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Topic: Urelement

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	Math Forum Discussions
		> >The simplest solution is to introduce a constant for each urelement.
		> >(E x)(x is a countable set &; (A y)(y in x -> y is an urelement)).
		> >(E z)(z is an urelement and ~(z in x)).
	www.mathforum.com /kb/thread.jspa?threadID=76529&messageID=349550 (326 words)

	Ur-element - Wikipedia, the free encyclopedia
		In set theory an ur-element or urelement is something which is not a set, but may itself be an element of a set.
		In the standard axiomatization of set theory known as Zermelo-Fraenkel set theory, there are no ur-elements.
		In systems, such as set theory with types, a ur-element is sometimes an object of type 0, hence the name "atom." In such theories, the axiom of extensionality requires special formalization and treatment.
	en.wikipedia.org /wiki/Ur-element (145 words)

	Kripke-Platek set theory with urelements - Wikipedia, the free encyclopedia
		The Kripke-Platek set theory with urelements (KPU) is an axiom system for set theory with urelements that is considerably weaker than the familiar system ZF.
		designate urelements, of which there may be none, whereas letters of the sort a,b,c,...
		Technically these are axioms that describe the partition of objects into sets and urelements.
	en.wikipedia.org /wiki/Kripke-Platek_set_theory_with_urelements (238 words)

	Clearing up the market cycle... best Urelement (Site not responding. Last check: 2007-11-01)
		If Periodia value is 26, that means that from the vary last maximum of its passed 26 ticks of time and we have "period length"/2-26 ticks of time to reverse point.
		Das Urelement Feuer in der Cheminée Skulptur strahlt eine wohltuend beruhigende Atmosphäre aus.
		Ett urelement är ett objekt som kan ingå som element i mängder men som själv inte är en mängd.
	ascot.pl /th/Fourier5/Urelement.htm (494 words)

	[No title]
		But there could also be urelemente that are abstract, while yet not beings sets.
		A pure set is one built up from the empty set by operations of set abstraction none of which involves recourse to any urelemente.
		An impure set is one which involves an urelement at some point in its membership pedigree.
	people.cohums.ohio-state.edu /tennant9/kaboom.doc (1871 words)

	Math Forum Discussions
		> > (E z)(z is an urelement and ~(z in x)).
		For every set of urelements, there is an urelement not in the set.
		Hence, there must be a proper class of urelements.
	www.mathforum.com /kb/thread.jspa?threadID=76529&messageID=349548 (374 words)

	[Phil-logic]Cantor's Rule (was pairs)
		This was in response to the question of whether or not the Cardinality Axiom or its equivalent was obvious or true or provable or anything of the sort.
		If we think that the urelement is my cat, then we might say there is one thing (since surely my cat has no subsets or members in the relevant sense).
		In particular, I think Cantor was trying to reproduce the structure of the natural numbers and the real numbers by modeling them with sets.
	philo.at /pipermail/phil-logic/2004-July/004569.html (897 words)

	Math Forum Discussions
		The simplest solution is to introduce a constant for each urelement.
		(E x)(x is a set &; (A y)(y in x -> y is an urelement)).
		(E z)(z is an urelement and ~(z in x)).
	www.mathforum.com /kb/thread.jspa?threadID=76529&messageID=349545 (260 words)

	Monadic Mathematics: Philosophy Forums (Site not responding. Last check: 2007-11-01)
		But as you see, in Monadic Mathematics framework, there are two kinds of cardinals, where one of them is the standard Quantitative Cardinal, but the second type of cardinal is what I call the Urelement Cardinal, which is based on the Length concept (which is the structural property of the Number concept).
		The state of a non-divisible element is called Urelement (http://mathworld.wolfram.com/Urelement.html) in SM and I call it a Monad in MM, which is the building block (not a number) of my numbers system, which are at least Set_AND_Multiset.
		Vectors (or the more genaral case of tensors) are not first order elements, and I am talking about first order elements, or in other words, the number system itself, and not about elements that their existence is based on these numbers.
	forums.philosophyforums.com /thread/11870 (2894 words)

	Cybernetic Ontology and Web Semantics
		The world as the place in which a historical event like the development of Aristotelian ontology is possible does not consists of ontological entities, neither Urelements nor sets.
		One of the basic distinctions of GOL is the distinction between urelements and sets.
		We assume the existence of both urelements and sets in the world and presuppose that both the impure sets and the pure sets constructed over the urelements belong to the world.
	www.thinkartlab.com /pkl/media/Dynamic/Semantic%20Web.html (4932 words)

	Orcmid's Lair
		Often, the axioms of set theory are modified to allow the presence of urelements for ease in representing something.
		This appears to be a typed set theory where urelements are injected (from somewhere) and identity of urelements is determined by their origins.
		Well, urelements (familiar to LISPers as whatever it is that atoms represent) do seem to have some popularity in computer science circles.
	nfocentrale.net /orcmid/blog/2004_01_11_lair-chive.asp (2402 words)

	Armageddon Online Archives - Monadic Mathematics (Site not responding. Last check: 2007-11-01)
		And that is still irrelevant, as your urelement cardinal is still analogous to the min and max functions.
		Nothing can be defined a singleton and a urelement, because a singleton is described as a set containing one element, and part of the definition of a urelement is that it contains no elements.
		The parallel thing in the physical world (which is analogous to my original interpretation of the Urelement definition) is what is called in biology "positive mutation", where some DNA code interpreted differently from the usual way, and constructs a better animal, which gradually substitutes the old animal.
	forums.armageddononline.org /history/topic/3349-1.html (7145 words)

	Math Forum Discussions
		> > >If you want a countable set of urelements (without names), you can say
		> > >Moss) where they use a huge number of urelements.
		> urelements is equiconsistent with ZF (or equivalent).
	www.mathforum.org /kb/thread.jspa?threadID=76529&messageID=349551 (330 words)

	Re: Axiom of Foundation (absymally stupid question)
		The point here is that speaking of intersections presupposes that the objects in question are sets, whereas set theories with urelements typically consider "epsilon" to be a relationship on the whole domain, including both urelements and sets.
		So saying there doesn't exist a z such that "z \in y and z \in B" allows for the possibility that y or B is an urelement (like a piece of fruit).
		Thus the same statement of the foundation axiom would suffice for a set theory with urelements of this kind.
	www.usenet.com /newsgroups/sci.math/msg20229.html (566 words)

	Anaximenes aus Milet
		Ihm ist die Seele wesensgleich mit dem göttlichen, ewig bewegten, alles aus sich erzeugenden Urelement der Luft [vgl.
		Anaximandros erklärte das Entstehen und Vergehen der Dinge aus den Gegensätzen, in die sich das Apeiron spalte.
		Anaximenes erweist sich nun als würdiger Schüler des Anaximandros, wenn er fragt, wie denn aus dem Urgrund, dem Urelement, dem keine qualitative Veränderung zukommt, die qualitative Vielfalt der Dinge zu erklären sei.
	www.philosophenlexikon.de /anaximen.htm (298 words)

	sciforums.com - The First Axioms of Complementary Set Theory
		c) There is no Urelement between {._.} and {__}.
		There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.
		An urelement contains no elements, belongs to some set, and is not identical with the empty set http://mathworld.wolfram.com/Urelement.html.
	www.sciforums.com /printthread.php?t=39203 (4646 words)

	Towards a Dynamic Semantic Web
		In other words, no Urelement can become a set and vice versa, simply because this ontology is mono-contextural, lacking any fundamental perspectivism and interactivity with diversity.
		The introduction of a plurality of first order ontologies, that is of poly-contexturality, opens up an interesting new interplay between the hierarchical order between core and upper level ontologies to a proemial exchange of their functionality.
		To the many possibilities of metamorphic interplays between basic ontologic terms of different fundamental ontologies, like sets vs. urelements, additionally core ontologies can change to upper and upper to core ontologies, ruled by proemial operators.
	www.stephenibaraki.com /semanticweb_techno_net.htm (5813 words)

	Identity and Individuality in Quantum Theory (Site not responding. Last check: 2007-11-01)
		The motivation behind such developments is the idea that collections of quantum objects cannot be considered as sets in the usual Cantorian sense of ‘… collections into a whole of definite, distinct objects of our intuition or of our thought.’ (Cantor 1955, p.
		Quasi-set theory incorporates two kinds of basic posits or ‘Urelemente’: m-atoms, whose intended interpretation are the quantal objects and M-atoms, which stand for the ‘everyday’ objects, and which fall within the remit of classical set theory with Ur- elements.
		Quasi-sets are then the collections obtained by applying the usual Zermelo- Fraenkel framework plus Urelement ZFU-like axioms to a basic domain composed of m- atoms, M-atoms and aggregates of them (Krause 1992; for a comparison of qua-set theory with quasi-set theory, see Dalla Chiara, Giuntini and Krause 1998).
	plato.stanford.edu /entries/qt-idind (5623 words)

	Taufe (Site not responding. Last check: 2007-11-01)
		Die Taufe und das Wasser bewirken nach christlicher Auffassung etwas, was innen und unsichtbar geschieht, durch die äußere sichtbare Handlung.
		Das Wasser ist ein Urelement, das Leben hervorbringt und erhält, aber auch Leben bedrohen und vernichten kann.
		Johannes der Täufer predigt die Umkehr und die Taufe zur Vergebung der Sünden (Mk 1, 4).
	members.tripod.com /seplatin/Taufe.htm (742 words)

	[No title]
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Unification Algorithm for set-terms with %%% urelements, as described in %%% A. Dovier, A. Formisano, and E. Omodeo, %%% Decidability Results with Atoms, %%% ACM TOCL, to appear.
		% We have used "/" to avoid conflicts with the list constructor) % Let us see a problem using properties of urelements % as self-singleton % s-unify :- {a,b/b} = {b,a/a}.
		%%% preproc(+term, -variable) %%% tranlates a term of the form %%% a: urelement %%% X: variable %%% {t_1,\dots,t_n / r} %%% into internal notation.
	www.dimi.uniud.it /~dovier/CST/ssing_prolog.txt (1203 words)

	urelement - OneLook Dictionary Search (Site not responding. Last check: 2007-11-01)
		We found one dictionary with English definitions that includes the word urelement:
		Tip: Click on the first link on a line below to go directly to a page where "urelement" is defined.
		Urelement : Eric Weisstein's World of Mathematics [home, info]
	onelook.com /?w=urelement (69 words)

	Wasserprojekt
		Wasser als Symbol und Urelement spielt in allen Religionen dieser Welt eine überaus wichtige Rolle.
		Zur Vorbereitung des Wasser-Projektes wurden im katholischen Religionsunterricht der Klasse WG 11/3 die in diesem Zusammenhang vielfältigen Deutungen und Auslegungen des biblischen Prophetenbuchs Jona erarbeitet.
		Der Weg der Sonne als Morgensonne Chepre, als Mittagssonne Re, als Abend- und Nachtsonne Atum, ihr Eintauchen in die Nacht und ihr Abstieg in die Unterwelt sind ein mythisches Bild für diesen Prozeß der Selbstwerdung.
	www.zum.de /Faecher/kR/BW/wagner/jona1.htm (539 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		It is said that [everything being a set] can be done in principle, although personally I'd like to rephrase it as 'everything can be modeled using sets' since when I talk about numbers or graphs or vectorspaces, I don't think of these things as sets at all.
		They're more like urelements; a primitive notion that doesn't decompose into sets, although we can model them in pure set theory.
		I'm not sure one can have more than one urelement if you don't have an equality relation, or maybe even irregular sets of all kinds fall apart without equality, I'd have to think about it.
	www.grahamkendall.net /Math/MathNewsgroups/mm-1374.txt (18114 words)

	Leibniz dream and my framework: Philosophy Forums (Site not responding. Last check: 2007-11-01)
		In other words, this absolute/relative system is the “story” of the interactions between Sets and Mutisets (as they defined by my axiomatic system) where each edge of {._.}, which is not= Urelement cardinal 0, has the potential to be the entire Urelement cardinals that are not= Urelement cardinal 0.
		I'll be glad to answer to any detailed quastion about their contents.
		In general, any arithmetic operation with a non-finite fractal or a collection of infinitely many elements, has no accurate result, and the best we can do is to compare between the ratios that stand in the basis of each never-ending arithmetic processes, for example:
	forums.philosophyforums.com /thread/10793 (3685 words)

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