

	Dedekind Sums: A Combinatorial-Geometric Viewpoint - Beck, Robins (ResearchIndex)
		In particular, there are some natural finite Fourier series which we call Fourier-Dedekind sums, and which form the building blocks of the number of partitions of an integer from a finite set of positive integers.
		In this expository paper we show that there is a common thread to many generalizations of Dedekind sums, namely through the study of lattice point enumeration of rational polytopes.
		Abstract: The literature on Dedekind sums is vast.
	citeseer.ist.psu.edu /292212.html (474 words)

	ABSTRACT ALGEBRA ON LINE: Ideal Theory of Commutative Rings
		An integral domain D is called a Dedekind domain if each proper ideal of D can be written as a product of a finite number of prime ideals of D. We will show in Theorem 12.2.4 that a Dedekind domain has some of the properties of a principal ideal domain.
		Let D be an integral domain with quotient field Q, and let F be a finite extension field of Q. If D* is the set of all elements of F that are integral over D, then D* is a Dedekind domain.
		Specifically, a Dedekind domain must be Noetherian, and any nonzero prime ideal of a Dedekind domain must be maximal.
	www.math.niu.edu /~beachy/aaol/commutative.html (2296 words)

	Dedekind-infinite set - Wikipedia, the free encyclopedia
		The existence of infinite, Dedekind-finite sets was studied by Bertrand Russell and Alfred North Whitehead in
		It is notable that this definition was the first definition of "infinite" which did not rely on the definition of the natural numbers.
		A set is infinite if it is not finite.
	en.wikipedia.org /wiki/Dedekind_infinite (2296 words)

	Infinite Ink: Cardinal Numbers
		A cardinal that is not a finite cardinal is an infinite cardinal.
		In ZF it is possible to have an infinite cardinal which is not a well-ordered cardinal.
		Any infinite cardinal that is also an ordinal is an infinite well-ordered cardinal.
	ii.best.vwh.net /math/cardinals (2296 words)

	ABSTRACT ALGEBRA ON LINE: Ideal Theory of Commutative Rings
		An integral domain D is called a Dedekind domain if each proper ideal of D can be written as a product of a finite number of prime ideals of D. We will show in Theorem 12.2.4 that a Dedekind domain has some of the properties of a principal ideal domain.
		Let D be an integral domain with quotient field Q, and let F be a finite extension field of Q. If D* is the set of all elements of F that are integral over D, then D* is a Dedekind domain.
		Let D be an integral domain with quotient field F, and let I be an ideal of D that is invertible when considered as a fractional ideal.
	www.math.niu.edu /~beachy/aaol/commutative.html (2296 words)

	AMCA: Dubrovin Valuation Rings and Orders in Central Simple Algebras by Joachim Grater
		Especially in the finite-dimensional case valuation rings are closely related to their centers and so the investigation of noncommutative valuation rings can be understood as the investigation of extensions of a given commutative valuation ring.
		This talk presents a survey of a general valuation theory for finite-dimensional division algebras as well as their matrix rings, i.e., central simple algebras.
		B holds for all nonzero x in D. With this notation the invariant valuation rings are exactly those which arise with classical valuations in the sense of Schilling.
	at.yorku.ca /cgi-bin/amca/cacv-21 (2296 words)

	Citations: The axiom of choice - Jech (ResearchIndex)
		....result that it is consistent with ZF that there is a Russell infinite set that is not Dedekind infinite.
		We have also proved: Corollary 15 It is consistent with ZF that there is a weakly Dedekind infinite but Dedekindfinite set.
		Without AC, if P(I) is Dedekind finite and ff is an ordinal number, then the range of each function f : I P(ff) is finite.
	citeseer.ist.psu.edu /context/334361/0 (2296 words)

	IIDB - Infinity
		For example, you could say that the universe is infinite in spatial distance, infinite in amount of energy, or simply infinite in terms of physcially discernable states or points of reference.
		All these sets are countably infinite, not meaning that you could complete the task of counting them, but that they contain no element to which you could not count by a finite process.
		It says that an infinite series of events that stretch out into the past has been traversed or crossed to bring us to the current event we are now experiencing.
	www.iidb.org /vbb/archive/index.php/t-36840.html (2296 words)

	Courses
		Dedekind rings, factorization in Dedekind rings, norms of ideals, splitting of prime ideals in field extensions, finiteness of the ideal class group and Dirichlet's theorem on units are treated in the second part.
		This is a very short introduction to local fields and local class field theory without using homological algebra.
		This course is an introduction to modules over rings, Noetherian modules, unique factorization domains and polynomial rings over them, modules over principal ideal domains, localization.
	www.maths.nott.ac.uk /personal/ibf/courses.html (2296 words)

	Math/CS, Emory Univ.
		Examples and classification of PDE's, initial and boundary value problems, well posed problems, the maximum principle, finite difference methods, variational formulations for elliptic PDE's, finite element methods, direct and iterative solution methods.
		Topics covered will include commutative algebra from Atiyah and McDonald: primary decomposition, integral dependence, valuations, Noetherian rings, Dedekind domains, discrete valuation rings, completions, and dimension theory.
		This is the third semester of the graduate algebra sequence.
	www.mathcs.emory.edu /Graduate/Course/Catalog-2002-2003.html (2296 words)

	Mikhail Bondarko
		[17] Finite flat commutative group schemes over complete discrete valuation fields: classification, structural results; application to reduction of Abelian varieties, Göttingen, 2004.
		[15] Finite flat commutative group schemes over complete discrete valuation rings II: good reduction of abelian varieties, (Russian), (2004).
		[7] The presence of idempotents in the endomorphism ring of an ideal in a $p$-extension of a complete discrete valuation field with a residue field of characteristic $p$ as a Galois module, (Russian)
	www.uni-math.gwdg.de /bondarko (2296 words)

	Order theory Details, Meaning Order theory Article and Explanation Guide
		Even infinite sets can sometimes be illustrated by similar diagrams, using an ellipsis (...) after drawing a sufficiently instructive finite sub-order.
		Again, in infinite posets maximal elements do not always exist - the set of all finite subsets of a given infinite set, ordered by subset inclusion, provides one out of many counterexamples.
		An additional simple but useful property leads to so-called well-orders, within which all non-empty subsets have a least element, or equivalently in which there is no infinite descending sequence of distinct elements.
	www.e-paranoids.com /o/or/order_theory.html (2296 words)

	Number Fields - Dedekind Domains
		Proof: Since a number ring is a free abelian group of finite rank, any ideal must also be a free abelian group of finite rank (because it is a additive subgroup) thus every ideal is finitely generated.
		Theorem: Every number ring is a Dedekind domain.
		Now suppose we are given a nonempty set
	rooster.stanford.edu /~ben/maths/numberfield/dedekind.php (2296 words)

	infinite-full.html
		Call that set N. An infinite set is now defined to be a set that has a subset equinumerous to N. Now we have the following Interesting Fact: To prove that every Dedekind-infinite set is infinite you need to assume the axiom of choice.
		Infinite activities are represented simply by the fact that they lack a begin or end point (or both).
		If I understand you, Austin, it seems that you are wanting to allow for infinite activities, but also suggesting that we assign such activities points at infinity as their begin and/or end points, where a point at infinity is a point that is infinitely distant from any other point.
	www.mel.nist.gov /psl/p3/semantics/issues-log/infinite-full.html (7973 words)

	234a
		This includes the Riemann zeta function, the Dedekind zeta function of a general number field, and the zeta function of a general algerbaic variety over a finite field (Dwork's theorem).
		If time permits, we shall also discuss the application of the algorithmic theory of zeta functions over finite fields to cryptography.
		For the fall quarter Math 234A, we shall cover the most basic parts of the theory of zeta functions.
	www.math.uci.edu /~dwan/234a (268 words)

	Joel Riou's mathematical home page.
		This last theorem involves the Dedekind zeta function of the number field and in some sense Borel proved the Lichtenbaum conjecture "up to a rational factor".
		The former dealt with character theory of linear representations of finite groups, the first goal being to count the number of isomorphism classes of irreducible representations of finite groups.
		The latter was devoted to harmonic polynomials of the sphere and the connexion with the representation theory of the special orthogonal group acting on the space of functions of the sphere, generalizing usual Fourier series.
	www.eleves.ens.fr /home/jriou/index.html.en (973 words)

	Dedekind, Richard
		12, 1916, was a German mathematician known for his study of CONTINUITY and definition of the real numbers in terms of Dedekind "cuts"; his analysis of the nature of number and mathematical induction, including the definition of finite and infinite sets; and his influential work in NUMBER THEORY, particularly in algebraic number fields.
		Dedekind's study of Dirichlet's work led to his own study of algebraic number fields, as well as his introduction of ideals.
		Dedekind also introduced such fundamental concepts as RINGS.
	euler.ciens.ucv.ve /English/mathematics/dedekind.html (121 words)

	Dedekind cuts of partial orderings
		Dedekind cuts are a clever trick for defining the reals given the rationals.
		When the partial ordering is finite and total the cuts add nothing of interest.
		Such a cut considers a set C of rationals such that if x is in C and y < x then y is in C. For any such cut there is just one real "between" C and its complement.
	www.cap-lore.com /MathPhys/Cuts.html (286 words)

	Lee Lady: Finite Rank Torsion Free Modules over Dedekind Domains (a book)
		And one becomes more aware of the fact that the theory of finite rank torsion free abelian groups is moving away from abelian group theory in general in much the same fashion that abelian group theory has moved away from general group theory.
		The transition from modules over principal ideal domains to modules over dedekind domains is actually a much smaller leap than that from the integers to an arbitrary PID.
		Moreover in the process of reformulating theorems and proofs to be valid over dedekind domains, one sees these results in a new and -- in your author's opinion -- "more correct" way.
	www.math.hawaii.edu /~lee/book (629 words)

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