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Topic: Perfect set property

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	PlanetMath: perfect set
		An non-trivial example of a perfect set is the middle-thirds Cantor set.
		This is version 4 of perfect set, born on 2002-12-24, modified 2005-08-04.
		A nonempty perfect subset of IR by Gorkem on 2003-11-15 23:13:06
	planetmath.org /encyclopedia/PerfectSet.html (141 words)

	Axiom of dependent choice - Wikipedia, the free encyclopedia
		In mathematics, the axiom of dependent choice, denoted DC, is a weak form of the axiom of choice which is still sufficient to develop most of real analysis.
		Unlike the full axiom of choice (AC), DC is insufficient to prove (given ZF) that there is a nonmeasurable set of reals, or that there is a set of reals without the property of Baire or without the perfect set property.
		DC is equivalent to the statement that every (nonempty) pruned tree has a branch.
	en.wikipedia.org /wiki/Axiom_of_dependent_choice (202 words)

	Fountain Forestry - Brokerage - Properties (Site not responding. Last check: 2007-10-11)
		The property is situated in the southeast portion of the Adirondack Park within the Township of Stony Creek.
		The balance of the property is forested, with the eastern end rising uphill from the road frontage to the southern boundary.
		The property is situated nearly equal distance between the village of Hardwick (6 miles to the east) and the larger village of Morrisville (7 miles to the west).
	www.fountainforestry.com /properties.htm (13159 words)

	Perfect numbers
		It is not known when perfect numbers were first studied and indeed the first studies may go back to the earliest times when numbers first aroused curiosity.
		Perfect numbers were studied by Pythagoras and his followers, more for their mystical properties than for their number theoretic properties.
		Today the usual definition of a perfect number is in terms of its divisors, but early definitions were in terms of the 'aliquot parts' of a number.
	www-gap.dcs.st-and.ac.uk /~history/HistTopics/Perfect_numbers.html (4360 words)

	Perfect Subset Property for co-analytic sets in ZF\P
		Perfect Subset Property for co-analytic sets in ZF\P
		A perfect set is the set of all paths through a perfect tree, and can be coded by such perfect tree.
		The constructible sets under the membership relation form a model L of ZFC\P. Suppose that every set is countable and perfect subset property holds for co-analytic sets.
	web.mit.edu /dmytro/www/other/PerfectSubsetsAndZFC.htm (917 words)

	Mossy Oak Properties (Site not responding. Last check: 2007-10-11)
		Perfect for the hunter are food plots set up with an abundance of cedar trees for stalking or hiding.
		superb for the duck hunter(100,s of ducks and geese flock this property) because of the proximity to the Tenn River flyway(3mi).
		The property has a landing strip and hanger that is a perfect set-up for a hunting lodge and storage.
	www.mossyoakproperties.com /2020applications/propertylistings.asp?province=TN&page_size=50 (584 words)

	A Large ... Set, Absolute for Set Forcings (ResearchIndex)
		We show how to obtain, by class-forcing over L, a set of reals X which is large in L(X) and has a \Pi 1 2 definiiton valid in all set-generic extensions of L(X).
		As a consequence we show that it is consistent for the perfect set property to hold for \Sigma 1 2 sets yet fail for some \Pi 1 2 set.
		Also it is consistent for the perfect set property to hold for \Sigma 1 2 sets and for there to be a long \Pi 1 2 well ordering.
	citeseer.ist.psu.edu /332799.html (213 words)

	Cornell Math - MATH 787, Fall 2000 (Site not responding. Last check: 2007-10-11)
		It may either be the usual basic introduction to axiomatic set theory (I) or a course in descriptive set theory (II).
		In particular, analyzes the structure of definable sets beginning with the hierarchy of Borel sets determined by beginning with the open sets and then iterating complementation and countable union.
		Typical issues considered are uniformization (of relations by functions of the same class in the hierarchy), measurability, perfect set property (which uncountable sets must conation perfect subsets) and the property of Baire (differing from an open set by a meager set).
	www.math.cornell.edu /~www/Courses/GradCourses/FA00/787.html (365 words)

	Phuket Property Development - Condominium for sale - Sunset Residence -
		It is a perfect blend of exquisite Thai and Western style architecture completed to our own high standards.
		Set in a quiet location surrounded by nature you can appreciate the magnificent views of the sunset and the enchanting sounds of the ocean.
		It offers you a residence amidst the resourcefulness of natural resource and the availability of all the conveniences offers by the Project that creats an atmosphere that is complete with an idyllic surrounding of plant species.
	www.sunsetresidence.com (265 words)

	Woodin cardinal - Wikipedia, the free encyclopedia
		In set theory, a Woodin cardinal is a cardinal number λ such that for all
		Woodin cardinals are important in descriptive set theory.
		The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses.
	en.wikipedia.org /wiki/Woodin_cardinal (130 words)

	Citebase - A large Pi-1-2 set absolute for set forcing
		Then there is a set of reals X, class-generic over L, such that L(X) and L have the same cardinals, X has size k in L(X) and some pi-1-2 formula defines X in all set-generic extensions of L(X).
		Two corollaries, both assuming the consistency of an inaccessible: It is consistent for the Perfect Set Property to hold for boldface sigma-1-2 sets, yet fail for some lightface pi-1-2 set.
		It is consistent that the Perfect Set Property holds for boldface sigma-1-2 sets yet some lightface pi-1-2 wellordering of some set of reals has length aleph-1000.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9212202 (279 words)

	[No title]
		Let V be a set containing exactly one representative from each equivalence class (possible by the Axiom of Choice).
		Subject: Re: Contructing a non-measurable set in R1 Date: 11 Nov 1999 18:42:55 +0100 Newsgroups: sci.math magidin@bosco.berkeley.edu (Arturo Magidin) writes: > Depends on what you mean by "construction." The standard definition of > the Vitali Set requires the Axiom of Choice and is non-constructive.
		As it is fairly typical in set theory that the first proof of a result uses large cardinals since it gives you more room to work with so to speak, originally people assumed that the hypothesis of IC could be avoided.
	www.math.niu.edu /~rusin/known-math/99/nonmbl (997 words)

	[No title]
		A stronger property, in the sense that for some models of ZFC a class of sets can satisfy CH but not satisfy what I'm about to describe, is that every uncountable set in the collection contains a nonempty perfect subset.
		One could argue that Cantor showed open sets satisfy CH in 1873-74 when he introduced cardinality, since it is easy to see that every open interval has the same cardinality as the reals.
		Souslin (1916, I think) showed that the CA sets are either countable, have cardinality aleph_1, or have cardinality c.
	www.math.niu.edu /~rusin/known-math/01_incoming/aleph1 (1073 words)

	Dr Benedikt Loewe: Infinite Games (Sommersemester 2004) (Site not responding. Last check: 2007-10-11)
		Properties of games: perfect information, perfect recall, number of players.
		Proof Sketch of Davis' theorem on the perfect set property.
		Basic properties of the Lipschitz hierarchy: successors are selfdual, countable limits are selfdual, selfdual degrees are countable limits.
	www.math.uni-bonn.de /people/logic/Lectures/SoSe2004/InfiniteGames.html (152 words)

	Amazon.com: Set Theory (Perspectives in Mathematical Logic): Books: Thomas J. Jech (Site not responding. Last check: 2007-10-11)
		Set Theory (Studies in Logic and the Foundations of Mathematics) by Kenneth Kunen
		This introduction to modern set theory covers all aspects of its two main general areas: classical set theory including large cardinals, infinitary combinatorics, desriptive set theory, and independence proofs starting with Goedel's proof around 1938 followed by Cohen's proof in 1963, whereby Cohen's method of forcing probably had a greater influence on mathematics.
		Intuitively, a set is a collection of all elements that satisfy a certain given property.
	www.amazon.com /exec/obidos/tg/detail/-/3540630481?v=glance (1572 words)

	Guardian Unlimited Money \| Property \| Pitch perfect for footballers' lives
		A property consultancy working on a confidential document for the FA claims footballers spend an estimated £85m a year on homes in Britain.
		As a result, a small industry of buying agents and support companies exists to service this growing, affluent and demanding niche of the property market - and with the football transfer window about to close, now is one of its busiest times of the year.
		'Steve McManaman married in Palma and bought a property in Son Vida, an exclusive area in the hills to the north-west of the city.
	money.guardian.co.uk /property/story/0,,1698144,00.html (692 words)

	DevX.com Forums - Get Property and function
		Properties are attributes of an object and Functions (Methods) typically act upon the object.
		Set procedures:Which are used to assign values to a property.
		In a Set procedure, the new value for a property is passed through the argument of the Set statement.
	forums.devx.com /showthread.php?threadid=144342 (422 words)

	Realty Times: That Condominium Will be the Perfect Rental Property
		Condominiums may appear to be the ideal way to purchase rental property, as their price is often lower than the price for a single family home.
		Condominiums are governed by a set of Condominium Documents or CC & R’s, and a Homeowners Association sees that the rules set down in these documents are followed.
		Properties located in popular resort areas may be better rented out as a “seasonal rental,” when the rents charged are typically higher than other times of the year.
	realtytimes.com /printrtpages/20000721_condo.htm (730 words)

	Amazon.com: Classical Descriptive Set Theory (Graduate Texts in Mathematics): Books: Alexander S. Kechris (Site not responding. Last check: 2007-10-11)
		Descriptive set theory has been one of the main areas of research in set theory for almost a century.
		This text provides a first basic course in classical descriptive set theory and covers material with which mathematicians interested in the subject for its own sake or those that wish to use it in their field should be familiar.
		A topological space is a pair (X, T), where X is a set and T a collection of subsets of X such that, X T and T is closed under arbitrary unions and finite intersections.
	www.amazon.com /exec/obidos/tg/detail/-/0387943749?v=glance (1106 words)

	Grimley - Other - Property Returns Set To Peak This Year
		Occupational property demand is strong and rents are now rising in real terms, the first time in this decade.
		The IPD all property return rose to 12.2% pa in June from 10.7% pa in March.
		We advise a range of leading property companies, investors and advisors on all aspects of public and media relations strategy and are one of the best known PR consultancies serving the real estate industry.
	www.propertymall.com /press/article/677 (646 words)

	Ruidoso Vacations - Vacation in New Mexico - Places to Stay in Ruidoso (Site not responding. Last check: 2007-10-11)
		Many properties offer hot tubs for a relaxing soak at the end of the day.
		All properties come furnished with kitchens and most with gas grills, if you want to cook for yourself.
		The towels and linens will be freshly laundered and we’ll leave the light on for you in the case of a late arrival time.
	www.ruidosoreservations.com (472 words)

	Math Forum Discussions
		of numbers from the Cantor set whose difference is exactly that number.
		is set to zero except in the places where i equals a factorial, in which
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /kb/thread.jspa?threadID=80542&messageID=380186 (313 words)

	Perfect property for the polo set - Horse&Hound online (Site not responding. Last check: 2007-10-11)
		The geography of stick-and-ball properties runs from Gloucestershire through to Berkshire and the M4 corridor down to Sussex, according to William Duckworth-Chad of Savills.
		Patrons at every level usually prefer to live close to their club and, since most polo clubs are situated in the south of England, a base in the Home Counties is a must.
		Availability also plays a part because "the not-so-grand properties come up a little bit more often than first-class establishments, some of which are sold privately".
	www.horseandhound.co.uk /best/396/68505.html (742 words)

	Jech's lectures (Site not responding. Last check: 2007-10-11)
		We give a proof of Goedel's Second Incompleteness Theorem stating that there is no proof in set theory of the consistency of set theory.
		We introduce the measure problem and show that the assumption of the existence of a nontrivial sigma additive measure implies the existence of inaccessible cardinals, and is therefore unprovable in set theory.
		We introduce infinite games and discuss their determinacy, with emphasis on its relation to Lebesgue measurability, the Baire property and the perfect set property.
	www.math.cas.cz /~krajicek/jech.html (191 words)

	Modest increase in Allegheny County home values (Site not responding. Last check: 2007-10-11)
		Allegheny County unveiled revised 2006 property assessment values on its Web site yesterday afternoon, and it appeared that most homeowners received only modest increases.
		That is in stark contrast to an earlier set of numbers released in March, when 500,000 residential properties saw an average increase of 19 percent, with a great many even higher.
		Biernacki has a client in Harmar who saw the assessed value of his property increase from $615,000 to $658,000, while the home of a neighbor increased from $480,000 to $515,000.
	www.post-gazette.com /pg/05265/575829.stm (1030 words)

	Find a Property - Pacific Heights Or Perfect Lodger?
		One in five people questioned couldn't afford their mortgage repayments; 36 per cent needed the extra cash; and four per cent had to factor in subletting a room to be able to get on the property ladder.
		Set up interviews with prospective tenants, and don't be afraid to call them back for a second interview.
		Although there is no legal requirement to have a written agreement between you and your lodger, it is a very good idea to have one.
	www.findaproperty.com /story.aspx?storyid=8376 (780 words)

	Compact and Perfect Sets
		A set S of real numbers is called compact if every sequence in S has a subsequence that converges to an element again contained in S.
		A subset of sets from the collection C that still covers the set S is called a subcovering of S.
		A set S is perfect if it is closed and every point of S is an accumulation point of S.
	pirate.shu.edu /projects/reals/topo/compact.html (732 words)

	[No title]
		God, under this view, exemplifies necessarily a maximally perfect set of compossible great-making properties, where a great making property is understood to be a property the possession of which is intrinsically better than the lacking of it.
		A property that is immemorial, enduring, or immutable is not necessarily essential.
		(8) A property is weakly enduring for an individual just in case that individual has that property, and there can be no time during the individual's existence when it will have ceased to have that property.
	www.homestead.com /philofreligion/files/MORRIS.htm (819 words)

	The Topology of the Real Line
		Our interest in perfect sets is motivated by the need to determine the cardinality of an uncountable closed set of reals.
		A set of real numbers is perfect if it is nonempty, closed, and has no isolated points.
		Exercise: Prove that the Cantor set is a perfect set
	br.endernet.org /~loner/settheory/reals2/reals2.html (535 words)

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