
	Where results make sense

Topic: Zermelo Frankel axioms

Related Topics

axiom of choice

Separation axioms

Peano axioms

probability axioms

axiom of extensionality

Zermelo Fraenkel axioms

Axiom of regularity

Kuratowski closure axioms

Axiom schema of specification

axiom of infinity

axiom of empty set

Axiom of pairs

In the News (Sun 27 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	PlanetMath: axiom
		Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge.
		In the modern understanding, a set of axioms is any collection of formally stated assertions from which other formally stated assertions follow by the application of certain well-defined rules.
		A set of axioms should be consistent; it should be impossible to derive a contradiction from the axiom.
	planetmath.org /encyclopedia/Axiom.html (1184 words)

	YourArt.com >> Encyclopedia >> axiom (Site not responding. Last check: 2007-11-02)
		An axiom is a sentence or proposition that is accepted as the first and last line of a one-line proof and is considered as obvious or as an initial necessary consensus for the theory building or acceptation.
		Explicit declaration of axioms is the necessary condition for the computationallity of a theory or model or method.
		Usually one takes as logical axioms at least some minimal set of tautologies that is sufficient for proving all tautologies in the language; in the case of predicate logic more logical axioms than that are required, in order to prove logical truths that are not tautologies in the strict sense.
	www.yourart.com /research/encyclopedia.cgi?subject=/axiom (3115 words)

	Axiom of pairing
		In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo-Fraenkel set theory.
		The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatization of set theory.
		Thus, one may use this as an axiom schema[?] in the place of the axioms of empty set and pairing.
	www.ebroadcast.com.au /lookup/encyclopedia/ax/Axiom_of_pairing.html (465 words)

	Axiom of empty set
		The axiom of empty set may also be seen as a special case of a generalisation of the axiom of pairing.
		In some formulations of ZF, the axiom of empty set is actually repeated in the axiom of infinity.
		Also, the ZF axioms can also written using a constant predicate[?] representing the empty set; then the axiom of infinity uses this predicate without requiring it to be empty, while the axiom of empty set is needed to state that it is in fact empty.
	www.ebroadcast.com.au /lookup/encyclopedia/ax/Axiom_of_empty_set.html (262 words)

	[No title]
		For example, Euclid's fifth postulate is independent of the other because there are models in which the axioms hold and the parallel postulate also holds ("space"=plane,"line"=line) and yet there are also models in which the axioms hold but the p.p.
		Again, consistency is proved by constructing a model in which the axioms hold (the Soundness Theorem assures us there cannot be a model in which inconsistent axioms hold).
		Oh, well, except in one case: if the axioms are _inconsistent_, then they prove a falsehood, and so, logically speaking, they prove any result at all -- including the consistency of the theory.
	www.math.niu.edu /~rusin/known-math/00_incoming/model_thy (295 words)

	Axiom (Site not responding. Last check: 2007-11-02)
		Axioms and postulates are the basic assumptions underlying a given
		consistent; it should be impossible to derive a contradiction from the axiom.
		Euclidean geometry, and the related demonstration of the consistency of those axioms.
	202.41.85.103 /manuals/planetmath/entries/03/Axiom/Axiom.html (968 words)

	set : QuicklyFind Info (Site not responding. Last check: 2007-11-02)
		For a discussion of the properties and axioms concerning the construction of sets, see the articles on naïve set theory and axiomatic set theory.
		In fact, the axiom of regularity is often called the foundation axiom since it can be proved within ZFC- (that is, ZFC without the axiom of regularity) that well-foundedness implies regularity.
		Three distinct anti-foundation axioms are well-known: # AFA (‘Anti-Foundation Axiom’) — due to M. Forti and F. Honsell; # FAFA (‘Finsler’s AFA’) — due to P. Finsler; # SAFA (‘Scott’s AFA’) — due to Dana Scott.
	www.quicklyfind.com /info/set.htm (1250 words)

	Logic & Proof
		As currently conceived and practiced, Mathematics is a tree whose foliage consists of various fields of study connected by branches of logical dependencies and history.
		A student's first acquaintance with logical structure is usually in Euclidean geometry, where the axioms are made explicit, and the student may be asked to write proofs.
		The axioms of arithmetic are Peano's axioms, and those of algebra are the field properties (really theorems) of the real numbers.
	www.stetson.edu /~mhale/logic (586 words)

	Axiomatic Set Theory. Zermelo-Fraenkel Axioms
		The axioms C1 and C2[F] (for all formulas F that do not contain x) and the axiom of choice define a formal set theory C which corresponds almost 100% to Cantor's intuitive set theory (of the "pre-paradox" period of 1873-94).
		The axiom of infinity completes the list of comprehension axioms, which are necessary for reconstruction of common mathematics, i.e.
		The set theory adopting the axiom of extensionality (C1), the separation axiom schema (C21), the pairing axiom (C22), the union axiom (C23), the power-set axiom (C24), the replacement axiom schema (C25), the axiom of infinity (C26) and the axiom of regularity (C3), is called Zermelo-Fraenkel set theory, and is denoted by ZF.
	linas.org /mirrors/www.ltn.lv/2001.03.27/~podnieks/gt2.html (7448 words)

	[No title]
		So with these definitions, we could restate the axiom of replacement as: whenever F is a class term and a is a set, then the class {y : y=F(x) for some x in a} is a set.
		Lemma: The axiom of pairing holds in V. Proof: Let x,y be in V. Then there are ordinals alpha and beta such that x is in V_alpha and y is in V_beta.
		Lemma: The axiom of foundation holds in V. Proof: Let a be a set and let F be a class term.
	br.endernet.org /~loner/settheory/evcummhier.txt (1964 words)

	Wikinfo \| Set (Site not responding. Last check: 2007-11-02)
		Cantor's theorem states that the cardinality of the set of all subsets of a set A is strictly greater than the cardinality of A itself.
		For a discussion of the properties and axioms concerning the construction of sets, see naive set theory and axiomatic set theory.
		is either not allowed (in the case of the Zermelo-Frankel axioms) or is considered to be a proper class (in the case of the von Neumann-Bernays-Godel axioms), and we have no paradox.
	www.wikinfo.org /wiki.php?title=set (1227 words)

	Poster Project, What Is Scientific Truth Poster (Site not responding. Last check: 2007-11-02)
		You start with a given set of axioms (axioms are statements which, for the purposes of the game, are assumed to be true) and a set of rules for manipulating these axioms.
		To say that any given statement is true is to say that it is possible to form the statement by manipulating the axioms according to the rules of logic.
		This means that it is not possible to derive both a statement and its converse from the axioms.
	www.math.sunysb.edu /posterproject/www/materials/truth/truth.html (1144 words)

	Mail-Jewish Volume 19 Number 17 (Site not responding. Last check: 2007-11-02)
		Avraham Frankel, a mathematician who made important contributions to set theory in the early part of this century (e.g.
		When he was growing up, Binyamin Frankel would hear his father and colleagues arguing about _why_ one plus one is two.
		On the other hand, I think Binymain Frankel would justifiably think it peculiar if he were giving a talk on x-ray diagnostics for tokamaks and someone in the audience made a comment on it, using the Zermelo-Frankel axioms to prove a point.
	www.emax.ca /mj_ht_arch/v19/mj_v19i17.html (3169 words)

	What makes mathematics? Text - Physics Forums Library
		The ZF axioms are basically useless to me because I have a problem with mathematical formalism at a far more fundamental level that that.
		Using the axioms of Euclidean geometry, you can prove that this system of hyperbolic things satisfies all of the axioms of Euclidean geometry except the parallel postulate, which it violated.
		Incidentally, one of the axioms of ZF (the axiom of infinity) says essentially "There is a set of all natural numbers"...
	www.physicsforums.com /archive/index.php/t-4033.html (18326 words)

	Set Theory. Zermelo-Fraenkel Axioms. Russell's Paradox. Infinity. By K.Podnieks
		The axioms C1, C1' and C2[F] (for all formulas F that do not contain x) and the axiom of choice define a formal set theory C which corresponds almost 100% to Cantor's intuitive set theory (of the "pre-paradox" period of 1873-94).
		To prove the existence of uncountable sets, the power-set axiom C24 must be applied additionally: by Cantor's Theorem, the set of all sets of natural numbers P(w) is uncountable.
		The set theory adopting the axiom of extensionality (C1), the axiom C1', the separation axiom schema (C21), the pairing axiom (C22), the union axiom (C23), the power-set axiom (C24), the replacement axiom schema (C25), the axiom of infinity (C26) and the axiom of regularity (C3), is called Zermelo-Fraenkel set theory, and is denoted by ZF.
	linas.org /mirrors/www.ltn.lv/2005.01.29/~podnieks/gt2.html (8496 words)

	AskPhilosophers.org
		The most popular modern approach to set theory is based on the axioms developed by Zermelo and Frankel, and the Zermelo-Frankel (ZF) axioms are formulated to avoid the paradox.
		The reason paradoxes in set theory are considered to be such a serious matter is that most mathematicians regard set theory as the foundation of all of mathematics.
		So if a new paradox were discovered that showed that the ZF axioms were flawed, I don't think anyone's confidence in 2+2=4 would be shaken.
	www.amherst.edu /askphilosophers/question/153 (384 words)

	Springer Online Reference Works
		Gödel and P.J. Cohen in the (unexpected) sense that the continuum hypothesis is independent of the Zermelo–Frankel axioms.
		For the Wightman axioms (also called Gårding–Wightman axioms) and the Osterwalder–Schrader axioms of quantum field theory see Constructive quantum field theory; Quantum field theory, axioms for.
		Currently (1998) there is a great deal of interest and activity in (the axiomatic approach represented by) topological quantum field theory and conformal quantum field theory; see e.g.
	eom.springer.de /h/h120080.htm (2418 words)

	LATEX of Lattice Embeddings
		\end{theorem} We note that Theorem \ref{thm1.6} has been proved by Friedman using axioms of large cardinals that are independent of the usual Zermelo-Frankel axioms of set theory plus the axiom of choice (ZFC).
		The assumption that there exists an uncountable inaccessible cardinal is the simplest example of a large cardinal axiom.
		A second type of large cardinal axiom would be to assume that there are weakly compact cardinals.
	www-cse.ucsd.edu /users/gill/Research/LatEmbLATEX.html (7343 words)

	Zermelo-Frankel Set Theory
		The Axiom of Extensionality remains the same, but the Comprehension Axiom is replaced by something of a hodgepodge of axioms that are not as intuitively obvious.
		The Axiom of Choice, although widely accepted, is controversial.
		So these two axioms together show that there is no set that is a member of itself.
	www.trinity.edu /cbrown/topics_in_logic/sets/node4.html (514 words)

	Graduate Courses
		Includes Zermelo-Frankel axioms for set theory, ordinal and cardinal numbers, the axiom of choice, and Zorn's Lemma.
		Studies of English translations of various mathematical classics from ancient to modern times, covering a wide range of various mathematical ideas relevant to the teaching of mathematics at the high school level.
		Topological spaces, continuous mappings, and homomorphisms, separation axioms, metric spaces and metrization, compact and connected spaces, product spaces and uniform spaces.
	www.math.fau.edu /Graduate_Students/courses.htm (879 words)

	Math 504 (Site not responding. Last check: 2007-11-02)
		In the first half of the century it was shown that most of mathematics can be formalized inside of set theory and that a simple set of axioms could be given so that every acceptable proof followed formally from these axioms.
		Godel's Incompleteness Theorem implies that there are mathematical truths not settled by these axioms.
		This course will start by introducting the axioms for set theory and developing the basic theory of cardinals and ordinals.
	www.math.uic.edu /~marker/504.html (218 words)

	Set Theory and Its Philosophy: A Critical Introduction (Site not responding. Last check: 2007-11-02)
		Now most treatments of set theory either hand down the standard Zermelo-Frankel (ZF) axioms as dogma, or just make them as an arbitrary starting point.
		But settling ZF's undecidable proposition will perforce require committing to new axioms, and justifying any such commitment will require the sort of philosophical nuance that Potter brings to unfolding the existing framework.
		This book features many nice touches: the distinction between fusion and collection; the bearing of second-order set theory on the continuum hypothesis; the distinct character for a philosophical point of view between the independence of the continuum hypothesis and that of the parallel postulate.
	www.booksmatter.com /b0199269734.htm (358 words)

	Highlights of 2005 Joint Math Meetings (Site not responding. Last check: 2007-11-02)
		Cohen, now 70 years old, received the Fields Medal in 1966 for showing that CH is independent of the Zermelo-Frankel axioms of set theory.
		Some from the audience thought that axioms must be simple and natural; Cohen agreed, while Martin and Woodin disagreed.
		"I want axioms to be what they are," said Martin, "and I don't know that they'll be simple." Woodin noted that while axioms may be simple, the way in which they reflect reality may not be simple at all--and that what's considered "simple" can vary greatly over time.
	www.ams.org /ams/jmm2005-highlights.html (5018 words)

	Read This: The Language Of Mathematics
		While doing this we brush past Pythagorean numbers and their applications in construction and we learn how a proof by induction works.
		In the next chapter we start by learning logic from Aristotle, and travel all the way to the Zermelo-Frankel axioms, just to find out that we now have the tools to solve such linguistic mysteries as "who wrote the Federalist papers?".
		It is all mathematics, everywhere we look around us.
	www.maa.org /reviews/langmath.html (841 words)

	Read This: Mathematical Mountaintops
		Is there a set which is bigger than the set of natural numbers but smaller than the set of real numbers?
		The standard Zermelo-Frankel axioms of mathematics are not strong enough to give an answer to this question.
		Does the standard way to stack oranges give the densest packing of spheres in three space?
	www.maa.org /reviews/mountaintops.html (1545 words)

	Duke Math News (Site not responding. Last check: 2007-11-02)
		The prerequisite for the course is MTH 104, though MTH 121 or 139 could prove helpful.
		Topics will include cardinal arithmetic, the Zermelo-Frankel axioms for set theory, and the axiom of choice.
		The topics will be developed in historical order.
	www.math.duke.edu /math_news/October97/October97.html (2672 words)

	Duke Mathematics Undergraduate Papers and Talks (Site not responding. Last check: 2007-11-02)
		Luis Von Ahn Models of the language of set theory and Zermelo Frankel axioms (2000) [with R. Hodel]
		David Jones Primality testing, factoring and continued fractions (1992) [with C. Schoen]
		Will Schneeberger The axiom diamond (1992) [with J. Shoenfield]
	www.math.duke.edu /news/awards/research.html (797 words)

Try your search on: Qwika (all wikis)