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Topic: Dedekind infinite

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	Dedekind cut Summary
		Dedekind used cuts to prove the completeness of the reals without using the axiom of choice (proving the existence of a complete ordered field to be independent of said axiom).
		The Dedekind cut resolves the contradiction between the continuous nature of the number line continuum and the discrete nature of the numbers themselves.
		The Dedekind cut is named after Richard Dedekind, who invented this construction in order to represent the real numbers as Dedekind cuts of the rational numbers.
	www.bookrags.com /Dedekind_cut (1240 words)

	Dedekind biography
		Dedekind and Dirichlet soon became close friends and the relationship was in many ways the making of Dedekind, whose mathematical interests took a new lease of life with the discussions between the two.
		Dedekind's work was quickly accepted, partly because of the clarity with which he presented his ideas and partly since Heinrich Weber lectured to Hilbert on these topics at the University of Königsberg.
		Dedekind's brilliance consisted not only of the theorems and concepts that he studied but, because of his ability to formulate and express his ideas so clearly, he introduced a new style of mathematics that been a major influence on mathematicians ever since.
	www-history.mcs.st-andrews.ac.uk /Biographies/Dedekind.html (2067 words)

	Dedekind biography
		Dedekind and Dirichlet soon became close friends and the relationship was in many ways the making of Dedekind, whose mathematical interests took a new lease of life with the discussions between the two.
		Dedekind's work was quickly accepted, partly because of the clarity with which he presented his ideas and partly since Heinrich Weber lectured to Hilbert on these topics at the University of Königsberg.
		Dedekind's brilliance consisted not only of the theorems and concepts that he studied but, because of his ability to formulate and express his ideas so clearly, he introduced a new style of mathematics that been a major influence on mathematicians ever since.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Dedekind.html (2067 words)

	20th WCP: The Model Theory Of Dedekind Algebras
		Each Dedekind algebra is associated with a cardinal value function called the confirmation signature which counts the number of configurations in each isomorphism type occurring in the decomposition of the algebra.
		A subalgebra, A, of the Dedekind algebra B is a small subalgebra of B provided the cardinality of the domain of A is strictly smaller than the cardinality of the domain of B. Let B be an infinite Dedekind algebra.
		THEOREM 9: All uncountable homogeneous Dedekind algebras are quasi-characterizable.
	www.bu.edu /wcp/Papers/Logi/LogiWeav.htm (3006 words)

	Infinity - Free Encyclopedia of Thelema
		Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part.
		An infinite set can simply be defined as one having the same size as at least one of its "proper" parts; this notion of infinity is called Dedekind infinite.
		Likewise, perpetual motion machines theoretically generate infinite energy by attaining 100% efficiency or greater, and emulate every conceivable open system; the impossible problem follows of knowing that the output is actually infinite when the source or mechanism exceeds any known and understood system.
	www.egnu.org /thelema/Infinity (2587 words)

	PlanetMath: infinite
		Assuming the Axiom of Choice (or the Axiom of Countable Choice), this definition of infinite sets is equivalent to that of Dedekind-infinite sets.
		This is version 14 of infinite, born on 2001-11-16, modified 2006-01-01.
		Object id is 881, canonical name is Infinite.
	planetmath.org /encyclopedia/Infinite.html (89 words)

	Julius Wihelm Richard Dedekind (Site not responding. Last check: 2007-11-05)
		Richard Dedekind attended school in Brunswick from the age of 7, and at this stage mathematics was not his main interest.
		Dedekind made a number of highly significant contributions to mathematics and his work would change the style of mathematics into what is familiar to us today.
		In the book Dedekind presented a logical theory of number and of complete induction, presented his principal conception of the essence of arithmetic, and dealt with the role of the complete system of real numbers in geometry in the problem of the continuity of space.
	www.stetson.edu /~efriedma/periodictable/html/Db.html (695 words)

	Cardinal number Summary
		He also proved that the set of all ordered pairs of natural numbers is denumerably infinite, and later that the set of all algebraic numbers is denumerably infinite.
		However when dealing with infinite sets it is essential to distinguish between the two — the two notions are in fact different for infinite sets.
		It is possible to use a different formal notion for number, called ordinals, based on the ideas of counting and considering each number in turn, and we discover that the notions of cardinality and ordinality are divergent once we move out of the finite numbers.
	www.bookrags.com /Cardinal_number (2645 words)

	Richard Dedekind - Wikipedia, the free encyclopedia
		Dedekind was elected to the Academies of Berlin (1880) and Rome, and to the Paris Académie des Sciences (1900).
		If there existed a one-to-one correspondence between two sets, Dedekind said that the two sets were "similar." He invoked similarity to give the first precise definition of an infinite set: a set is infinite when it is "similar to a proper part of itself," in modern terminology, is equinumerous to one of its proper subsets.
		Dedekind's study of Dirichlet's work was what led him to his later study of algebraic number fields and ideals.
	en.wikipedia.org /wiki/Richard_Dedekind (1187 words)

	Dedekind's Real Numbers
		Dedekind regards counting as the simplest arithmetic act, and describes this as ``the successive creation [Schöpfung] of the infinite series of positive integers in which each individual is defined by the one immediately preceeding'' (p.
		One reason that Dedekind gives in a letter to Weber from January 24 1888 for not identifying the irrational numbers with the cuts themselves is that one might attribute certain properties with the cuts, that would sound odd if they were attributed to the numbers.
		Dedekind's general aim is the reconstruction of the systems of numbers starting with the natural numbers, in such a way, that the building blocks of each system are only elements belonging to previously obtained systems.
	www.colorado.edu /StudentGroups/PhilosophyClub/reals.htm (1860 words)

	From Frege To Godel: von Heijenoort (Site not responding. Last check: 2007-11-05)
		Brouwer decried the use of excluded middle in infinite situations and suggested (demanded?) a move towards intuitionism.
		Keferstein confused an equivalence relation on sets with identity of sets; he thought two different definitions of infinite sets were in conflict but were, in fact, merely stylistic variants; he proposed a new definition of simply infinite sets that avoided chains but abandoned mathematical induction.
		Dedekind's letter explains the development and justification of his ideas on the notion of natural number.
	www.andrew.cmu.edu /~cebrown/notes/vonHeijenoort.html (8419 words)

	Reference.com/Encyclopedia/Infinity
		Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part.
		An infinite set can simply be defined as one having the same size as at least one of its " proper" parts; this notion of infinity is called Dedekind infinite.
		Likewise, perpetual motion machines theoretically generate infinite energy by attaining 100% efficiency or greater, and emulate every conceivable open system; the impossible problem follows of knowing that the output is actually infinite when the source or mechanism exceeds any known and understood system.
	www.reference.com /browse/wiki/Infinite (3901 words)

	PlanetMath: Dedekind-infinite
		A Dedekind-infinite set is clearly infinite, and in ZFC it can be shown that a set is Dedekind-infinite if and only if it is infinite.
		It is consistent with ZF that there is an infinite set that is not Dedekind-infinite.
		However, the existence of such a set requires the failure not just of the full Axiom of Choice, but even of the Axiom of Countable Choice.
	planetmath.org /encyclopedia/DedekindInfinite.html (120 words)

	untitled
		Dedekind came to the conclusion that the essence of the continuity of a line segment is not due to a vague hang-togetherness, but to an exactly opposite property--the nature of the division of the segment into two parts by a point on the segment (Boyer, 1968, p.
		Dedekind asked us to imagine a blade with an infinitely thin blade (i.e., it has no thickness at all) which can be used to cut the continuum into two segments.
		Peirce was especially proud of this discovery, claiming that "the proposition that finite and infinite collections are distinguished by the applicability to the former of the syllogism of transposed quantity ought to be regarded as the basal one of scientific arithmetic" (CP6.114).
	www.angelfire.com /super/magicrobin/peirce.htm (7996 words)

	The Continuum Hypothesis
		With his theory of sets and his introduction of the concept of infinite nu mbers, Cantor broke through the barriers of previous generations, and has allowed for the further exploration of areas that were previously unattainable.
		Dedekind was another mathematician whose thoughts were similar to Cantor's.
		Richard Dedekind was a teacher at the Polytechni c School in Zurich.
	www.math.rutgers.edu /courses/436/436-s00/Papers2000/brazza.html (2485 words)

	quick question...
		A set is Dedekind infinite iff there exists a bijection from it to a proper subset.
		Erm, a set is dedekind infinite if there is an injection from it to a proper subset of itself, just like i said, and just like you said.
		It's easy to show this is the same as Dedekind infinite provided you use a certain technical axiom that some people feel is best avoided.
	www.physicsforums.com /showthread.php?t=45805 (632 words)

	Dedekind-infinite set - Wikipedia, the free encyclopedia
		This definition of "infinite set" should be compared and contrasted to the usual definition: a set A is finite if A is empty, or if there is a positive integer n such that A is equinumerous to the set {1, 2, 3,..., n}.
		Since every infinite, well-ordered set is Dedekind-infinite, and since the AC is equivalent to the well-ordering theorem stating that every set can be well-ordered, clearly the general AC implies that every infinite set is Dedekind-infinite.
		It is notable that this definition was the first definition of "infinite" which did not rely on the definition of the natural numbers.
	en.wikipedia.org /wiki/Dedekind-infinite_set (867 words)

	[No title]
		The existence of an infinite set, the existence of a (necessarily unique) model of the Peano axioms, and the existence of a (necessarily unique) universal dynamical system are all logically equivalent (the existence of any one of these entities implies the existence of the others).
		Dedekind's axiomatic characterizations of the major number systems accomplished for analysis what Euclid had accomplished for geometry some two thousand years earlier.
		Though partly based on a negative and skeptical view of infinite mathematics, constructivism is seen by its proponents as a positive and vigorous philosophy of mathematics.
	bahai-library.com /?file=hatcher_foundations_mathematics (14902 words)

	Set theory
		Bolzano gave examples to show that, unlike for finite sets, the elements of an infinite set could be put in 1-1 correspondence with elements of one of its proper subsets.
		Numerous letters between the two in the years 1873-1879 are preserved and although these discuss relatively little mathematics it is clear that Dedekind's deep abstract logical way of thinking was a major influence on Cantor as his ideas developed.
		Dedekind was working independently on irrational numbers and Dedekind published Continuity and irrational numbers.
	www-gap.dcs.st-and.ac.uk /~history/HistTopics/Beginnings_of_set_theory.html (2182 words)

	[No title]
		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.
		And one you are stuck with infinite time, you have to add awkward qualifications like, for every timepoint ?t there is another before, unless of course ?t is -inf, etc etc. This all I take to count as harm in allowing infinite activities.
		Infinite activities are represented simply by the fact that they lack a begin or end point (or both).
	www.mel.nist.gov /psl/p3/semantics/issues-log/infinite-full.html (7973 words)

	Nicholas of Cusa and the Infinite
		The former is a view from the finite upward toward the unattainable and incomprehensible infinite, while the latter is an incomprehensible view from the infinite downward toward the finite that is identical with the infinite.
		Thus, the actual infinite ultimately had to be explicitly affirmed in mathematics in order to provide a foundation for the numbers used in both analytic geometry and calculus.
		The history of the Infinite thus reveals in both mathematics and philosophy a development of increasingly subtle thought in the form of a dialectical dance around the ineffable and incomprehensible Infinite.
	www.integralscience.org /cusa.html (5490 words)

	AskPhilosophers.org
		Its infinitude doesn't consist in the fact that there are actually an infinite number of chairs (for there never is actually more than a finite number of chairs), but rather in the fact that we can keep on generating more chairs forever.
		Most people accepted Dedekind's proof, but it was clear from the outset that there was something strange about it.
		But it can be proven that, without the axiom of choice (more precisely, without countable choice), one cannot prove that every infinite set is Dedekind infinite: If the axiom of (countable) choice is false, then there may be a Dedekind finite set that is not finite.
	www.amherst.edu /askphilosophers/topic/Mathematics&page=3 (3405 words)

	Amazon.com: Essays on the Theory of Numbers: Books: Richard Dedekind (Site not responding. Last check: 2007-11-05)
		Richard Dedekind (1831-1916) is recognized as one of the great pioneers in the logical and philosophical analysis of the foundations of mathematics.
		Dedekind completed his doctoral studies under Gauss, was a friend of Cantor and Riemann, and worked under Dirichlet.
		Richard Dedekind is one of the fathers of modern mathematical proofs.
	www.amazon.com /Essays-Theory-Numbers-Richard-Dedekind/dp/0486210103 (1653 words)

	Content
		Dedekind cut: any real number corresponds to a Dedekind cut of the set of rational numbers.
		A Dedekind cut in an ordered field is a partition of it, (A, B), such that A is closed downwards (meaning that whenever a is in A and x ≤ a, then x is in A as well), B is closed upwards and A has no maximum.
		Every bounded infinite set has an accumulation point; that is, every bounded sequence must have a convergent subsequence.
	www.cs.cmu.edu /~dpwu/books/math/analysis/RealAnalysis.html (3085 words)

	[No title]
		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.
		It is also strong enough to consider the existence of models of any first-order theory we can express from a Platonist standpoint; if one is committed to pocket set theory, one believes that there is an answer to the question as to whether it is consistent that there is a measurable cardinal.
	math.boisestate.edu /~holmes/holmes/pocket.txt (910 words)

	Infinite Ink: Cardinal Numbers
		A cardinal that is not a finite cardinal is an infinite cardinal.
		Any infinite cardinal that is also an ordinal is an infinite well-ordered cardinal.
		In ZF it is possible to have an infinite cardinal which is not a well-ordered cardinal.
	www.ii.com /math/cardinals (1276 words)

	More on Infinity
		In mathematics, some articles relevant to the subject can be found at limit (mathematics), aleph number, class (set theory), Dedekind infinite, large cardinal, Russell's paradox, hyperreal numbers, projective geometry, extended real number and absolute infinite.
		It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of classes, of which one is correlated with the other, but which are never related as class and subclass.
		Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null (\aleph_0), the cardinality of the set of natural numbers.
	www.artilifes.com /infinity.htm (2548 words)

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