Axiom schema of specification - Revision history

en>R.e.b.: Undid revision 625154377 by R.e.b. (talk)

2014-09-11T22:19:27Z

Undid revision 625154377 by R.e.b. (talk)

← Older revision		Revision as of 00:19, 12 September 2014
Line 1:		Line 1:
	~~{{no footnotes\|date=March 2013}}~~
	In [[axiomatic set theory]], the '''axiom of empty set''' is an [[axiom]] of [[Kripke–Platek set theory]] and the variant of [[general set theory]] that Burgess (2005) calls "ST," and a demonstrable truth in [[Zermelo set theory]] and [[Zermelo–Fraenkel set theory]], with or without the [[axiom of choice]].

	~~== Formal statement ==~~
	~~In the [[formal language]] of the Zermelo–Fraenkel axioms, the axiom reads:~~
	~~:<math>\exist x\, \forall y\, \lnot (y \in x)</math>~~
	~~or in words:~~
	~~:[[Existential quantification\|There is]] a [[Set (mathematics)\|set]] such that no set is a member of it.~~

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	~~We can use the [[axiom of extensionality]] to show that there is only one empty set. Since it is unique we can name it. It is called the ''[~~[~~empty set]]'' (denoted by { } or ∅). The axiom, stated in natural language, is in essence:~~
	:~~''An empty set exists''~~.

	~~The axiom of empty set is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatisation of set theory~~.

	~~In some formulations of ZF, the axiom of empty set is actually repeated in the [[axiom of infinity~~]]. However, there are other formulations of that axiom that do not presuppose the existence of an empty set. The ZF axioms can also be written using a [[First-order logic#Non-logical symbols\|constant symbol]] representing the empty set; then the axiom of infinity uses this symbol without requiring it to be empty, while the axiom of empty set is needed to state that it is ~~in fact empty~~.

	~~Furthermore,~~ one ~~sometimes considers set theories in which there are no infinite sets, and then the axiom of empty set may still be required~~. ~~That said, any axiom of set theory or logic that implies the existence of any set will imply the existence of the empty set, if one has the~~ [~~[axiom schema of separation]]. This is true, since the empty set is a subset of any set consisting of those elements that satisfy a contradictory formula~~.

	~~In many formulations of first-order predicate logic, the existence of at least one object is always guaranteed~~. ~~If the axiomatization of set theory is formulated in such a [[logical system~~]] with the [[axiom schema of separation]] as axioms, and if the theory makes no distinction between sets and other kinds of objects (which holds for ZF, KP, and similar theories), then the existence of the empty set is a theorem.

	~~If separation is not postulated as an axiom schema,~~ but ~~derived as a theorem schema from the schema of replacement (as is sometimes done), the situation is more complicated, and depends on the exact formulation of the replacement schema~~. ~~The formulation used in the~~ [[axiom schema of replacement]] article only allows to construct the image ''F''[''a''] when ''a'' is contained in the domain of the class function ''F''; then the derivation of separation requires the axiom of empty set. On the other hand, the constraint of totality of ''F'' is often dropped from the replacement schema, in which case it implies the separation schema without using the axiom of empty set (or any other axiom for that matter).

	=~~= References ==~~
	*Burgess, John, 2005. '~~'Fixing Frege''. Princeton Univ. Press.~~
	*[[Paul Halmos]], ''Naive set theory''. Princeton, NJ: D. ~~Van Nostrand Company, 1960. Reprinted by Springer~~-~~Verlag, New York, 1974~~. ~~ISBN 0-387-90092-6 (Springer-Verlag edition)~~.
	*[[~~Thomas Jech\|Jech, Thomas]], 2003. ''Set Theory~~: ~~The Third Millennium Edition, Revised and Expanded''~~. ~~Springer. ISBN 3-540~~-~~44085-2~~.
	*[[Kenneth Kunen\|Kunen, Kenneth]], 1980. ~~''Set Theory: An Introduction to Independence Proofs''. Elsevier. ISBN 0-444-86839-9.~~

	~~[[Category:Axioms of set theory]]~~
	~~[[Category:Nothing]]~~

	~~[[de:Zermelo-Fraenkel-Mengenlehre#Die Axiome von ZF und ZFC]~~]

en>Njol: Re-added the arguments to the formula and changed the other occurrence of \phi to \varphi

2013-11-05T15:57:19Z

Re-added the arguments to the formula and changed the other occurrence of \phi to \varphi

← Older revision		Revision as of 17:57, 5 November 2013
Line 1:		Line 1:
			{{no footnotes\|date=March 2013}}
			In [[axiomatic set theory]], the '''axiom of empty set''' is an [[axiom]] of [[Kripke–Platek set theory]] and the variant of [[general set theory]] that Burgess (2005) calls "ST," and a demonstrable truth in [[Zermelo set theory]] and [[Zermelo–Fraenkel set theory]], with or without the [[axiom of choice]].

			== Formal statement ==
			In the [[formal language]] of the Zermelo–Fraenkel axioms, the axiom reads:
			:<math>\exist x\, \forall y\, \lnot (y \in x)</math>
			or in words:
			:[[Existential quantification\|There is]] a [[Set (mathematics)\|set]] such that no set is a member of it.

	~~Wordpress has become increasingly popular over~~ the ~~years and is now being widely adopted by many webmasters. It has come~~ to ~~be known as an efficient blogging tool.<br><br>Wordpress has become increasingly popular over the years and~~ is ~~now being widely adopted by many webmasters~~. ~~It has come to be known as an efficient blogging tool. In addition,~~ it is ~~now being used to build websites since~~ it is ~~a wonderful content management system~~. ~~When building websites~~, ~~you should explore this new concept of psd to wordpress conversion~~, ~~since it will give you a website that~~ is ~~much easier to maintain~~. ~~Instead~~ of ~~just thinking about psd to html and xhtml conversions~~, ~~you should consider switching your web design to wordpress~~ and ~~you will see amazing results.<br><br>Wordpress has become very popular with today�s web designers because~~ it ~~has amazing features and great functionality support~~. ~~Wordpress~~ is ~~constantly updated and every new version comes with additional interesting features~~. ~~You should make a point~~ of ~~understanding~~ the ~~reasons~~ of ~~converting psd to wordpress~~. ~~All files and documents that are made~~ using ~~Adobe Photoshop~~ [~~https://www.flickr.com/search/?q=software software~~] ~~can be saved using~~ the ~~format .psd. Such files and documents can~~ be ~~exported to and from several other programs to create designs and special effects~~, ~~including psd to wordpress theme for [http://Websites.org websites] and other media. It~~ is ~~such qualities that have made psd~~ to ~~wordpress conversion so popular.<br><br>There are several professional companies~~ that ~~offer valuable services~~ in ~~psd to wordpress theme coding~~. ~~All you have to do is to send them your psd files including all the creative work~~ and ~~they will do~~ the ~~technical conversion for you~~. ~~Using these professional companies~~ will ~~help you come up with a website that is attractive to~~ the ~~eye and is very user-friendly. Instead~~ of ~~just using~~ one of ~~the many psd to wordpress templates found online~~, the ~~professional company will come up with~~ a ~~unique design for you~~ that ~~reflects the true identity of your website and suits your needs perfectly~~.~~<br><br>There are~~ many ~~advantages to be gained with psd to wordpress conversion. Wordpress has templates that contain widgets~~, ~~which can be rearranged without~~ the ~~need to edit PHP or HTML code, and also inbuilt friendly features for search engines~~. It is ~~also~~ a ~~comfortable executing~~ system ~~that is light in weight. With a professional company providing~~ the ~~best psd to wordpress service~~, ~~you will get to enjoy services like cross browser compatibility~~, ~~a completely dynamic wordpress CMS or blog~~, ~~SEO friendly coding~~, ~~widget ready side bar and efficient technical support. You should identify~~ a ~~company that offers all these services as one package~~.~~<br><br>~~If ~~you are looking for~~ a ~~professional website or blog that will be distinct~~ from the ~~rest~~ of the ~~competition~~, ~~then wordpress theme coding is~~ the ~~best option to take~~. The ~~latest versions~~ of ~~wordpress have made psd~~ to ~~wordpress conversion more competitive and powerful~~. ~~Wordpress conversions and coding are very important~~ for ~~a website to get good visibility in search engines and also to enhance its smooth navigation~~ by ~~search engine bots~~. ~~Professional web designers are now able to use a customized strategic approach to create unique websites suited for different purposes~~.~~<br><br>If you have any questions regarding where and exactly how to use~~ [~~http~~:~~//www~~.~~yourdomain~~.~~com domain~~], ~~you can contact us at our own site~~.		== Interpretation ==
			We can use the [[axiom of extensionality]] to show that there is only one empty set. Since it is unique we can name it. It is called the ''[[empty set]]'' (denoted by { } or ∅). The axiom, stated in natural language, is in essence:
			:''An empty set exists''.

			The axiom of empty set is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatisation of set theory.

			In some formulations of ZF, the axiom of empty set is actually repeated in the [[axiom of infinity]]. However, there are other formulations of that axiom that do not presuppose the existence of an empty set. The ZF axioms can also be written using a [[First-order logic#Non-logical symbols\|constant symbol]] representing the empty set; then the axiom of infinity uses this symbol without requiring it to be empty, while the axiom of empty set is needed to state that it is in fact empty.

			Furthermore, one sometimes considers set theories in which there are no infinite sets, and then the axiom of empty set may still be required. That said, any axiom of set theory or logic that implies the existence of any set will imply the existence of the empty set, if one has the [[axiom schema of separation]]. This is true, since the empty set is a subset of any set consisting of those elements that satisfy a contradictory formula.

			In many formulations of first-order predicate logic, the existence of at least one object is always guaranteed. If the axiomatization of set theory is formulated in such a [[logical system]] with the [[axiom schema of separation]] as axioms, and if the theory makes no distinction between sets and other kinds of objects (which holds for ZF, KP, and similar theories), then the existence of the empty set is a theorem.

			If separation is not postulated as an axiom schema, but derived as a theorem schema from the schema of replacement (as is sometimes done), the situation is more complicated, and depends on the exact formulation of the replacement schema. The formulation used in the [[axiom schema of replacement]] article only allows to construct the image ''F''[''a''] when ''a'' is contained in the domain of the class function ''F''; then the derivation of separation requires the axiom of empty set. On the other hand, the constraint of totality of ''F'' is often dropped from the replacement schema, in which case it implies the separation schema without using the axiom of empty set (or any other axiom for that matter).

			== References ==
			*Burgess, John, 2005. ''Fixing Frege''. Princeton Univ. Press.
			*[[Paul Halmos]], ''Naive set theory''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
			*[[Thomas Jech\|Jech, Thomas]], 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. ISBN 3-540-44085-2.
			*[[Kenneth Kunen\|Kunen, Kenneth]], 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier. ISBN 0-444-86839-9.

			[[Category:Axioms of set theory]]
			[[Category:Nothing]]

			[[de:Zermelo-Fraenkel-Mengenlehre#Die Axiome von ZF und ZFC]]

en>Gabbe: changed hyphens to dashes per WP:ENDASH

2012-08-21T20:53:40Z

changed hyphens to dashes per WP:ENDASH

New page

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