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Topic: ZFC

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	Zermelo–Fraenkel set theory - Wikipedia, the free encyclopedia
		ZFC consists of a single primitive ontological notion, that of set, and a single ontological assumption, namely that all individuals in the universe of discourse (i.e., all mathematical objects) are sets.
		ZFC is a first-order theory; hence ZFC includes axioms whose background logic is first-order logic.
		Thus, to the extent that ZFC is identified with ordinary mathematics, the consistency of ZFC cannot be demonstrated in ordinary mathematics.
	en.wikipedia.org /wiki/ZFC (1372 words)

	TA Holding::ZFC Ltd
		ZFC Ltd is the largest local manufacturer and distributor of fertilizer and the third largest distributor of agrochemicals in Zimbabwe.
		ZFC has distribution agreements with several of the world's leading principal manufacturers of agricultural chemicals such as Bayer, FMC, Monsanto and Syngenta.
		It is ZFC's objective to continually improve the distribution of its products to promote fertilizer use in order to improve agricultural yields and the general produce output in the country.
	www.ta-holdings.com /investments/zfc.htm (205 words)

	PlanetMath: von Neumann-Bernays-Gödel set theory
		The primary difference between ZFC and NBG is that NBG has proper classes among its objects.
		This theory is essentially stronger than ZFC or NBG, as it can prove their consistency (in addition to everything they already prove).
		Thus the comprehension scheme of ZFC can be replaced with a finite number of axioms, provided we allow for proper classes.
	planetmath.org /encyclopedia/VonNeumannBernausGodelSetTheory.html (801 words)

	ZFC website
		The ZFC is short for the Zylstro Females Club and it is, as you would have guessed from the name, a club for females.
		In the ZFC, the activities in the mailings are compulsory; the activities in the magazine are not.
		ZFC Smiles is the name given to the club magazine.
	www.geocities.com /zfc_rulz/steph.html (501 words)

	Beerman Thesis Section 3 (Site not responding. Last check: 2007-10-13)
		The peak at 8 K in the ZFC curve of Figure 10 was unexpected, but is observed for spherical samples that utilize a surfactant mixture comprised of trictylphosphine oxide (TOPO) and oleic acid (OA) during synthesis.
		However a variation of the ZFC measurement may be utilized to characterize the dipole-dipole interaction.
		The difference curve between the ZFC and FC suggest that there is dynamic behavior at a characteristic temperature of about 50 K. measurements were carried out at other temperatures, however, 50 K showed the largest difference peak.
	students.washington.edu /mbeerman/Research/Section3.htm (1948 words)

	ZFC and Russell's Paradox
		Here, you're getting into some sticky territory: in order to talk about what ZFC can and cannot do, you have to adopt some external theory which is capable of talking about such things, which begs the question of why the external theory is valid...
		It seems to me that in ZFC there's no really effective way to prove that something is not a set.
		Well, in ZFC, Russel's paradox is a proof that the class of sets that do not contain themselves is not, itself, a set.
	www.physicsforums.com /showthread.php?t=51980 (1846 words)

	List of statements undecidable in ZFC - Wikipedia, the free encyclopedia
		Martin's Axiom implies that there exists a function on the unit square whose iterated integrals are not equal, while as a variant of Freiling's Axiom of Symmetry implies that in fact a strong Fubini type theorem for [0, 1] does hold, and whenever the two iterated integrals exist they are equal.
		The consistency of ZFC was the first statement shown to be undecidable in ZFC.
		Martin's axiom together with the negation of the continuum hypothesis is undecidable in ZFC.
	en.wikipedia.org /wiki/List_of_statements_undecidable_in_ZFC (497 words)

	Antimeta: Consistency and Platonism (Site not responding. Last check: 2007-10-13)
		In addition, there seems to be another problem with consistency for platonists, in that whatever system of axioms is "true" (according to the platonists) may turn out not to be consistent.
		If the model of ZFC that thinks ZFC is inconsistent is a set model within the actual universe, then it's clear that this proof must "actually" either be infinitely long or use some steps or axioms that the model mistakenly thinks are valid but "actually" aren't.
		If "ZFC is consistent" is interpreted in its natural language sense, we can believe it to be literally true while remaining non-committal about the set-theoretic claim Con(ZFC).
	www.antimeta.org /blog/archives/2005/02/consistency_and.html (700 words)

	Major features of ZFC - Freeware File Encryption
		ZFC is Very easy to use, there is no unnecessary jargon.
		ZFC is a true 32bit multi threaded application.
		ZFC is able to provide a slide show of images either in windowed or full screen mode.
	www.baroufasoft.net /major_features__of_zfc.htm (488 words)

	[No title]
		The reason that these models of ZFC have nonstandard natural numbers is that there are inductive subsets of the "set of naturals" in the model which do not correspond to any set of the model.
		In other words, the reason that a model of first-order ZFC may have nonstandard natural numbers is that it fails to be a model of second-order ZFC -- its "sets of natural numbers" are not all the sets of its "natural numbers".
		Not only can ZFC formalize all mathematical proofs, but in addition, ZFC is the strongest axiomatic theory that we can write down in the language {epsilon,=} that is directly based on certain informal insights or naive intuitions about the universe of set theory (informal replacement, informal axiom of choice, etc).
	math.boisestate.edu /~holmes/holmes/fomletter13.txt (2094 words)

	Infinite Ink: The Continuum Hypothesis FAQ
		Even though these different versions of CH are equivalent in ZFC they have different flavors and help us to think about the problem, and its possible solution, in different ways.
		Mathematicians suspected that CH was undecidable in ZFC but it took until 1963 until this was proved.
		So this means that either CH or ~CH could be added as an axiom of ZFC but since neither of these seem axiomatic or `self evident,' mathematicians have instead tried to find another axiom that will tell us the cardinality of the continuum.
	www.ii.com /math/ch/faq (520 words)

	The History of ZFC (Site not responding. Last check: 2007-10-13)
		Those members who were invited to join us were asked not to reveal the meaning of the ZFC rating, so as to increase the curiosity factor and to refer the enquirer to the ZFC co-ordinator with an email, and the result was that we have over one hundred and fifty members.
		We decided to create the ZFC A category to accommodate all those below fifty years old, all the lady members, and all the rest of the fella's who are shy about revealing their age...
		We have formed a small steering committee, known as the ZFC Politburo, to set up ZFC properly and will be asking (coercing, arm twisting) certain members with expertise in various fields for their assistance.
	www.crashnet.org.uk /zfc/zfc.html (384 words)

	[No title]
		One reason ZFC (and other similar systems) is interesting is that it is a candidate for providing a basis for mathematics, and in particular arithmetic.
		What is not so commonly appreciated is that even if ZFC is consistent, there might still be statements about the natural numbers that are theorems of ZFC but nevertheless false.
		In other words, consistency doesn't buy you everything that you might want from a formal system---it tells you that models of the system exist, but it doesn't tell you that there are any models where the "integers" are isomorphic to the standard integers, so you can't necessarily "trust" theorems of the system.
	www-math.mit.edu /~tchow/mathstuff/piddling (542 words)

	[No title]
		Models of ZFC Perhaps the most famous theorem proved using forcing is the independence of the continuum hypothesis (CH).
		Goedel had already proved that if ZFC is consistent then ~CH, the negation of CH, is not a theorem of ZFC (using his concept of "constructible sets").
		The rest of the axioms of ZFC are similarly self-evident when M = V. The catch, of course, is that V is too large to actually be a set, and we stipulated that M had to be a set.
	www-math.mit.edu /~tchow/mathstuff/forcingdum (4040 words)

	SUICIDAL TENDENCIES (Site not responding. Last check: 2007-10-13)
		ZFC: What are some of the political views that you take seriously?
		He is one of the best skaters and he's been involved in skating for a long time.
		ZFC: Most of us have suicidal tendencies at least once, some of us have them every day.
	hometown.aol.com /zero4cndct/myhomepage/warriors.html (883 words)

	ZFC Axioms Without Distinct Variable Conditions - Metamath Proof Explorer
		Strictly speaking, each of our standard ZFC "axioms" is really an axiom scheme or template that represents an infinite number of actual axioms (see our note on the axioms).
		Although these conditionless axioms are provably equivalent as a group to the standard ZFC axioms, it should be noted that there is a sense in which they are not "pure" individually.
		So for some purposes, such studying the independence of the standard ZFC axioms, they are not suitable as a replacement for the standard ones.
	metamath.planetmirror.com /mpegif/mmzfcnd.html (1251 words)

	Antimeta: Large Cardinals and their Justifications
		Now, by Gödel's Completeness theorem (which requires some fragment of ZFC to prove), we know that if Con(T) is the case, then there is actually some set and a collection of relations and functions on that set that can be used as the interpretation of the symbols in T to make it come out true.
		The axioms of ZFC are postulated to be true in the fiction, but beyond that, we have a variety of means of coming to know what else is true in the fiction.
		Since it is also true in ZFC that N is a submodel of anything modeling PA it follows that proving you have a model of Th(N) entails you have proved Con(T) is true in the actual w.
	www.ocf.berkeley.edu /~easwaran/blog/2005/12/large_cardinals_and_their_just.html (5467 words)

	Gödel's theorem
		Since all the theorems ordinarily proved in mathematics can be proved in ZFC, and since the consistency of ZFC cannot be proved in ZFC (unless ZFC is inconsistent), it is often concluded that we cannot expect to prove, and therefore can't know, that ZFC is consistent.
		Consistency proofs for ZFC are essentially proofs by reflection, meaning that we note, in some way or another, that since the axioms of ZFC are true, they are consistent.
		If ZFC is consistent, then the new theory ZFC' is consistent (by Gödel's second theorem), even though it falsely proves that ZFC is inconsistent.
	www.sm.luth.se /~torkel/eget/godel/second.html (721 words)

	Zuilens Fanfare Corps - Utrecht (Site not responding. Last check: 2007-10-13)
		Op zaterdag 5 november 2005 vierde het ZFC haar 105 jarig jubileum met een spetterend jubileumconcert in de Bethelkerk in Zuilen, die was omgebouwd tot een heuse concertzaal.
		Voor het ZFC was het een goede gelegenheid de concoursstukken voor publiek uit te proberen.
		Op zaterdag 5 juni 2004 was het ZFC te bewonderen op de regionale zender RTV-Utrecht in het kader van 50 jaar Zuilen bij Utrecht.
	www.zfc-utrecht.nl /gebeurt.htm (2241 words)

	Antimeta: Set Theory Archives (Site not responding. Last check: 2007-10-13)
		However, all of these axioms in PA and ZFC are in the end accepted for primarily intrinsic reasons - the reason their necessity for these arguments is decisive is that these arguments had already been accepted, and thus the axioms had been implicitly as well.
		While Frege's inconsistent system also had intrinsic justification, it turned out to be inconsistent, and ZFC was a weakening that seemed to have similar intrinsic justification and seemed likely to be consistent as well.
		Thus, believing that the axioms of ZFC are true (in a literal platonist sense) gives one strong reason to believe in addition that there are inaccessible, Mahlo, hyper-Mahlo, etc. cardinals.
	www.antimeta.org /blog/archives/set_theory/index.html (8868 words)

	Instructor Class Description (Site not responding. Last check: 2007-10-13)
		Indeed, “inside” any model of ZFC would reside a model of ZFC + V=L (namely the constructible sets of that model of ZFC).
		He also proved that CH holds in any model of ZFC + V=L. Godel had previously proved that any collection of axioms in first-order logic powerful enough to do arithmetic was logically incomplete, so ZFC was known to be logically incomplete.
		With this method Cohen was able to prove that if ZFC is consistent, then ZFC with the failure of CH is consistent.
	www.washington.edu /students/icd/S/phil/570dumas.html (363 words)

	Variants of ZFC (Site not responding. Last check: 2007-10-13)
		The usual definition of "transitive closure" of a set as a certain countable union needs an instance of the replacement axiom.
		The (easy) answer is no. The short note tcl gives 2 models of ZC in which tcl does not exist.
		Note: There is a book by E.Forster that discusses several variants of "set theory with a universal set", most of them in the context of NF.
	info.tuwien.ac.at /goldstern/papers/notes/zfc.html (186 words)

	ZFC - OneLook Dictionary Search
		ZFC : Stammtisch Beau Fleuve Acronyms [home, info]
		ZFC : Free On-line Dictionary of Computing [home, info]
		Phrases that include ZFC: list of statements undecidable in zfc, zfc set theory
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=ZFC (98 words)

	ZFC by Jerry E. Vaughan (Site not responding. Last check: 2007-10-13)
		The formal system ZF with the addition of the axiom of choice (denoted ZFC) is generally held to encompass essentially all of mathematics.
		In topology (and elsewhere) there are simple statements that can neither be proved nor disproved (in ZFC).
		A discussion on ZFC is given in Section 6.4.
	at.yorku.ca /i/a/a/i/68.htm (193 words)

	zForum - Zope Based Message Boards
		zfc (first row in the above section) was not indexed, even with cat_id being the primary key for that table, that is undertandable since the zfc (forum categories) has to be sweeped 100% anyway, but the yielding performance is way better, just there 1.0.5 has to be faster in generate the main page.
		Now, We have to look at other options as far as the optmization goes, the way it is now, it is as optimize as it can be, I'd be surprised if this query takes more than a couple of seconde on a 100,000+ records system.
		select zfc., zf.forum_id, zf.forum_title, zf.forum_desc, zf.moderation_flag, zf.anonymous_viewaccess, zf.add_postings_access_roles, zf.admin_min_access_roles, sum(zt.hits) as hits from zf_forum_category as zfc* left join zf_forum as zf on zfc.cat_id = zf.cat_id left join zf_topic as zt on zt.forum_id = zf.forum_id group by zf.forum_id, zf.forum_title, zf.forum_desc, zf.moderation_flag, zf.anonymous_viewaccess, zf.add_postings_access_roles, zf.admin_min_access_roles, zfc.cat_id, zfc.cat_name, zfc.cat_desc, zfc.cat_visible_to
	www.zforum.org /view_topic?topic_id=437 (2262 words)

	Math Forum Discussions
		> notation, that is) but ZFC alone does not.
		a < b iff a < b & b is not a proof of contradiction in ZFC \/ a>b &
		b is a proof of contradiction in ZFC
	mathforum.org /kb/thread.jspa?threadID=1330911&messageID=4279327 (271 words)

	ZFC:: Web Design and Hosting - Boca Raton, FL
		ZFC:: Web Design and Hosting - Boca Raton, FL www.
		The key to a successful web presence is a website your visitors will remember.
		ZFC designs unique original concepts that will differentiate you from the millions of other sites out there.
	www.zfc.com (63 words)

	ZFC-Meuselwitz
		Spieltag (Bezirksliga 2005/06) Der ZFC Meuselwitz behauptet sich Fünf Fragen an ZFC-Coach Damian Halata ZFC ohne Weinert nach Bad Blankenburg Bundesliga-Junioren beim ZFC Hallenturnier Testsieg und drei Sorgenkinder Sieg gegen Saalfeld — ZFC ohne Fünf zum Hallenturnier Nachwuchsturniere 5.
		ZFC Hallenturnier Ergebnisse ZFC vs. Altenburg Zwickau Gera vs. ZFC ZFC Meuselwitz vs. Borna Hallenturniere des ZFC Meuselwitz ZFC vs. Wolfen ZFC vs. Weida Heimspiel gegen Bautzen ZFC Meuselwitz vs. Bayern Hof Oldie-Cup Vorschau Sachsen-Leipzig (Rückrunde) 19.
		Spieltag (Oberliga 2005/06) Vierte Spielabsage für den ZFC Meuselwitz Halata verlässt den ZFC Meuselwitz Vorschau Plauen (Rückrunde) Polizeiauswahl 20.
	www.zfc.de /zfc-web/?&PRINT=1 (1060 words)

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