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en>NameloCmaS: /* Electrolytic failure from disuse */

2015-01-08T23:05:26Z

Electrolytic failure from disuse

← Older revision		Revision as of 01:05, 9 January 2015
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	~~Golda~~ is what~~'s created on my birth certification even though it~~ is ~~not~~ the ~~name on my beginning certification~~. ~~Doing ballet is something~~ she ~~would by no means give up~~. ~~Ohio~~ is ~~exactly where his home is and his family loves it. Credit authorising is how he tends to make cash~~.<br><br>my ~~web blog~~: [~~https~~://~~www~~.~~machlitim~~.~~org~~.il/~~subdomain~~/~~megila~~/~~end~~/~~node~~/~~12300 tarot card readings~~]		Andera is what you can contact her but she never really liked that title. To perform lacross is the thing I love most of all. For years she's been living in Kentucky but her spouse desires them to move. Invoicing is my profession.<br><br>Feel free to visit my homepage: real psychic ([http://hknews.classicmall.com.hk/groups/some-simple-tips-for-personal-development-progress/ http://hknews.classicmall.com.hk])

en>Materialscientist: Reverted edits by 202.5.157.183 (talk) to last version by ClueBot NG

2014-02-28T09:25:21Z

Reverted edits by 202.5.157.183 (talk) to last version by ClueBot NG

← Older revision		Revision as of 11:25, 28 February 2014
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	'~~''Set theory of the real line'''~~ is ~~an area of [[mathematics]] concerned with the application of [[set theory]] to aspects of~~ the ~~[[real numbers]]~~.		Golda is what's created on my birth certification even though it is not the name on my beginning certification. Doing ballet is something she would by no means give up. Ohio is exactly where his home is and his family loves it. Credit authorising is how he tends to make cash.<br><br>my web blog: [https://www.machlitim.org.il/subdomain/megila/end/node/12300 tarot card readings]

	~~For example, one knows that all countable sets of reals are [[Null set\|null]], i~~.~~e. have [[Lebesgue measure]] 0; one might therefore ask the least possible size of a set~~
	~~which~~ is ~~not Lebesgue null. This invariant~~ is ~~called the uniformity of the [[Ideal (set theory)\|ideal]] of null sets, denoted <math>non(\mathcal{N})</math>. There are many such [[cardinal invariant\|invariants]] associated with this~~ and ~~other ideals, e~~.~~g. the ideal of [[Meagre set\|meagre]] sets, plus more which do not have a characterisation in terms of ideals. If the [[continuum hypothesis]] (CH) holds, then all such invariants are equal~~ to ~~<math>\aleph_1</math>, the least uncountable [[Cardinal number\|cardinal]]~~. ~~For example, we know <math>non(\mathcal{N})~~<~~/math~~> ~~is uncountable, but being the size of some set of reals under CH it can be at most~~ <~~math~~>~~\aleph_1</math>.~~

	~~On the other hand, if one assumes~~ [~~[Martin's Axiom]] (MA) all common invariants are "big", that is equal to <math>\mathfrak{c}<~~/~~math>, the [[cardinality of the continuum]]. Martin's Axiom is consistent with <math>\mathfrak{c}>\aleph_1<~~/~~math>~~. In fact one should view Martin's Axiom as a [[forcing (mathematics)\|forcing]] axiom that negates the need to do specific forcings of a certain class (those satisfying the [[countable chain condition\|ccc]], since the consistency of MA with large continuum is proved by doing all such forcings (up to a certain size shown to be sufficient). Each invariant can be made large by some ccc forcing, thus each is big given MA.

	If one restricts to specific forcings, some invariants will become big while others remain small. Analysing these effects is the major work of the area, seeking to determine which inequalities between invariants are provable and which are inconsistent with ZFC. The inequalities among the ideals of [[Measure (mathematics)\|measure]] (null sets) and [[Category (mathematics)\|category]] (meagre sets) are captured in [[Cichon's diagram]]. ~~Seventeen models (forcing constructions) were produced during the 1980s, starting with work of Arnold Miller, to demonstrate that no other inequalities are provable~~. ~~These are analysed in detail in the book by Tomek Bartoszynski and Haim Judah, two of the eminent workers in the field.~~

	~~One curious result is that if you can cover the real line with <math>\kappa<~~/~~math> meagre sets (where <math>\aleph_1 \leq \kappa \leq \mathfrak{c}<~~/~~math>) then <math>non(\mathcal{N}) \geq \kappa</math>; conversely if you can cover the real line with <math>\kappa</math> null sets then the least non-meagre set has size at least <math>\kappa<~~/~~math>; both of these results follow from the existence of a decomposition of <math>\mathbb{R}<~~/~~math> as the union of a meagre set and a null set.~~

	~~One of the last great unsolved problems of the area was the consistency of~~

	~~:<math>\mathfrak{d} < \mathfrak{a},<~~/~~math>~~

	~~proved in 1998 by [[Saharon Shelah]].~~

	~~==See also==~~

	* [[Cichoń's diagram]]
	* [[Cardinal invariant]]

	~~==References==~~

	* [[Tomek Bartoszyński\|Bartoszynski, Tomek]] & Judah, Haim ''Set theory: On the structure of the real line'' A.. K. Peters Ltd. (1995). ISBN 1-56881-044-X
	* Miller, Arnold ''Some properties of measure and category'' Transactions of the American Mathematical Society, 266(1):93-114, (1981)

	~~[[Category:Set theory]~~]

en>ClueBot NG: Reverting possible vandalism by 74.70.226.199 to version by JaunJimenez. False positive? Report it. Thanks, ClueBot NG. (1682918) (Bot)

2014-02-03T21:19:11Z

Reverting possible vandalism by 74.70.226.199 to version by JaunJimenez. False positive? Report it. Thanks, ClueBot NG. (1682918) (Bot)

← Older revision		Revision as of 23:19, 3 February 2014
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	~~I would like~~ to ~~introduce myself~~ to ~~you~~, ~~I am Jayson Simcox~~ but ~~I don~~'~~t like when people use my full name. One~~ of the ~~things she enjoys most~~ is ~~canoeing and she~~'s ~~been~~ doing ~~it for fairly~~ a ~~whilst~~. ~~I am presently a journey agent~~. ~~Her family members life~~ in ~~Ohio but her spouse desires them~~ to ~~move~~.<br><br>~~Check out my web blog email psychic readings [[http:~~//~~brazil.amor~~-~~amore~~.~~com~~/~~irboothe http~~:~~//brazil~~.~~amor~~-~~amore.com/irboothe~~]]		'''Set theory of the real line''' is an area of [[mathematics]] concerned with the application of [[set theory]] to aspects of the [[real numbers]].

			For example, one knows that all countable sets of reals are [[Null set\|null]], i.e. have [[Lebesgue measure]] 0; one might therefore ask the least possible size of a set
			which is not Lebesgue null. This invariant is called the uniformity of the [[Ideal (set theory)\|ideal]] of null sets, denoted <math>non(\mathcal{N})</math>. There are many such [[cardinal invariant\|invariants]] associated with this and other ideals, e.g. the ideal of [[Meagre set\|meagre]] sets, plus more which do not have a characterisation in terms of ideals. If the [[continuum hypothesis]] (CH) holds, then all such invariants are equal to <math>\aleph_1</math>, the least uncountable [[Cardinal number\|cardinal]]. For example, we know <math>non(\mathcal{N})</math> is uncountable, but being the size of some set of reals under CH it can be at most <math>\aleph_1</math>.

			On the other hand, if one assumes [[Martin's Axiom]] (MA) all common invariants are "big", that is equal to <math>\mathfrak{c}</math>, the [[cardinality of the continuum]]. Martin's Axiom is consistent with <math>\mathfrak{c}>\aleph_1</math>. In fact one should view Martin's Axiom as a [[forcing (mathematics)\|forcing]] axiom that negates the need to do specific forcings of a certain class (those satisfying the [[countable chain condition\|ccc]], since the consistency of MA with large continuum is proved by doing all such forcings (up to a certain size shown to be sufficient). Each invariant can be made large by some ccc forcing, thus each is big given MA.

			If one restricts to specific forcings, some invariants will become big while others remain small. Analysing these effects is the major work of the area, seeking to determine which inequalities between invariants are provable and which are inconsistent with ZFC. The inequalities among the ideals of [[Measure (mathematics)\|measure]] (null sets) and [[Category (mathematics)\|category]] (meagre sets) are captured in [[Cichon's diagram]]. Seventeen models (forcing constructions) were produced during the 1980s, starting with work of Arnold Miller, to demonstrate that no other inequalities are provable. These are analysed in detail in the book by Tomek Bartoszynski and Haim Judah, two of the eminent workers in the field.

			One curious result is that if you can cover the real line with <math>\kappa</math> meagre sets (where <math>\aleph_1 \leq \kappa \leq \mathfrak{c}</math>) then <math>non(\mathcal{N}) \geq \kappa</math>; conversely if you can cover the real line with <math>\kappa</math> null sets then the least non-meagre set has size at least <math>\kappa</math>; both of these results follow from the existence of a decomposition of <math>\mathbb{R}</math> as the union of a meagre set and a null set.

			One of the last great unsolved problems of the area was the consistency of

			:<math>\mathfrak{d} < \mathfrak{a},</math>

			proved in 1998 by [[Saharon Shelah]].

			==See also==

			* [[Cichoń's diagram]]
			* [[Cardinal invariant]]

			==References==

			* [[Tomek Bartoszyński\|Bartoszynski, Tomek]] & Judah, Haim ''Set theory: On the structure of the real line'' A.. K. Peters Ltd. (1995). ISBN 1-56881-044-X
			* Miller, Arnold ''Some properties of measure and category'' Transactions of the American Mathematical Society, 266(1):93-114, (1981)

			[[Category:Set theory]]

en>Jim1138: Reverted edits by Salako1999 (talk) addition of unsourced content (HG)

2012-09-01T09:53:56Z

Reverted edits by Salako1999 (talk) addition of unsourced content (HG)

New page

I would like to introduce myself to you, I am Jayson Simcox but I don't like when people use my full name. One of the things she enjoys most is canoeing and she's been doing it for fairly a whilst. I am presently a journey agent. Her family members life in Ohio but her spouse desires them to move.<br><br>Check out my web blog email psychic readings [[http://brazil.amor-amore.com/irboothe http://brazil.amor-amore.com/irboothe]]