Where results make sense

About us   |   Why use us?   |   Reviews   |   PR   |   Contact us

Topic: Dedekind-infinite set

    Note: these results are not from the primary (high quality) database.

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)

Category:Set theory - Wikipedia, the free encyclopedia Internal set theory is an axiomatic extension of set theory that supports a logically consistent identification of illimited (enormously large) and infinitesimal elements within the real numbers. Naive set theory is the original set theory developed by mathematicians at the end of the 19th century, treating sets simply as collections of things. Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's paradox) in naive set theory. en.wikipedia.org /wiki/Category:Set_theory   (181 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)

Infinite Sets Together with the trivial truth that no finite set can be put into one-to-one correspondence with any of its proper subsets, this theorem establishes the important result that all and only infinite sets possess the property that they can be put into one-to-one correspondence with at least one of their proper subsets. We know from the proof of Theorem 6 that every infinite set has an infinite proper subset which consists of the original set minus denumerably many members. In Theorems 1 and 2 we saw that two kinds of infinite addition to an infinite set did not increase the cardinality of the original set. www.mathpath.org /concepts/infinity.htm   (6833 words)

Set Theory The smallest infinite cardinal is the cardinality of a countable set. Cantor observed that many infinite sets of numbers are countable: the set of all integers, the set of all rational numbers, and also the set of all algebraic numbers. Set Theory is the mathematical science of the infinite. plato.stanford.edu /entries/set-theory   (3302 words)

Infinite Number [Definition] Dedekind's approach was essentially to adopt the idea of one-to-one correspondenceIn mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective ("one-to-one") and surjective ("onto"), and therefore bijections are also called one-to-one and onto. 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. It appeared, by this reasoning, as though a set which is naturally smaller than the set of which it is a part (since it does not contain all the members of that set) is in some sense the same size. www.wikimirror.com /Infinite_number   (5324 words)

PlanetMath: infinite Assuming the axiom of choice, this definition of infinite sets is equivalent to that of dedekind-infinite sets. This is version 9 of infinite, born on 2001-11-16, modified 2005-08-26. is infinite if it is not finite; that is, there is no planetmath.org /encyclopedia/InfiniteSet.html   (82 words)

PlanetMath: Dedekind infinite A Dedekind infinite set is certainly infinite, and if the axiom of choice is assumed, then an infinite set is Dedekind infinite. However, it is consistent with the failure of the axiom of choice that there is a set which is infinite but not Dedekind infinite. This is version 4 of Dedekind infinite, born on 2002-01-03, modified 2002-02-03. planetmath.org /encyclopedia/DedekindInfinite.html   (107 words)

SIMMER February 1998 Presentation Topic The sets of positive integers and positive even integers belong to an important class of infinite sets, the denumerable sets. So, according to our definition, all we have proved is that the positive integers form an infinite set, which is as it should be. The smallest infinite number aleph_0 is the cardinal number of the set N of positive integers. www.math.toronto.edu /mathnet/plain/simmer/topic.feb98.html   (1218 words)

Foundations of Mathematics The remaining axioms of class/set theory affirm that the null class is a set, posit the existence of an infinite set, and establish several basic ways of generating new sets from existing sets. The experience of several generations of mathematicians with set theory has restored a certain confidence in infinite mathematics, and the principles of class/set theory have come to be viewed generally (but not universally) as intuitively natural. 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). bahai-library.com /?file=hatcher_foundations_mathematics   (14860 words)

Finite and Infinite Sets and Alphabets A set that is equinumerous with N is said to be countably infinite; a set that is either countably infinite or finite is said to be countable. Notice that all countably infinite sets can be listed in a sequence that starts with the first element of the set, then passes to the second, and then the third, and so on; we can thus say that such a set is enumerable. We will instead say simply that infinite sets are those sets that are not finite. www.rpi.edu /~faheyj2/SB/LCU/lcu.driver/node13.html   (665 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)

Set Theory: Cantor Cantor's last two papers on set theory, Contributions to the foundations of infinite set theory, 1895/1897, give his most polished study of cardinal and ordinal numbers and their arithmetic. By the end of the nineteenth century Cantor was aware of the paradoxes one could encounter in his set theory, e.g., the set of everything thinkable leads to contradictions, as well as the set of all cardinals and the set of all ordinals. The first textbook explicitly devoted to the subject of Cantor's set theory was published in 1906 in England by the Youngs, a famous husband and wife team. www.math.uwaterloo.ca /~snburris/htdocs/scav/cantor/cantor.html   (1069 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)

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)

Infinite Hotel Richard Dedekind defined an infinite set to be one which could be put in one-to-one correspondence with a proper subset of itself. When that was over, all of the original guests, along with the baseball guy, were all on the right-hand side of the hotel, in the even-numbered rooms (of which there are an infinite number) and that left all the odd-numbered rooms vacant. Seems Dylan was playing nearby, and the van outside was carrying an infinite number of Dylan freaks, all ready to catch their man in action. scidiv.bcc.ctc.edu /Math/InfiniteHotel.html   (1590 words)

MAT246Y Course Material Related Links Use Dedekind's Theorem to show that the set of integers Z and the interval of real numbers between 0 and 2, [0, 2], are both infinite(which is of course not surprising). Let N be the set of natural numbers, Z be the set of all integers. Show that all polynomials (with integer coefficients) are countable by writing that set as a countable union of countable sets. www.math.toronto.edu /jkorman/Math246Y/links.htm   (335 words)

Dedekind Dedekind was perhaps the first mathematician to put the real numbers on a firm foundation, constructing them from the rationals via his famous Dedekind cuts. Richard Dedekind completed his doctorate in 1852 at Göttingen under the supervision of Gauss; he was to be Gauss' last student. Gauss is, of course, one of the most admired and influential figures in mathematics, but he was also known as a very unpopular teacher. www.math.fau.edu /schonbek/Modern_Analysis/calcmath21.html   (100 words)

Amazon.com: Everything and More: A Compact History of Infinity (Great Discoveries): Books The task Wallace (author of the bestseller Infinite Jest and other fiction) has set himself is enormously challenging: without radically compromising the complexity of the philosophy, metaphysics, or mathematics that underlies the evolving concept of infinity, present the material to a lay audience in a manner that is entertaining. Nor is it a narrative fixated on the cultish fear of--and obsession with--the infinite that has seemingly driven mathematicians insane over the centuries. But, unlike his previous works, this one deals with extremely (towards the end) technical mathematics which the author is obliged to gloss over.-Quite a contrast to, say, Infinite Jest. www.amazon.com /exec/obidos/tg/detail/-/0393003388?v=glance   (2551 words)

PPT Slide HOL postulates an (Dedekind) infinite set of individuals, ind. num is the least set containing 0 and closed under successor. It is specified by a function s : ind ? www.cs.cornell.edu /html/Nuprl/DARPA_LPE/Oct98PaloAlto/tsld017.htm   (64 words)

infinite set from FOLDOC Nearby terms: Infinite Impulse Response « infinite loop « Infinite Monkey Theorem « infinite set » infinity » infix notation » infix syntax A set with an infinite number of elements. gd.tuwien.ac.at /study/foldoc/foldoc.cgi?infinite+set   (32 words)

gradcdfall05.html the notion of a Dedekind-infinite set) and methods (e.g. In tracing out the origins of the modern preference for first-order logic, we will also take a look at the development of axiomatic set theory, and at the emergence of the modern "semantic" (model-theoretic) approach to logic. We will encounter various philosophical questions concerning the significance of these advances in logic. www.nd.edu /~ndphilo/gradcdfall05.html   (1845 words)

infinite set Computer Encyclopedia Enterprise Resource Directory Complete Guide to Internet A set with an infinite number of elements. infinite set Computer Encyclopedia Enterprise Resource Directory Complete Guide to Internet www.jaysir.com /computer-encyclopedia/i/infinite-set-computer-terms.htm   (29 words)

quick question... - Information Technology Services 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. A set is Dedekind infinite iff there exists a bijection from it to a *proper* subset. 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 /archive/topic/t-45805_quick_question....html   (29 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   (29 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   (29 words)

Supplementary Material Using what are now called Dedekind Cuts, Dedekind defined a given real number to be a partition of the rationals into two sets--those that are less than the given real number, and those that are greater than the real. For example, the set of negative integers is countable infinite, since we can define a mapping from the naturals to their negative counterpart. A countably-infinite set is defined as a set with infinitely many elements, such that a bijection can be established between the natural numbers to the infinite set. www.columbia.edu /~cs3203/supp.html   (29 words)

PlanetMath: Dedekind infinite A Dedekind infinite set is certainly infinite, and if the axiom of choice is assumed, then an infinite set is Dedekind infinite. However, it is consistent with the failure of the axiom of choice that there is a set which is infinite but not Dedekind infinite. This is version 4 of Dedekind infinite, born on 2002-01-03, modified 2002-02-03. planetmath.org /encyclopedia/DedekindInfinite.html   (29 words)

Try your search on: Qwika (all wikis)

About us   |   Why use us?   |   Reviews   |   Press   |   Contact us   Copyright © 2005-2007 www.factbites.com Usage implies agreement with terms.