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Topic: Axiom of comprehension

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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

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	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.
		The axiom schema of specification is characteristic of systems of axiomatic set theory related to the usual set theory ZFC, but does not usually appear in radically different systems of alternative set theory.
		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).
		In theories which make no distinction between objects and sets (such as ZF), this formulation leads to Russell's paradox, however in stratified theories this is not a problem (for example second order arithmetic includes the axiom of comprehension).
		This is version 6 of comprehension axiom, born on 2002-08-17, modified 2006-08-16.
	planetmath.org /encyclopedia/ComprehensionAxiom.html (213 words)

	PlanetMath: permutation model
		A permutation model is a model of the axioms of set theory in which there is a non trivial automorphism of the set theoretic universe.
		Such models are used to show the consistency of the negation of the Axiom of Choice (AC).
		Cross-references: completes, function, closed under, axiom of comprehension, universal, transitive, symmetric sets, class, symmetric, fixes, permutations, group, rank, induction, infinite set, atoms, axiom of foundation, axiom of choice, negation, universe, automorphism, set theory, axioms
	planetmath.org /encyclopedia/GodelOperations.html (151 words)

	April 3, 2000
		We will start with an unusual set of axioms which I chose because of their elegance; we will prove from these a set of theorems (which I will formally list) which could be taken as axioms in a more usual development.
		It seems natural to propose this axiom to say what sets there are: Axiom of Comprehension: For every property P, there is a set {x : Px} such that for all y, y E {x : Px} iff Py.
		Axiom of Subsets We define x c= y (x is a subclass of y) as (Az.
	math.boisestate.edu /~holmes/M387syllabus/node50.html (893 words)

	Axiomatic Theories of Truth (Stanford Encyclopedia of Philosophy)
		The reductions of second-order theories (i.e., theories of properties or sets) to axiomatic theories of truth may be conceived as forms of reductive nominalism, for they replace existence assumptions for sets or properties (e.g., comprehension axioms) by ontologically innocuous assumptions, in the present case by assumptions on the behaviour of the truth predicate.
		Axiom 5 says that a universally quantified sentence of the language of arithmetic is true if and only if all its numerical instances are true.
		If these axioms are to be formulated for a language like set theory that lacks names for all objects, then (5) and (6) require the use of a satisfaction relation rather than a unary truth predicate.
	plato.stanford.edu /entries/truth-axiomatic (6010 words)

	FOLDOP search (Site not responding. Last check: 2007-09-10)
		A formula containing variables of the metalanguage which becomes an axiom when its variables are instantiated to wffs of the formal language.
		An axiom that is a logically valid wff of the language of the system.
		An axiom that is not a logically valid wff of the language of the system (but is a closed wff).
	www.swif.it /foldop/dizionario.php?find=axiom (112 words)

	When is one thing equal to some other thing? \| Lambda the Ultimate
		The author does not profer an axiom of comprehension, and explicitly sets aside notions of quantifiers, but is at pains (during the parts that I have read) to separate the concept of 'class' from the concept of 'set', so I do not understand how the theory he presents suffers from Russel's antimony.
		ZF is one example with its axiom of separation/subset/restricted comprehension schema, NBG is another example which admits a finite axiomatization.
		He did not say explicitely that his "bare sets" allow unrestricted comprehension, but unless such restriction is clearly stated, the default is to assume the naive set theory with unrestricted comprehension.
	lambda-the-ultimate.org /node/1338 (5350 words)

	Alternative Axiomatic Set Theories (Stanford Encyclopedia of Philosophy)
		The confidence of at least some mathematicians in their understanding of this subject (or in its coherence as a subject at all) was shaken by the discovery of paradoxes in "naive" set theory around the beginning of the 20th century.
		Further, the introduction of the Foundation Axiom identifies the set theories of this class as the theories of a particular class of structures (the well-founded sets) of which the Zermelo axioms certainly seem to hold (whether Replacement holds so evidently is another matter).
		An Axiom of Infinity would be wanted to ensure that an interpretation of Heyting arithmetic could be embedded in the theory; it might be simplest to provide type 0 with the primitives of HA (just as the earliest versions of TST had the primitives of classical arithmetic provided for type 0).
	plato.stanford.edu /entries/settheory-alternative (17285 words)

	3. NFU and related set theories
		In NF the axiom schema of separation used in ZF is replaced by a stratified axiom schema of comprehension.
		According to NF, the axiom schema of comprehension Axiom 1 holds for all stratified formulas.
		Though NF as originally defined consists of these two axioms alone, the acronym ``NFU'' is usually used in later literature to refer to the combination of these two axioms with the Axiom of Infinity and the Axiom of Choice.
	www.hf.uio.no /ifikk/filosofi/njpl/vol4no1/ruskap/node3.html (933 words)

	The Proper Axioms of the Theory
		Then the above instance of the comprehension axiom for abstract objects asserts that there is an abstract object that encodes just those properties F that satisfy these sentences.
		Then this instance of comprehension asserts that there is an abstract object that encodes both of those properties and no others.
		The present instance of comprehension merely asserts that there is an abstract object that encodes roundness and squareness; it is logically compatible with the above law of geometry for it does not follow from the fact that an object encodes a property that it also exemplifies that property.
	mally.stanford.edu /tutorial/proper.html (907 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).
		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 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)

	Subspaces in abstract Stone duality
		The construction is done first by formally adjoining certain equalisers that $\Sigma^{(-)}$ takes to coequalisers, then using Eilenberg-Moore algebras, and finally presented as a lambda calculus similar to the axiom of comprehension in set theory.
		The comprehension calculus has a normalisation theorem, by which every type can be embedded as a subspace of a type formed without comprehension, and terms also normalise in a simple way.
		Keywords: axiom of comprehension; subtype; typed lambda calculus; Stone duality; subspace topology; locally compact spaces; nucleus of a locale; injective object; monadic adjunction; Beck’s theorem.
	www.tac.mta.ca /tac/volumes/10/13/10-13abs.html (248 words)

	More on Reverse Mathematics
		This system is closely related to first-order arithmetic (or first-order Peano axioms), defined as the basic axioms, plus the first order induction axiom scheme (for all formulas φ involving no class variables at all, bound or otherwise), in the language of first order arithmetic (which does not permit class variables at all).
		Arithmetical comprehension is the comprehension axiom scheme for arithmetical formulae: that is, it allows us to form the set of natural numbers satisfying an arbitrary arithmetical formula (one with no bound class variables).
		For the reverse, the idea is as follows: to prove arithmetical comprehension it is actually sufficient to prove Σ1 comprehension (because that allows us to remove a quantifier, and we then use induction on the number of quantifiers since we can certainly pass to the complement).
	www.artilifes.com /reverse-mathematics.htm (3606 words)

	FAQ: Epistemology, Axioms, Reality, Consciousness, the Universe and Everything (Site not responding. Last check: 2007-09-10)
		The Axiom of identity states that a thing is something, it has a nature, an identity.
		From the axioms of union and pairing we are able to prove that the universal membership predicate is infinite and that all other sets are subsets of the universal membership predicate through the axiom of subset (comprehension).
		You are specifically warned that study of documents produced by the Church of Virus may lead to a permanent change in your attitudes or behavior as a result of exposure to the memeplexii and component memes embedded in such documents.
	www.nemorathwald.com /6AxiomsEpistemology.htm (1655 words)

	Axiomatic Set Theory
		This axiom means that a set is determined by its elements.
		1.3.5 Axiom of power: for every set x there exists a set y the elements of which are the subsets of x.
		1.3.6 Axiom of comprehension or separation : For every set x and every formula there exits a set whose elements are exactly those of x for which holds.
	www.mathresource.iitb.ac.in /project/Axiom_pg2.html (242 words)

	C[omp]UTE (Site not responding. Last check: 2007-09-10)
		Since the axiom of regularity implies that no set contains itself as a member, it can be tempting for the non-expert to think that the presence of the axiom of regularity in Zermelo-Fraenkel set theory (ZF) has something to do with the way in which ZF resolves Russell's paradox.
		The axiom of regularity is irrelevant to the resolution of Russell's paradox.
		And if you look at the Separation Axiom then the "architect approach" is pretty clear - "If X is a set and P is a condition on sets, there exists a set Y whose members are precisely the members of X satisfying P".
	www.acooke.org /cute/ACA0Russel0.html (365 words)

	Bertrand Russell / Biography
		The axiom gives form to the intuition that any coherent condition may be used to determine a set (or class).
		Using the vicious circle principle also adopted by Henri Poincaré, together with his so-called "no class" theory of classes, Russell was then able to explain why the unrestricted comprehension axiom fails: propositional functions, such as the function "x is a set", should not be applied to themselves since self-application would involve a vicious circle.
		Although the axiom successfully lessened the vicious circle principle's scope of application, many claimed that it was simply too ad hoc to be justified philosophically.
	www.cooperativeindividualism.org /russellbio.html (1395 words)

	The First Three Axioms (Site not responding. Last check: 2007-09-10)
		This is actually an axiom scheme, rather than a single axiom.
		We wouldn't want set theory to become inconsistent after only 3 axioms, thus y cannot be a free variable in f.
		Combine the set x in axiom 0 with the formula z ≠ Z using comprehension.
	www.mathreference.com /set-zf,ax0.html (298 words)

	Positive set theory - Wikipedia, the free encyclopedia
		The closure conditions for the various constructions allowed in building positive formulas are readily motivated (and one can further justify the use of universal quantifiers bounded in sets to get generalized positive comprehension): the justification of the existential quantifier seems to require that the topology be compact.
		The axiom of closure: for every formula φ(x), a set exists which is the intersection of all sets which contain every x such that φ(x); this is called the closure of
		The axiom of infinity: the von Neumann ordinal ω exists.
	en.wikipedia.org /wiki/Positive_set_theory (436 words)

	Charming Python: Getting version 2.0
		In any case, list comprehension is a way of doing much of what Python's functional built-ins do, but in a much more compact way that is simultaneously easier to read and understand.
		This is often useful if you do not want a list comprehension that uses a complete permutation of lists, but merely one that utilizes corresponding elements of multiple lists.
		As with list comprehensions, no fundamentally new capability is added, but the expression of some common chores is clearer and more concise.
	www-106.ibm.com /developerworks/linux/library/l-py20.html (2227 words)

	Axiom Software - Discovery Series
		Axiom's Discus range of profilers have always been at the forefront of personality assessment.
		Axiom's new 'Discovery' series is designed to do just that.
		A test designed to test your general level of reading comprehension, as well as looking at your aptitude for inferring and deducing information from written text.
	www.axiomsoftware.com /products/discovery.asp (431 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).
		He proposed to restrict the comprehension axiom schema by adopting only of those axioms, which are really necessary for reconstruction of common mathematics.
		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)

	ru - Metamath Proof Explorer
		This contradiction was discovered by Russell in 1901 (published in 1903), invalidating Comprehension and leading to the collapse of Frege's system.
		In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom funimaex 3290 (whose modern formalization is due to Skolem, also in 1922).
		An advantage of NBG is that it is finitely axiomatizable - the Axiom of Replacement can be broken down into a finite set of formulas that eliminate its wff metavariable.
	us.metamath.org /mpegif/ru.html (438 words)

	The Logic of Complex Predicates
		Remark: Notice that in the last instance of Axiom 1, we have an example of a 3-place relation that is exemplified just in case a 2-place relation is exemplified.
		This is a logical theorem schema that can be derived from Axiom 1 in n+2 simple steps: apply Universal Generalization to Axiom 1 n times, beginning with the variable x_n and ending with the variable x_1; then apply the Rule of Necessitation; finally apply the rule of Existential Generalization to the complex predicate.
		These definitions, together with the comprehension principle for relations, constitute a mathematically precise theory of relations and properties, for they are explicit existence and identity conditions for these entities.
	mally.stanford.edu /tutorial/complex-predicates.html (1210 words)

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