Distribution (differential geometry) - Revision history

en>Policron: Added an external link (EOM).

2014-09-27T23:20:41Z

Added an external link (EOM).

← Older revision		Revision as of 01:20, 28 September 2014
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	~~In [[mathematics]]~~, ~~specifically in [[axiomatic set theory]], a '''Hartogs number''' is a particular kind of [[cardinal number]]~~. ~~It was shown by [[Friedrich Hartogs]] in 1915, from [[Zermelo-Fraenkel set theory\|ZF]] alone (that~~ is~~, without using the [[axiom of choice]]), that there is a least [[well-ordered]] [[cardinal number\|cardinal]] greater than a given well-ordered cardinal~~.		Hi there, I am Andrew Berryhill. To climb is some thing she would never give up. North Carolina is exactly where we've been living for many years and will never move. Credit authorising is how he tends to make cash.<br><br>Also visit my weblog - psychic phone ([http://appin.co.kr/board_Zqtv22/688025 appin.co.kr])

	~~To define the Hartogs number of a set it~~ is not in fact necessary that the set be well-orderable: If ''X'' is any set, then the Hartogs number of ''X'' is the least [[ordinal number\|ordinal]] α such that there is no [[Injective function\|injection]] from α into ''X''. ~~If ''X'' cannot be well-ordered, then we can no longer say that this α~~ is ~~the least well-ordered cardinal ''greater'' than the cardinality of ''X'', but it remains the least well-ordered cardinal ''not less than or equal~~ to~~'' the cardinality of ''X''. The [[map (mathematics)\|map]] taking ''X'' to α is sometimes called '''Hartogs' function'''.~~

	~~==Proof==~~
	~~Given some basic theorems of set theory, the proof is simple~~. ~~Let~~ <~~math~~>~~\alpha = \{\beta \in \textrm{Ord}\| \exists i: \beta \hookrightarrow X\}~~<~~/math~~>~~. First, we verify that α is a set.~~
	~~#''X'' × ''X'' is a set, as can be seen in [[axiom of power set#Consequences\|axiom of power set]].~~
	~~# The [[power set]] of ''X'' × ''X'' is a set, by the [[axiom of power set]].~~
	~~# The class ''W'' of all [[reflexive relation\|reflexive]] well~~-~~orderings of subsets of ''X'' is a definable subclass of the preceding set, so it is a set by the [[axiom schema of separation]].~~
	~~# The class of all [[order type]]s of well-orderings in ''W'' is a set by the [[axiom schema of replacement]], as~~
	~~#::([[Domain (mathematics)\|Domain]](''w''), ''w'') <math>\cong</math>~~ (~~β, ≤)~~
	~~#:can be described by a simple formula.~~

	~~But this last set is exactly α.~~

	Now because a [[transitive set]] of ordinals is again an ordinal, α is an ordinal. Furthermore, if there were an injection from α into ''X'', then we would get the contradiction that α ∈ α. It is claimed that α is the least such ordinal with no injection into ''X''. Given β < α, β ∈ α so there is an injection from β into ''X''.

	~~==References==~~
	*{{Cite journal
	~~\| last = Hartogs~~
	~~\| first = Fritz~~
	~~\| author-link =~~
	~~\| title = Über das Problem der Wohlordnung~~
	~~\| journal =~~ [~~[Mathematische Annalen]]~~
	~~\| language = [[German language\|German]]~~
	~~\| volume = 76~~
	~~\| pages =438–443~~
	~~\| year = 1915~~
	~~\| url =~~ http://~~www.digizeitschriften.de/dms/img/?PPN=GDZPPN002266105~~
	~~\| doi = 10.1007/BF01458215~~
	~~\| id =~~
	~~\| jfm = 45.0125.01~~
	~~\| issue = 4~~
	~~\| postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a "~~.~~", as necessary~~. ~~-->{{inconsistent citations}}~~
	~~}}. Available at the [http:~~//~~www~~.~~digizeitschriften~~.~~de/ DigiZeitschriften~~].
	* {{cite book\|authorlink=Thomas Jech\|author=Jech, Thomas\|title=Set theory, third millennium edition (revised and expanded)~~\|publisher=Springer\|year=2002\|isbn=3-540-44085-2}}~~
	* {{cite web \| title=Axiomatic set theory \| work=Course Notes \| author=Charles Morgan \| publisher=University of Bristol \| url=http://www.ucl.ac.uk/~ucahcjm/ast/ast_notes_4.pdf \| accessdate =2010-04-10 }}

	~~[[Category:Set theory]]~~
	~~[[Category:Cardinal numbers]]~~

	~~{{settheory-stub}}~~

en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q564835

2013-03-14T19:22:07Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q564835

← Older revision		Revision as of 21:22, 14 March 2013
Line 1:		Line 1:
	~~Marvella~~ is ~~what you~~ can ~~call her~~ but it's not the ~~most~~ [~~http~~:/~~/www~~.~~Amazon~~.~~com/Chlamydia-Female-Instant-Result-Clamidia/dp/B00B19DVAK feminine title]~~ ~~at home~~ [~~http://www~~.~~healthboards~~.~~com/boards/herpes/784722~~-~~herpes-anus~~.~~html std testing~~] ~~out std testing at home~~ [~~http~~:/~~/www~~.~~animecontent~~.~~com/blog/349119 over the counter std test~~] there. ~~He used to be unemployed but now he~~ is ~~a meter reader~~. ~~To collect coins~~ is ~~what his family and him enjoy~~. ~~California~~ [http://www.~~karachicattleexpo~~.~~com~~/~~blog~~/~~56 http:~~//~~www~~.~~karachicattleexpo.com/blog/56] is where I've usually been residing and home std test kit I adore each working day living right here~~.<~~br><br>My web~~-~~site~~ ... [http://www.xn--~~80aeerfhjjg1bv8c~~.~~xn--p1ai~~/~~%D0%A1%D0%BE%D0%B4%D0%B5%D1%80%D0%B6%D0%B0%D0%BD%D0%B8%D0%B5~~/~~cures~~-~~yeast~~-~~infection~~-~~tips-use-now home std test kit]~~		In [[mathematics]], specifically in [[axiomatic set theory]], a '''Hartogs number''' is a particular kind of [[cardinal number]]. It was shown by [[Friedrich Hartogs]] in 1915, from [[Zermelo-Fraenkel set theory\|ZF]] alone (that is, without using the [[axiom of choice]]), that there is a least [[well-ordered]] [[cardinal number\|cardinal]] greater than a given well-ordered cardinal.

			To define the Hartogs number of a set it is not in fact necessary that the set be well-orderable: If ''X'' is any set, then the Hartogs number of ''X'' is the least [[ordinal number\|ordinal]] α such that there is no [[Injective function\|injection]] from α into ''X''. If ''X'' cannot be well-ordered, then we can no longer say that this α is the least well-ordered cardinal ''greater'' than the cardinality of ''X'', but it remains the least well-ordered cardinal ''not less than or equal to'' the cardinality of ''X''. The [[map (mathematics)\|map]] taking ''X'' to α is sometimes called '''Hartogs' function'''.

			==Proof==
			Given some basic theorems of set theory, the proof is simple. Let <math>\alpha = \{\beta \in \textrm{Ord}\| \exists i: \beta \hookrightarrow X\}</math>. First, we verify that α is a set.
			#''X'' × ''X'' is a set, as can be seen in [[axiom of power set#Consequences\|axiom of power set]].
			# The [[power set]] of ''X'' × ''X'' is a set, by the [[axiom of power set]].
			# The class ''W'' of all [[reflexive relation\|reflexive]] well-orderings of subsets of ''X'' is a definable subclass of the preceding set, so it is a set by the [[axiom schema of separation]].
			# The class of all [[order type]]s of well-orderings in ''W'' is a set by the [[axiom schema of replacement]], as
			#::([[Domain (mathematics)\|Domain]](''w''), ''w'') <math>\cong</math> (β, ≤)
			#:can be described by a simple formula.

			But this last set is exactly α.

			Now because a [[transitive set]] of ordinals is again an ordinal, α is an ordinal. Furthermore, if there were an injection from α into ''X'', then we would get the contradiction that α ∈ α. It is claimed that α is the least such ordinal with no injection into ''X''. Given β < α, β ∈ α so there is an injection from β into ''X''.

			==References==
			*{{Cite journal
			\| last = Hartogs
			\| first = Fritz
			\| author-link =
			\| title = Über das Problem der Wohlordnung
			\| journal = [[Mathematische Annalen]]
			\| language = [[German language\|German]]
			\| volume = 76
			\| pages =438–443
			\| year = 1915
			\| url = http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002266105
			\| doi = 10.1007/BF01458215
			\| id =
			\| jfm = 45.0125.01
			\| issue = 4
			\| postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}
			}}. Available at the [http://www.digizeitschriften.de/ DigiZeitschriften].
			* {{cite book\|authorlink=Thomas Jech\|author=Jech, Thomas\|title=Set theory, third millennium edition (revised and expanded)\|publisher=Springer\|year=2002\|isbn=3-540-44085-2}}
			* {{cite web \| title=Axiomatic set theory \| work=Course Notes \| author=Charles Morgan \| publisher=University of Bristol \| url=http://www.ucl.ac.uk/~ucahcjm/ast/ast_notes_4.pdf \| accessdate =2010-04-10 }}

			[[Category:Set theory]]
			[[Category:Cardinal numbers]]

			{{settheory-stub}}

en>777sms at 03:30, 14 April 2012

2012-04-14T03:30:55Z

New page

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← Older revision		Revision as of 01:20, 28 September 2014
Line 1:		Line 1:
	~~In [[mathematics]]~~, ~~specifically in [[axiomatic set theory]], a '''Hartogs number''' is a particular kind of [[cardinal number]]~~. ~~It was shown by [[Friedrich Hartogs]] in 1915, from [[Zermelo-Fraenkel set theory\|ZF]] alone (that~~ is~~, without using the [[axiom of choice]]), that there is a least [[well-ordered]] [[cardinal number\|cardinal]] greater than a given well-ordered cardinal~~.		Hi there, I am Andrew Berryhill. To climb is some thing she would never give up. North Carolina is exactly where we've been living for many years and will never move. Credit authorising is how he tends to make cash.<br><br>Also visit my weblog - psychic phone ([http://appin.co.kr/board_Zqtv22/688025 appin.co.kr])

	~~To define the Hartogs number of a set it~~ is not in fact necessary that the set be well-orderable: If ''X'' is any set, then the Hartogs number of ''X'' is the least [[ordinal number\|ordinal]] α such that there is no [[Injective function\|injection]] from α into ''X''. ~~If ''X'' cannot be well-ordered, then we can no longer say that this α~~ is ~~the least well-ordered cardinal ''greater'' than the cardinality of ''X'', but it remains the least well-ordered cardinal ''not less than or equal~~ to~~'' the cardinality of ''X''. The [[map (mathematics)\|map]] taking ''X'' to α is sometimes called '''Hartogs' function'''.~~

	~~==Proof==~~
	~~Given some basic theorems of set theory, the proof is simple~~. ~~Let~~ <~~math~~>~~\alpha = \{\beta \in \textrm{Ord}\| \exists i: \beta \hookrightarrow X\}~~<~~/math~~>~~. First, we verify that α is a set.~~
	~~#''X'' × ''X'' is a set, as can be seen in [[axiom of power set#Consequences\|axiom of power set]].~~
	~~# The [[power set]] of ''X'' × ''X'' is a set, by the [[axiom of power set]].~~
	~~# The class ''W'' of all [[reflexive relation\|reflexive]] well~~-~~orderings of subsets of ''X'' is a definable subclass of the preceding set, so it is a set by the [[axiom schema of separation]].~~
	~~# The class of all [[order type]]s of well-orderings in ''W'' is a set by the [[axiom schema of replacement]], as~~
	~~#::([[Domain (mathematics)\|Domain]](''w''), ''w'') <math>\cong</math>~~ (~~β, ≤)~~
	~~#:can be described by a simple formula.~~

	~~But this last set is exactly α.~~

	Now because a [[transitive set]] of ordinals is again an ordinal, α is an ordinal. Furthermore, if there were an injection from α into ''X'', then we would get the contradiction that α ∈ α. It is claimed that α is the least such ordinal with no injection into ''X''. Given β < α, β ∈ α so there is an injection from β into ''X''.

	~~==References==~~
	*{{Cite journal
	~~\| last = Hartogs~~
	~~\| first = Fritz~~
	~~\| author-link =~~
	~~\| title = Über das Problem der Wohlordnung~~
	~~\| journal =~~ [~~[Mathematische Annalen]]~~
	~~\| language = [[German language\|German]]~~
	~~\| volume = 76~~
	~~\| pages =438–443~~
	~~\| year = 1915~~
	~~\| url =~~ http://~~www.digizeitschriften.de/dms/img/?PPN=GDZPPN002266105~~
	~~\| doi = 10.1007/BF01458215~~
	~~\| id =~~
	~~\| jfm = 45.0125.01~~
	~~\| issue = 4~~
	~~\| postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a "~~.~~", as necessary~~. ~~-->{{inconsistent citations}}~~
	~~}}. Available at the [http:~~//~~www~~.~~digizeitschriften~~.~~de/ DigiZeitschriften~~].
	* {{cite book\|authorlink=Thomas Jech\|author=Jech, Thomas\|title=Set theory, third millennium edition (revised and expanded)~~\|publisher=Springer\|year=2002\|isbn=3-540-44085-2}}~~
	* {{cite web \| title=Axiomatic set theory \| work=Course Notes \| author=Charles Morgan \| publisher=University of Bristol \| url=http://www.ucl.ac.uk/~ucahcjm/ast/ast_notes_4.pdf \| accessdate =2010-04-10 }}

	~~[[Category:Set theory]]~~
	~~[[Category:Cardinal numbers]]~~

	~~{{settheory-stub}}~~