en>Addbot: Bot: Removing Orphan Tag (Nolonger an Orphan) (Report Errors)

2013-01-09T00:23:21Z

Bot: Removing Orphan Tag (Nolonger an Orphan) (Report Errors)

← Older revision		Revision as of 02:23, 9 January 2013
Line 1:		Line 1:
	~~Friends contact her Claude Gulledge~~. ~~Alabama~~ is ~~where he~~ and ~~his wife live and he has every thing~~ that ~~he needs there. What she loves performing~~ is ~~taking part~~ in ~~croquet and she~~ is ~~attempting~~ to ~~make it a profession~~. ~~Bookkeeping is what he~~ does.<br><br>~~Also visit my web blog - car warranty~~ ([~~http~~://~~www~~.~~miketoerner~~.~~com~~/~~ActivityFeed~~/~~MyProfile~~/~~tabid~~/60/~~userId~~/~~35329~~/~~Default~~.~~aspx Going Listed here~~])		{{more footnotes\|date=September 2012}}
			'''Ackermann set theory''' is a version of [[axiomatic set theory]] proposed by [[Wilhelm Ackermann]] in 1956.

			== The language==
			Ackermann set theory is formulated in [[first-order logic]]. The language <math>L_A</math> consists of one binary relation <math>\in</math> and one constant <math>V</math> (Ackermann used a predicate <math>M</math> instead). We will write <math>x \in y</math> for <math>\in(x,y)</math>. The intended interpretation of <math>x \in y</math> is that the object <math>x</math> is in the class <math>y</math>. The intended interpretation of <math>V</math> is the class of all sets.

			== The axioms==
			The axioms of Ackermann set theory, collectively referred to as A, consists of the [[Universal quantifier\|universal closure]] of the following formulas in the language <math>L_A</math>

			1) [[Axiom of extensionality]]:
			:<math>\forall x \forall y ( \forall z (z \in x \leftrightarrow z \in y)
			\rightarrow x = y).</math>

			2) [[Axiom of separation\|Class construction axiom schema]]: Let <math>F(y,z_1, \dots, z_n)</math> be any formula which does not contain the variable <math>x</math> free.
			:<math>\forall y (F(y, z_1, \dots, z_n) \rightarrow y \in V) \rightarrow \exists x \forall y (y \in x \leftrightarrow F(y,z_1, \dots, z_n) )</math>

			3) Reflection axiom schema: Let <math>F(y,z_1, \dots, z_n)</math> be any formula which does not contain the constant symbol <math>V</math> or the variable <math>x</math> free. If <math> z_1, \dots, z_n \in V</math> then
			:<math>\forall y (F(y, z_1, \dots, z_n) \rightarrow y \in V) \rightarrow \exists x (x \in V \land \forall y (y \in x \leftrightarrow F(y, z_1, \dots, z_n))).</math>

			4) Completeness axioms for <math>V</math>
			:<math>x \in y \land y \in V \rightarrow x \in V</math>
			:<math>x \subseteq y \land y \in V \rightarrow x \in V.</math>

			5) [[Axiom of regularity\|Axiom of regularity for sets]]:
			:<math>x \in V \land \exists y ( y \in x) \rightarrow \exists y ( y \in x \land \lnot \exists z (z \in y \land z \in x)).</math>

			==Relation to Zermelo–Fraenkel set theory==
			Let <math>F</math> be a [[First-order logic\|first-order formula]] in the language <math>L_\in = \{\in\}</math> (so <math>F</math> does not contain the constant <math>V</math>). Define the "restriction of <math>F</math> to the universe of sets" (denoted <math>F^V</math>) to be the formula which is obtained by recursively replacing all [[sub-formulas]] of <math>F</math> of the form <math>\forall x G(x,y_1\dots, y_n)</math> with <math>\forall x (x \in V \rightarrow G(x,y_1\dots, y_n))</math> and all sub-formulas of the form <math>\exists x G(x,y_1\dots, y_n)</math> with <math>\exists x (x \in V \land G(x,y_1\dots, y_n))</math>.

			In 1959 [[Azriel Levy]] proved that if <math>F</math> is a formula of <math>L_\in</math> and A proves <math>F^V</math>, then [[Zermelo–Fraenkel set theory\|ZF]] proves <math>F</math>

			In 1970 [[William Reinhardt (mathematician)\|William Reinhardt]] proved that if <math>F</math> is a formula of <math>L_\in</math> and ZF proves <math>F</math>, then A proves <math>F^V</math>.

			== Ackermann set theory and Category theory ==
			The most remarkable feature of Ackermann set theory is that, unlike [[Von Neumann–Bernays–Gödel set theory]], a [[proper class]] can be an element of another proper class (see Fraenkel, Bar-Hillel, Levy(1973), p. 153).

			An extension (named ARC) of Ackermann set theory was developed by F.A. Muller(2001), who stated that ARC "founds Cantorian set-theory as well as category-theory and therefore can pass as a founding theory of the whole of mathematics".

			== See also ==
			* [[Zermelo set theory]]

			==References==
			* [[Wilhelm Ackermann\|Ackermann, Wilhelm]] "Zur Axiomatik der Mengenlehre" in Mathematische Annalen, 1956, Vol. 131, pp. 336--345''.
			* [[Azriel Levy\|Levy, Azriel]], "On Ackermann's set theory" ''Journal of Symbolic Logic Vol. 24, 1959 154--166''
			* [[William Reinhardt\|Reinhardt, William]], "Ackermann's set theory equals ZF" ''Annals of Mathematical Logic Vol. 2, 1970 no. 2, 189--249''
			* A.A.Fraenkel, Y. Bar-Hillel, A.Levy, 1973. ''Foundations of Set Theory'', second edition, North-Holand, 1973.
			* F.A. Muller, "Sets, Classes, and Categories" ''British Journal for the Philosophy of Science 52 (2001) 539-573''.


			[[Category:Systems of set theory]]

en>TutterMouse: general fixes, typos fixed: Krakow → Kraków using AWB

2012-01-15T02:50:43Z

general fixes, typos fixed: Krakow → Kraków using AWB

New page

Friends contact her Claude Gulledge. Alabama is where he and his wife live and he has every thing that he needs there. What she loves performing is taking part in croquet and she is attempting to make it a profession. Bookkeeping is what he does.<br><br>Also visit my web blog - car warranty ([http://www.miketoerner.com/ActivityFeed/MyProfile/tabid/60/userId/35329/Default.aspx Going Listed here])

Current differencing transconductance amplifier - Revision history

en>Addbot: Bot: Removing Orphan Tag (Nolonger an Orphan) (Report Errors)

en>TutterMouse: general fixes, typos fixed: Krakow → Kraków using AWB