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Topic: Axiom of specification

	Axiom schema of specification - Wikipedia, the free encyclopedia
		In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, or axiom schema of separation, or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory.
		This axiom schema was tacitly used in the early days of naive set theory, before a strict axiomatisation was adopted.
		Most of the other Zermelo-Fraenkel axioms (but not the axiom of extensionality or the axiom of regularity) then became necessary to serve as an additional replacement for the axiom schema of comprehension; each of these axioms states that a certain set exists, and defines that set by giving a predicate for its members to satisfy.
	en.wikipedia.org /wiki/Axiom_schema_of_specification (1012 words)

	PlanetMath: comprehension axiom
		The names specification and separation are sometimes used in place of comprehension, particularly for weakened forms of the axiom (see below).
		CA, -CA, comprehension, comprehension axiom, axiom of comprehension, separation, separation axiom, axiom of separation, specification, specification axiom, axiom of specification, Aussonderungsaxiom
		This is version 6 of comprehension axiom, born on 2002-08-17, modified 2006-08-16.
	planetmath.org /encyclopedia/Specification.html (213 words)

	The Book of THoTH (Leaves of Wisdom) - Axiom
		The word axiom comes from the Greek word αξιωμα (axioma), which means that which is deemed worthy or fit or that which is considered self-evident.
		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.
		The axioms are referred to as "4 + 1" because for nearly two millennia the fifth (parallel) postulate ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four.
	www.book-of-thoth.com /thebook/index.php/Axiom (3028 words)

	[No title]
		The generation of valid expressions from axiom instances is governed by the simple rule that if all the conditions of an axiom instance are valid expressions, then the conclusion is a valid expression.
		These axioms generate expressions such as '(the sort of ((* * ) () ()) is (() () ( * *)))', which states that the sort of (3 0 1) is (0 1 3).
		Axioms are defined the same as in regular axiomatic language and the rules for generating valid expressions from axioms likewise correspond.
	www.axiomaticlanguage.org /sigplan82.html (3902 words)

	Chapter II
		Axioms are comprehensions, propositions, declarations, which are initially set in place AS IF they stand of their own power and dignity, AS IF they were capable of, but do not need, a verification or confirmation from another source.
		The Axiom of the Mediation consequently does not only point to the worldly side, but also to the non-worldly side of the beingness, insofar it is the expression of the transcendence of Consciousness.
		For this reason we can name the axiom of formation also as the axiom of specification [2], the axiom of construction, of shaping, of imagination.
	www.freezone.de /scientologie/e_34c02.htm (1785 words)

	Axiom schema of replacement - Wikipedia, the free encyclopedia
		In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory which essentially asserts that the image of a set under a mapping is also a set.
		The axiom was independently discovered by Thoralf Skolem later in the same year, and it is in fact Skolem's final version of the axiom list that we use today -- but he usually gets no credit since each individual axiom was developed earlier by either Zermelo or Fraenkel.
		But in this case, the set B required for the axiom of specification is the empty set, so the axiom schema follows in general using also the axiom of empty set.
	en.wikipedia.org /wiki/Axiom_schema_of_replacement (1636 words)

	Axiom UK - Project Management
		Axiom's attitude towards project management is based on sound system design.
		When the installation starts Axiom will appoint a key individual to your project and he / she will be responsible for all aspects of the installation, ensuring continuity right through until project hand over.
		Axiom and their customers work together as a team to accommodate 3rd party issues.
	www.axiomgb.com /ProjectManagementPage.html (273 words)

	JML Reference Manual: Method Specifications
		is intended to tell the reader that this specification is in addition to some specifications of the method that are given in the superclass of the class, one of the interfaces it implements, or in another file in the refinement sequence.
		So, for example, a public specification case may only refer to public members, a protected specification case may refer to both public and protected members, as long as the protected members are otherwise accessible according to Java's rules, etc. See section 2.4 Privacy Modifiers and Visibility, for more details and examples.
		Multiple signals clauses in a specification case mean the same as a single signals clause whose exceptional postcondition predicate is the conjunction of the exceptional postcondition predicates in the given signals clauses.
	www.cs.iastate.edu /~leavens/JML/jmlrefman/jmlrefman_9.html (6544 words)

	8.3 State of the art
		By the antidecomposition axiom, whenever a parent class is modified, it is necessary to retest all its descendant classes, and when we add a new descendant class, we must retest the methods inherited from its parent classes, because it has an enlarged set of methods and instance variables: the vector
		They differentiate two cases: the case where testing the descendant against its own specification is enough, and the case where it is also necessary to test the descendant against the specification of its parent.
		When testing the descendant against the specification of its parent, they state that it may be unnecessary to perform a complete regression test, but that it is only necessary to execute the test cases that involve methods that have been redefined.
	lglwww.epfl.ch /Team/SB/PhD/html/phd.34.html (3933 words)

	Java Web services, Part 2: Digging into Axis2: AXIOM
		AXIOM is one of the major innovations behind Axis2, and one of the reasons Axis2 offers the potential for substantially better performance than the original Axis.
		AXIOM provides a virtual document model it expands on demand, building only as much of the tree structure document model representation as has been requested by the client application.
		Given that AXIOM is really designed around the on-demand expansion of the tree, the timings for the two larger documents are not a big concern since they're at least close to the other document models.
	www-128.ibm.com /developerworks/webservices/library/ws-java2 (4830 words)

	SetTheory - PineWiki
		With just the axiom of extensionality, it is possible that no sets at all exist; the axiom of existence gives us at least one set (but possibly no others).
		In reading the statement of the axiom, you should think of x as the set we are taking a union over, y as the union ∪x, z as some element that might or might not be in y, and q as an element of x that contains z and thus puts z in the union.
		Finally, we have one final axiom, which is generally accepted by mathematicians because it's convenient but is in a very real sense optional, in that the rest of the axioms are consistent with either it or its negation.
	pine.cs.yale.edu /pinewiki/SetTheory (4112 words)

	More on Axiomatically
		By constructing a valid model for a subsystem without a specific axiom, we show that the omitted axiom is independent if its correctness does not necessarily follow from the subsystem.
		This traditional approach, in which axioms were supposed to be self-evident and so indisputable, was swept away during the course of the nineteenth century, by the development of Non-Euclidean geometry, the foundations of real analysis, Cantor's set theory and Frege's work on foundations, and Hilbert's 'new' use of axiomatic method as a research tool.
		Once the axioms were clarified (that inverse elements should be required, for example), the subject could proceed autonomously, without reference to the transformation group origins of those studies.
	www.artilifes.com /axiomatically.htm (820 words)

	6moons audio reviews: Axiom M80 v2 & EP600
		Axiom claims that their amplifier is unique in several respects.
		Axiom's website is very short on explanation as to how their DSP works but claims that the chip provides a digital roadmap for the amp by which it monitors (and then corrects) various parameters of performance to control linearity and keep the EP600 performing within specification.
		Axiom reports that this is only an issue with the M80s and smaller models with less bass output don't interact in the same way.
	www.6moons.com /audioreviews/axiom/axiom_2.html (2677 words)

	Axiom 25 Product Description
		Axiom 25™ is an easy to apply, one component sealant that cures to a durable, flexible silicone rubber upon exposure to atmospheric moisture.
		Axiom 25™ was formulated for use with all solid surface products, engineered stone, cultured marble, and other non-porous surfaces.
		Axiom 25™ is a highly flexible, fast-curing silicone sealant which accommodates joint movement of +/- 25 percent.
	www.axiom25.com /prod_desc.htm (418 words)

	Univeral Resource Identifiers -- Axioms of Web architecture
		The aim of thes article is to summarize in one place the axioms of Web architecture: those invariant aspects of Web design which are implied or stated in various specifications or in some cases simply part of the folk law of how the Web ought to be used.
		That is a very important axiom, and it leaves open the "in some sense" behind which is a very complicated discussion of the concept of identity.
		As a result of this axiom the many parts of metadata, that information about the object that a client might be tempted to infer from the actual sting value of the URI but can't, have to be made available through the HTTP protocol.
	www.w3.org /DesignIssues/Axioms.html (5269 words)

	MainFrame:Axioms for galactic set theory.
		The axioms of extensionality and well-foundedness may be thought of as telling us what kind of thing a set is (later axioms tell us how many of these sets are to be found in our domain of discourse).
		The axiom of well-foundedness asserts the requirement that the elements of ('a)GS are a subset of the cumulative heirarchy of sets formed by iteration of set formation beginning with the empty set.
		The remaining axioms are intended to ensure that the subset is a large and well-rounded subset of the cumulative heirarchy.
	www.rbjones.com /x-logic/pp/gst/gst-axioms-m.html (2612 words)

	Get the most out of XML processing with AXIOM
		AXIOM goes beyond existing XML processing methodologies as it combines deferred building with a customizable object model that is fast and lightweight.
		But AXIOM's memory footprint is better than most of the existing object models that depend on SAX and/or DOM for input and output.
		AXIOM has many more powerful features, and I encourage you to download the latest source from the Axis 2 source repository (see Resources) and explore more about AXIOM.
	www-128.ibm.com /developerworks/xml/library/x-axiom (2818 words)

	Article: Introducing Axiom
		Axiom is an XML Object model, just like DOM, jDOM, but better than them in terms of memory and perfomance.
		In this panel, moderated by Ted Neward at TSSJS in March, Cameron Purdy, Rod Johnson, Bruce Snyder, Bruce Tate, Floyd Marinescu and Ari Zilka engage in a lively discussion highlighting the current state of the Java platform, its strengths and shortcomings and where they think the language is headed.
		With lightweight domain specific modeling, developers can implement and use their own code generators to avoid repetitive manual coding and increase efficiency.
	www.theserverside.com /news/thread.tss?thread_id=41361 (4600 words)

	Audioholics On-Spot Location: Factory Tour #2 of Axiom Audio (Site not responding. Last check: 2007-11-05)
		Based on the excellent digital amp which is the heart of the EP500 and EP600 subwoofers, Axiom has taken this design concept to the next level by developing one of the industry’s first high-performance full-range digital amplifiers.
		In discussing this amplifer design with Tom, he informed me that the design group at Axiom carefully selected the poles of the LPF so there would be no phase or frequency response shifts in the audio passband while driving 1-ohm to 8-ohm reactive loudspeaker loads.
		Axiom’s massive power supply and efficient amplifier topology allows the amplifier to maintain a very high power factor of greater than 90%.
	www.audioholics.com /news/editorials/AxiomAudioTour2.php (1694 words)

	Set theory
		Axiom of pair: If A and B are sets, then there exists a set F, that contains the set A and set B. Axiom of specification: Let U be any set and P any statement.
		According to the axiom of specification, there is a set that contains only A and B. That set is called a pair and is denoted by {A, B}.
		Definition: If certain axioms imply their own duals, then the dual of any theorem that is a consequence of the axioms is also a consequence of the axioms.
	library.thinkquest.org /C0126820/algebra.html (1608 words)

	RELAX NG Specification
		This is the definitive specification of RELAX NG, a simple schema language for XML, based on [RELAX] and [TREX].
		Axioms and inference rules are implicitly universally quantified over the variables they contain.
		The possibility that an inference rule or axiom may contain more than one occurrence of a particular variable requires that an identity relation be defined on each kind of object over which a variable can range.
	www.oasis-open.org /committees/relax-ng/spec-20011203.html (4969 words)

	Java.net - the Source for Java(tm) Technology Collaboration
		Axiom is much more memory-efficient and performant than the existing object models available today, due to the usage of StAX as its XML parsing technology.
		Apache Axiom: The XML object model that uses StAX as its underlying XML parsing methodology
		Eran Chinthaka is a pioneering member of Apache Axis2, AXIOM and Synapse projects, working fulltime with WSO2 Inc..
	today.java.net /lpt/a/305 (1322 words)

	MathAction and Axiom ('', 'mathaction', 'TuplesProductsAndRecords')
		Axiom is a strongly typed language with object oriented roots.
		In the case of Axiom I think that it is highly significant that "The category Cat of all small categories (with functors as morphisms) is cartesian closed;".
		I think Axiom's definition of Mapping is wrong to admit more than two arguments (mapping should be semantically equivalent to exponentiation in a cartesian closed category).
	wiki.axiom-developer.org /TuplesProductsAndRecords (6929 words)

	SENG 609.03 Object Theory: Review of Wegner Part II
		Subsection two "Specification of Inheritance" examines the issues associated with the problem of modeling behavior (types) with syntactic constructs (signatures).
		A class is a implementation dependent specification made up of a signature and a body where the body is code.
		In these cases the behavior of a subtype may not be compatible with the behavior of a specific supertype but the difference between them should be deducible by the name conflicts algorithm used.
	sern.ucalgary.ca /courses/SENG/609.03/W98/nhhennin1/review.htm (2561 words)

	Aldor Compiler User Guide: Section III, Chapter 18: Using Aldor with AXIOM (Site not responding. Last check: 2007-11-05)
		This course of redefining AXIOM operations should be a last resort used in order to gain efficiency, because any changes to the representation of a domain must be reflected in both the AXIOM library, and in the Aldor and Lisp code which defines the extension.
		AXIOM may not be available on the machine on which you have Aldor installed.
		This is needed by AXIOM and during the compilation of other files that depend on the contents of the current file.
	www.csd.uwo.ca /~watt/aldor/UserGuide/chap18.html (5299 words)

	Axiom 25 (Site not responding. Last check: 2007-11-05)
		These values are not intended for use in preparing specification.
		In no event shall manufacturer or seller be responsible for incidental or consequential damages arising out of or in connection with the use of this product by buyer, buyer’s users, or any other person.
		Notification of any claim must be sent in writing to Axiom.
	www.kampelent.com /axiom25.htm (419 words)

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