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Topic: Axiom of replacement

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

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	Axiom schema of replacement - Wikipedia, the free encyclopedia
		The axiom of replacement then states that given a set A, we can find a set B whose members are precisely the values of F at the members of A.
		Replacement is difficult to express at all in foundations built upon topos theory, so it's usually left out there as well.
		Nevertheless, replacement is not controversial in the sense that some people find its consequences to be necessarily false (a sense in which the axiom of choice, for example, is controversial); it's just that they find it unnecessary.
	en.wikipedia.org /wiki/Axiom_schema_of_replacement (1636 words)

	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)

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	mmtheorems46 - Metamath Proof Explorer (Site not responding. Last check: 2007-10-13)
		Axiom of Replacement, reproved from conditionless ZFC axioms.
		Axiom of Union, reproved from conditionless ZFC axioms.
		Axiom of Infinity, reproved from conditionless ZFC axioms.
	metamath.planetmirror.com /mpegif/mmtheorems46.html (717 words)

	New Axioms for Set Theory
		The theory of sets is canonical and the axiom schema appears to be simply a canonical schema stating that the universe is endless and that an extension would be nonrigid, and providing powerful reflection principles for set theory.
		Also, the axiom schema with measurable replaced by inaccessible can be approximated by allowing greater expressiveness in the formulas in the replacement axiom schema and be viewed as the natural limit of such approximations.
		GCH is a natural strengthening of the Axiom of Choice: GCH implies the Axiom of Choice over ZF and the “unintuitive” consequences of GCH can be viewed as the natural strengthening of the consequences of the axiom of choice.
	web.mit.edu /dmytro/www/ProposedAxioms.htm (1541 words)

	Amazon.ca: Axiomatic Set Theory: Books: Patrick Suppes (Site not responding. Last check: 2007-10-13)
		The axiom schema that is used explicitly in the book is the "axiom schema of separation" due to Ernst Zermelo, which he formulated in order to make precise the notion of a statement as being "definite".
		The axiom of infinity is brought in to permit the construction of arithmetical operations as certain sets.
		The author shows that the use of this axiom allows one to prove that an infinite set has a denumerable subset, and he shows the equivalence of the axiom of choice with the numeration theorem, the well-ordering theorem, Zorn's lemma, and the law of trichotomy.
	www.amazon.ca /Axiomatic-Set-Theory-Patrick-Suppes/dp/0486616304 (1706 words)

	MA10126 - Set Theory - Axioms of Set Theory (ZFC)
		Axiom of Pairing - For all a and b, there exists a set X whose elements are exactly a and b.
		Axiom of the Power Set - For all sets X, there exists a set Y = P(X) whose elements are all the subsets of X. Georg Cantor used this axiom to prove that not all infinite sets have the same size (cardinality).
		Stated like this, the Axiom of Choice appears intuitively obvious - but the axiom is equivalent to a number of statements that are strongly counter-intuitive - primarily the Well-Ordering Theorem, and the Banach-Tarski Paradox.
	www.bath.ac.uk /~njs25/axioms.html (607 words)

	PlanetMath: Zermelo-Fraenkel axioms
		Ernst Zermelo and Abraham Fraenkel proposed the following axioms as a foundation for what is now called Zermelo-Fraenkel set theory, or ZF.
		If this set of axioms are accepted along with the Axiom of Choice, it is often denoted ZFC.
		This is version 14 of Zermelo-Fraenkel axioms, born on 2001-10-18, modified 2006-10-25.
	planetmath.org /encyclopedia/ZermeloFraenkelAxioms.html (215 words)

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	Michael Potter - Set Theory and Its Philosophy: a Critical Introduction - Reviewed by Timothy Bays, University of Notre ...
		Replacement is introduced as one of several axioms which govern the height of the set-theoretic hierarchy (the other main ones are the previously mentioned "axiom of ordinals" and an axiom scheme of reflection, although modern large cardinal axioms do receive a brief mention).
		There are well-known difficulties with generating these axioms from the iterative conception (for instance, the very fact that the axioms don't follow from axiomatizations of the kind Potter gives in section one).
		I think, for instance, that replacement looks more intuitive when it's considered in conjunction with the axiom of infinity, and that regressive arguments for choice and replacement together work better than arguments for either of them by itself (because, for instance, of the nice structure they jointly put on the classes of cardinals and ordinals).
	ndpr.nd.edu /review.cfm?id=2141 (2338 words)

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	MainFrame:Axioms for galactic set theory.
		The replacement axiom embodies the "limitation of size" principle (or its converse) by stating that any collection which is no larger than a known set is also a set.
		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)

	The Weblog: Single Post View
		Once induction #1 is made, and once set theory is selected as the mathematical language, axioms follow promptly - and not for any reason other than those that led Zermelo down that path in the first place.
		Although he never stated them, all of Cantor's operations are said to be derivable from three axioms: the axiom of extensionality, the axiom of abstraction, and the axiom of choice.
		Although Zermelo's axiom schema of separation gets supplanted by Fraenkel's axiom schema of replacement, the important thing to note here is that the first step toward axiomatization developed directly in response to early paradoxes.
	www.adamkotsko.com /weblog/2006/02/laicity-of-axioms.html (396 words)

	zfaxioms.html
		Axiom ZF1 - Sets with the same members are equal - (Extensionality).
		Axiom ZF2 - The "Empty Set" is a set.
		Axiom ZF9 - There are not Russell's Paradox like Sets (Regularity).
	www.umsl.edu /~siegel/SetTheoryandTopology/zfaxioms.html (286 words)

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	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)

	Storm Knights: Magic Axiom (Site not responding. Last check: 2007-10-13)
		Building a complete Magic axiom required the developement of a comprehensive magical theory and associated game mechanics.
		The Magic axiom covers a number of new or revised topics: cantrips, wishes, rites, superstitions, knacks, hexes, curses, magical creatures, and more.
		Each of these articles covers one of these topics, and includes the pertinent magical theory, the game mechanics, and the axiom entries for the subject in question.
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	[No title] (Site not responding. Last check: 2007-10-13)
		Axiom of Choice, it tells you that it can be done, but it doesn't tell you which value to choose.
		Even with this approach, there may still be reason to prefer the iota-based axioms to the epsilon-ones, at least in some cases, because it's probably easier to mix sets and HOL in the iota world.
		What it says is you don't need the axiom of replacement to work for U. As you say, that means that U is contained in R1(omega+omega).
	ghilbert.org /choice.txt (11721 words)

	Induction, recursion, replacement and the ordinals (Site not responding. Last check: 2007-10-13)
		Section 9.5 of the book concludes with a sketch of how extensional well founded coalgebras may be used to characterise ordinal iteration of functors in an elementary way, and thereby formulate a version of the axiom-scheme of replacement.
		Mostowski's theorem and the rank for some of the notions of ordinal are formulated and proved without the axiom of replacement, but this seems to be unavoidable for the plump rank.
		Models of these axioms based on partial equivalence relations have received much attention, but there are also very simple sheaf models based on classical domain theory.
	www.cs.man.ac.uk /~pt/ordinals (1056 words)

	The axioms of ZF
		The axiom of replacement allows us to make a new set v from any statement in the language of ZF (with any fixed parameters) that defines y uniquely as a function of x and any set u.
		This is the axiom that defines sets of higher cardinality or at least seems to.
		The important thing to understand about the axioms is that they are comparatively simple precise rules for deducing new statements from existing ones.
	www.mtnmath.com /book/node53.html (456 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		Axiom 7) The Axiom of ReplacementGiven any set A and a function f defined on A, the image f(A) is a set.
		Axiom 8) The Axiom of InfinityThere exists a set A such that EMBED Equation.2 EMBED Equation.2 , and whenever a EMBED Equation.2 , it follows that a EMBED Equation.2 EMBED Equation.2 .
		Axiom 9) The Axiom of RegularityGiven any non-empty set A, there exists an a EMBED Equation.2 such that EMBED Equation.2 A = EMBED Equation.2 .
	faculty.kutztown.edu /mcloughl/255ax.doc (506 words)

	bnd - Metamath Proof Explorer (Site not responding. Last check: 2007-10-13)
		Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 3324), derived from the Collection Principle
		does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement.
		This theorem was proved from axioms: ax-1 4 ax-2 5
	us.metamath.org /mpegif/bnd.html (62 words)

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	Axiom scheme of replacement
		The general axiom scheme for building up complex sets like the ordinals is called replacement.
		These axioms could be defined by a single finite expression, but they are usually defined as an easily generated sequence.
		This axiom schema came about because previous attempts to axiomatize mathematics were too general and led to contradictions like the Barber Paradox
	www.mtnmath.com /whatth/node35.html (203 words)

	ru - Metamath Proof Explorer (Site not responding. Last check: 2007-10-13)
		In 1908 Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ssex 2518 asserting that
		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)

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