Axiom schema of replacement - Revision history

en>Moosbrugger: Minor changes: specification to the more common separation; irrelevant allusion to Feferman cut; added credit to Mirimanoff.

2014-11-15T14:13:26Z

Minor changes: specification to the more common separation; irrelevant allusion to Feferman cut; added credit to Mirimanoff.

← Older revision		Revision as of 16:13, 15 November 2014
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	~~{{no footnotes\|date=March 2013}}~~		I'm Adolfo (27) from Harelbeke, Belgium. <br>I'm learning Italian literature at a local college and I'm just about to graduate.<br>I have a part time job in a college.<br>[http://bandirmasebzecilerodasi.com/index.php/ziyaretci-defteri?5 legal transcription company]
	~~In [[mathematics]], the~~ '~~''axiom of power set''' is one of the [[Zermelo–Fraenkel axioms]] of [[axiomatic set theory]].~~

	~~In the [[formal language]] of the Zermelo–Fraenkel axioms, the axiom reads:~~

	~~:<math>\forall A \, \exists P \, \forall B \, [B \in P \iff \forall C \,~~ (~~C \in B \Rightarrow C \in A~~)~~]</math>~~

	~~where ''P'' stands for the power set of ''A''~~, ~~<math>\mathcal{P}(A)</math>~~. ~~In English, this says:~~

	~~:[[Given any]] [[Set (mathematics)\|set]] ''A'', [[Existential quantification\|there is]] a set <math>\mathcal{P}(A)~~<~~/math~~> ~~such that, given any set ''B~~'~~', ''B'' is~~ a ~~member of <math>\mathcal{P}(A)</math> [[if~~ and ~~only if]]~~ '~~'B'' is a [[subset]] of ''A''. (Subset is not used in the formal definition above because the axiom of power set is an axiom that may need to be stated without reference~~ to ~~the concept of subset.)~~

	~~By the [[axiom of extensionality]] this set is unique~~.
	~~We call the set~~ <~~math~~>~~\mathcal{P}(A)</math> the ''[[power set]]'' of ''A''. Thus, the essence of the axiom is that every set has~~ a ~~power set.~~

	~~The axiom of power set appears~~ in ~~most axiomatizations of set theory. It is generally considered uncontroversial, although [[constructive set theory]] prefers~~ a ~~weaker version to resolve concerns about predicativity~~.

	~~== Consequences ==~~

	~~The Power Set Axiom allows a simple definition of the [[Cartesian product]] of two sets~~ <~~math~~>~~X</math> and <math>Y</math>:~~

	~~:<math> X \times Y = \{ (x, y)~~ : ~~x \in X \land y \in Y \}. <~~/~~math>~~

	~~Notice that~~
	~~:<math>x, y \in X \cup Y </math>~~
	~~:<math>\{ x \}, \{ x, y \} \in \mathcal{P}(X \cup Y) </math>~~
	~~:<math>(x, y) = \{ \{ x \}, \{ x, y \} \} \in \mathcal{P}(\mathcal{P}(X \cup Y)) <~~/~~math>~~

	~~and thus the Cartesian product is a set since~~

	~~:<math> X \times Y \subseteq \mathcal{P}(\mathcal{P}(X \cup Y))~~. </~~math>~~

	~~One may define the Cartesian product of any [[finite set\|finite]] [[class (set theory)\|collection]] of sets recursively:~~

	~~:<math> X_1 \times \cdots \times X_n = (X_1 \times \cdots \times X_{n-1}) \times X_n~~. </~~math>~~

	~~Note that the existence of the Cartesian product can be proved without using the power set axiom, as in the case of the [[Kripke–Platek set theory]].~~

	~~== References ==~~
	*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).
	*Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. ISBN 3-540-44085-2.
	*Kunen, Kenneth, 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier. ISBN 0-444-86839-9.

	~~{{PlanetMath attribution\|id=4399\|title=Axiom of power set}}~~

	~~{{Set theory}}~~

	~~[[Category:Axioms of set theory]]~~

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

en>Martinkunev: formatting

2014-01-05T23:44:06Z

formatting

← Older revision		Revision as of 01:44, 6 January 2014
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	~~The author~~ is ~~known by~~ the ~~name~~ of ~~Figures Wunder~~. ~~Years~~ in the ~~past we moved to Puerto Rico and my family enjoys it~~. ~~For many years I~~'~~ve been operating as~~ a ~~payroll clerk. It~~'~~s not~~ a ~~common thing but what she likes performing~~ is ~~base jumping and now she~~ is ~~trying~~ to ~~earn cash with it~~.<br><br>~~Feel free to surf to my web-site~~ ... ~~healthy food delivery~~ (~~[http~~://~~tomjones1190~~.~~Livejournal~~.~~com~~/ ~~look what i found~~])		{{no footnotes\|date=March 2013}}
			In [[mathematics]], the '''axiom of power set''' is one of the [[Zermelo–Fraenkel axioms]] of [[axiomatic set theory]].

			In the [[formal language]] of the Zermelo–Fraenkel axioms, the axiom reads:

			:<math>\forall A \, \exists P \, \forall B \, [B \in P \iff \forall C \, (C \in B \Rightarrow C \in A)]</math>

			where ''P'' stands for the power set of ''A'', <math>\mathcal{P}(A)</math>. In English, this says:

			:[[Given any]] [[Set (mathematics)\|set]] ''A'', [[Existential quantification\|there is]] a set <math>\mathcal{P}(A)</math> such that, given any set ''B'', ''B'' is a member of <math>\mathcal{P}(A)</math> [[if and only if]] ''B'' is a [[subset]] of ''A''. (Subset is not used in the formal definition above because the axiom of power set is an axiom that may need to be stated without reference to the concept of subset.)

			By the [[axiom of extensionality]] this set is unique.
			We call the set <math>\mathcal{P}(A)</math> the ''[[power set]]'' of ''A''. Thus, the essence of the axiom is that every set has a power set.

			The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although [[constructive set theory]] prefers a weaker version to resolve concerns about predicativity.

			== Consequences ==

			The Power Set Axiom allows a simple definition of the [[Cartesian product]] of two sets <math>X</math> and <math>Y</math>:

			:<math> X \times Y = \{ (x, y) : x \in X \land y \in Y \}. </math>

			Notice that
			:<math>x, y \in X \cup Y </math>
			:<math>\{ x \}, \{ x, y \} \in \mathcal{P}(X \cup Y) </math>
			:<math>(x, y) = \{ \{ x \}, \{ x, y \} \} \in \mathcal{P}(\mathcal{P}(X \cup Y)) </math>

			and thus the Cartesian product is a set since

			:<math> X \times Y \subseteq \mathcal{P}(\mathcal{P}(X \cup Y)). </math>

			One may define the Cartesian product of any [[finite set\|finite]] [[class (set theory)\|collection]] of sets recursively:

			:<math> X_1 \times \cdots \times X_n = (X_1 \times \cdots \times X_{n-1}) \times X_n. </math>

			Note that the existence of the Cartesian product can be proved without using the power set axiom, as in the case of the [[Kripke–Platek set theory]].

			== References ==
			*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).
			*Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. ISBN 3-540-44085-2.
			*Kunen, Kenneth, 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier. ISBN 0-444-86839-9.

			{{PlanetMath attribution\|id=4399\|title=Axiom of power set}}

			{{Set theory}}

			[[Category:Axioms of set theory]]

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

160.45.152.6: There is a single German Wikpedia article (Ersetzungsaxiom).

2012-08-10T14:21:54Z

There is a single German Wikpedia article (Ersetzungsaxiom).

New page

The author is known by the name of Figures Wunder. Years in the past we moved to Puerto Rico and my family enjoys it. For many years I've been operating as a payroll clerk. It's not a common thing but what she likes performing is base jumping and now she is trying to earn cash with it.<br><br>Feel free to surf to my web-site ... healthy food delivery ([http://tomjones1190.Livejournal.com/ look what i found])