Definable real number - Revision history

en>Melchoir: needs more scary tags, see talk page as well

2014-07-23T03:13:35Z

needs more scary tags, see talk page as well

← Older revision		Revision as of 05:13, 23 July 2014
Line 1:		Line 1:
	~~A [[real number]] ''a'' is '''first-order definable in the language of set theory, without parameters''', if there is a formula ''φ'' in the language of [[set theory]]~~, ~~with~~ one ~~[[free variable]], such that ''a'' is the unique real number such that ''φ(a)'' holds in the standard model~~ of ~~set theory (see Kunen 1980:153).~~		If you were young, maybe you were one of those fortunate persons that didnt need fat loss aid. Maybe we were chasing kids, working at an outside job, cooking, cleaning plus living an active lifestyle. Its not which guys over 50 plus post-menopausal ladies are not active. However frequently they are less active. This may be considering of bodily limitations, including arthritis, or because they have simply gotten utilized to a more sedentary life-style inside retirement or even when they are still working.<br><br>My point is I understand I'm not obese plus I recognize I am a healthy individual on my cholesterol plus blood pressure, thus there are aspects which provide to my weight that a calculator can't account for.<br><br>This post was written August 15, 2013 and Gabriel Cousens is 70 years of age. So in the event you plan on going to his center, I advise that you do it shortly. As of 2011, over 11% of Americans (adults) have diabetes. The rate of prediabetes (not quite significant enough to be diabetes) in America is 3 occasions because excellent because this.<br><br>So being the girl I am and having a need to understand form of notice, I found an online [http://safedietplans.com/bmi-calculator bmi calculator women] plus place in my numbers. After my friend Sally's experience I was certain which mine would at the least state I was at the high-end of usual. Oh yet no, my online BMI result was that I was underweight. WTF? Again I know that everyone has their own perception of what is heavy plus was is not however, when I were to truthfully judge where I think my weight could fall I would have said average, specifically with all the knowledge that Sally's returned found on the high-end of obese.<br><br>Economist Giorrio Brunello from the University of Padova, Italy said which BMI affects wages negatively inside Europe. Moreover, the scale of the effect is greater for guys in comparison to women.<br><br>Another limitation is the fact that age is not considered whenever calculating BMI. So for elderly people, whether or not the BMI indication is usual, the fact remains which most muscles have been lost because of age.<br><br>So, this would require time management. There are a lot of exercise programs available even online for we to do and accomplish. It's all up to we. If inside case as of the moment we never have the time to do it, then possibly doing a little adjustment inside a daily activities would be a advantageous begin.

	~~For the purposes of this article, such reals will be called simply '''''definable numbers'''''. This should not be understood to be standard terminology.~~

	~~Note~~ that ~~this definition cannot be expressed in the language of set theory itself~~.

	~~==General facts==~~

	~~Assuming they form a set~~, ~~the definable numbers form a [[field (mathematics)\|field]] containing all the familiar real numbers such as [[Zero\|0]]~~, ~~[[One\|1]]~~, ~~[[Pi\|{{pi}}]], [[E_(mathematical_constant)\|''e'']], et cetera~~. ~~In particular, this field contains all the numbers named in the [[mathematical constant]]s article, and all [[algebraic number]]s (and therefore all [[rational number]]s)~~. However~~, most real numbers~~ are ~~not definable: the [[Set (mathematics)\|set]]~~ of ~~all definable numbers is [[countably infinite]] (~~because ~~the set of all logical formulas is) while the set of real numbers is [[uncountable set\|uncountably infinite]] (see [[Cantor's diagonal argument]]). As~~ a ~~result, [[almost all\|most]] real numbers have no description (in the same sense of "most" as 'most real numbers~~ are ~~not rational')~~.

	~~The field of definable numbers is not [[complete space\|complete]]; there exist convergent [[sequence]]s of definable numbers whose [[limit of a sequence\|limit]]~~ is not ~~definable (since every real number is the limit of~~ a ~~sequence of rational numbers). However~~, ~~if the sequence itself is definable in the sense~~ that we can ~~specify a single formula~~ for ~~all its terms, then its limit will necessarily be a definable number~~.

	While every [[computable number]] is definable, the converse is not true: the numeric representations of the [[Halting problem]], [[Chaitin's constant]], the truth set of first order arithmetic, and [[Zero sharp\|0<~~sup~~>#<~~/sup~~>~~]] are examples of numbers that are definable but not computable. Many other such numbers are known.~~

	~~One may also wish to talk about definable [[complex number]]s: complex numbers which are uniquely defined by a logical formula. However~~, ~~whether this~~ is ~~possible depends on how the field~~ of ~~complex numbers is derived~~ in the ~~first place: it may not be possible~~ to ~~distinguish a complex number from its conjugate (say~~, ~~3+i from 3-i), since~~ it ~~is impossible to find a property~~ of ~~one that is not also a property~~ of ~~the other, without falling back on the underlying set-theoretic definition~~. ~~Assuming we can define at least one nonreal complex number, however, a [[complex number]]~~ is ~~definable if and only if both its real part~~ and ~~its imaginary part are definable. The definable complex numbers also form~~ a ~~field if they~~ form ~~a set.~~

	~~The related concept~~ of ~~"standard" numbers~~, ~~which can only be defined within a finite time and space, is used to motivate axiomatic~~ [~~[internal set theory]], and provide a workable formulation for [[illimited number\|illimited]] and [[infinitesimal number]]~~. ~~Definitions of the hyper~~-~~real line within [[non-standard analysis~~]~~] (the subject area dealing with such~~ numbers~~) overwhelmingly include the usual, uncountable set of real numbers as a subset~~.

	~~==Notion does not exhaust "unambiguously described" numbers==~~
	~~Not every number that we~~ would ~~informally say has been unambiguously described, is definable in~~ the ~~above sense. For example, if we can enumerate all such definable numbers by~~ the ~~[[Gödel number]]s of their defining formulas then we can use [[Cantor's diagonal argument]] to find a particular real that is not first~~-~~order definable in the same language. The argument can be made as follows:~~

	~~Suppose that in a mathematical language L, it is possible to enumerate all~~ of ~~the defined numbers in L~~. ~~Let this enumeration be defined by the function G: W → R~~, ~~where G(n) is the real number described by the nth description in the sequence. Using the diagonal argument, it is possible to define a real number x, which is not equal to G(n) for any n~~. ~~This means~~ that ~~there~~ is ~~a language L' that defines x, which~~ is ~~undefinable in L.~~

	~~==Other notions of definability==~~
	~~The notion of definability treated in this article has been chosen primarily for definiteness,~~ not ~~on the grounds that it's more useful or interesting than other notions. Here we treat a few others:~~

	~~===Definability in other languages or structures===~~
	~~====Language of arithmetic====~~
	~~The [[language of arithmetic]] has symbols for 0, 1, the successor operation~~, ~~addition, and multiplication, intended~~ to ~~be interpreted in the usual way over the [[natural number]]s. Since no variables of this language range over the [[real number]]s~~, ~~we cannot simply copy~~ the ~~earlier definition of definability. Rather, we say~~ that ~~a real~~ '~~'a'' is '''''definable in~~ the language of arithmetic''''' (or '''''[[arithmetical hierarchy\|arithmetical]]''''') if its [[Dedekind cut]] can be defined as a [[Predicate (logic)\|predicate]] in that language; that is, if there is a first-~~order formula ''φ'' in the language~~ of ~~arithmetic, with two free variables, such that~~

	:<~~math~~>~~\forall m \, \forall n \, (\varphi(n,m)\iff\frac{n}{m}~~<~~a).</math~~>

	~~====2nd-order language~~ of ~~arithmetic====~~
	~~The second-order language of arithmetic is the same as the first-order language~~, ~~except that variables and quantifiers are allowed to range over sets of naturals~~. ~~A real that is second-order definable in~~ the ~~language~~ of ~~arithmetic~~ is ~~called '''''[[analytical hierarchy\|analytical]]'''''.~~

	~~===Definability with ordinal parameters===~~
	~~Sometimes it is of interest~~ to ~~consider definability ''with parameters''; that is, to give a definition relative to another object that remains undefined~~. ~~For example, a real ''a'' (or for that matter, any set ''a'')~~ is ~~called '''''ordinal definable''''' if there is a first-order formula ''φ'' in~~ the ~~language of set theory, with ''two'' free variables, and an [[Ordinal number\|ordinal]] γ, such~~ that ~~''a''~~ is ~~the unique object such that ''φ''(''a'',γ) holds (in V).~~

	~~The other sorts of definability thus far~~ considered ~~have only countably many defining formulas, and therefore allow only countably many definable reals~~. ~~This is not true~~ for ~~ordinal definability~~, ~~because an ordinal definable real is defined~~ not ~~only by~~ the ~~formula ''φ''~~, ~~but also by~~ the ~~ordinal γ~~. ~~In fact it is consistent with [[ZFC]] that ''all'' reals are ordinal-definable~~, ~~and therefore that there are uncountably many ordinal-definable reals~~. ~~However it is also consistent with ZFC that there~~ are ~~only countably many ordinal-definable reals~~.

	~~== See also ==~~
	* [[Berry paradox]]
	* [[Computable number]]
	* [[Constructible number]]
	* [[Constructible universe]]
	* [[Entscheidungsproblem]]

	~~==References==~~

	* {{Citation \| last1=Kunen \| first1=Kenneth \| author1-link=Kenneth Kunen \| title=[[Set Theory: An Introduction to ~~Independence Proofs]] \| publisher=North-Holland \| location=Amsterdam \| isbn=978-0-444-85401-8 \| year=1980}}~~

	* [http://rothkamm.~~com/download.cfm/turing/Turing.pdf Alan Turing, "On Computable Numbers, With An Application to~~ the ~~Entscheidungsproblem", ''Proceedings of~~ the ~~London Mathematical Society''~~, ~~1936] ([[Alan Turing\|Turing's]] original paper distinguishing computable and definable numbers)~~

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

en>Mr. Granger: Repairing links to disambiguation pages - You can help!

2013-12-19T03:39:45Z

Repairing links to disambiguation pages - You can help!

← Older revision		Revision as of 05:39, 19 December 2013
Line 1:		Line 1:
	~~I am 28 years old~~ and ~~my name~~ is ~~Phillipp Bussau~~. ~~I life~~ in ~~Neuchatel~~ (~~Switzerland~~).<br><br>~~Feel~~ free to ~~surf~~ to ~~my webpage~~ [http://~~Perysa~~.com/~~noticias~~/~~noticia-1~~.~~html fifa 15 coin Hack~~]		A [[real number]] ''a'' is '''first-order definable in the language of set theory, without parameters''', if there is a formula ''φ'' in the language of [[set theory]], with one [[free variable]], such that ''a'' is the unique real number such that ''φ(a)'' holds in the standard model of set theory (see Kunen 1980:153).

			For the purposes of this article, such reals will be called simply '''''definable numbers'''''. This should not be understood to be standard terminology.

			Note that this definition cannot be expressed in the language of set theory itself.

			==General facts==

			Assuming they form a set, the definable numbers form a [[field (mathematics)\|field]] containing all the familiar real numbers such as [[Zero\|0]], [[One\|1]], [[Pi\|{{pi}}]], [[E_(mathematical_constant)\|''e'']], et cetera. In particular, this field contains all the numbers named in the [[mathematical constant]]s article, and all [[algebraic number]]s (and therefore all [[rational number]]s). However, most real numbers are not definable: the [[Set (mathematics)\|set]] of all definable numbers is [[countably infinite]] (because the set of all logical formulas is) while the set of real numbers is [[uncountable set\|uncountably infinite]] (see [[Cantor's diagonal argument]]). As a result, [[almost all\|most]] real numbers have no description (in the same sense of "most" as 'most real numbers are not rational').

			The field of definable numbers is not [[complete space\|complete]]; there exist convergent [[sequence]]s of definable numbers whose [[limit of a sequence\|limit]] is not definable (since every real number is the limit of a sequence of rational numbers). However, if the sequence itself is definable in the sense that we can specify a single formula for all its terms, then its limit will necessarily be a definable number.

			While every [[computable number]] is definable, the converse is not true: the numeric representations of the [[Halting problem]], [[Chaitin's constant]], the truth set of first order arithmetic, and [[Zero sharp\|0<sup>#</sup>]] are examples of numbers that are definable but not computable. Many other such numbers are known.

			One may also wish to talk about definable [[complex number]]s: complex numbers which are uniquely defined by a logical formula. However, whether this is possible depends on how the field of complex numbers is derived in the first place: it may not be possible to distinguish a complex number from its conjugate (say, 3+i from 3-i), since it is impossible to find a property of one that is not also a property of the other, without falling back on the underlying set-theoretic definition. Assuming we can define at least one nonreal complex number, however, a [[complex number]] is definable if and only if both its real part and its imaginary part are definable. The definable complex numbers also form a field if they form a set.

			The related concept of "standard" numbers, which can only be defined within a finite time and space, is used to motivate axiomatic [[internal set theory]], and provide a workable formulation for [[illimited number\|illimited]] and [[infinitesimal number]]. Definitions of the hyper-real line within [[non-standard analysis]] (the subject area dealing with such numbers) overwhelmingly include the usual, uncountable set of real numbers as a subset.

			==Notion does not exhaust "unambiguously described" numbers==
			Not every number that we would informally say has been unambiguously described, is definable in the above sense. For example, if we can enumerate all such definable numbers by the [[Gödel number]]s of their defining formulas then we can use [[Cantor's diagonal argument]] to find a particular real that is not first-order definable in the same language. The argument can be made as follows:

			Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G(n) is the real number described by the nth description in the sequence. Using the diagonal argument, it is possible to define a real number x, which is not equal to G(n) for any n. This means that there is a language L' that defines x, which is undefinable in L.

			==Other notions of definability==
			The notion of definability treated in this article has been chosen primarily for definiteness, not on the grounds that it's more useful or interesting than other notions. Here we treat a few others:

			===Definability in other languages or structures===
			====Language of arithmetic====
			The [[language of arithmetic]] has symbols for 0, 1, the successor operation, addition, and multiplication, intended to be interpreted in the usual way over the [[natural number]]s. Since no variables of this language range over the [[real number]]s, we cannot simply copy the earlier definition of definability. Rather, we say that a real ''a'' is '''''definable in the language of arithmetic''''' (or '''''[[arithmetical hierarchy\|arithmetical]]''''') if its [[Dedekind cut]] can be defined as a [[Predicate (logic)\|predicate]] in that language; that is, if there is a first-order formula ''φ'' in the language of arithmetic, with two free variables, such that

			:<math>\forall m \, \forall n \, (\varphi(n,m)\iff\frac{n}{m}<a).</math>

			====2nd-order language of arithmetic====
			The second-order language of arithmetic is the same as the first-order language, except that variables and quantifiers are allowed to range over sets of naturals. A real that is second-order definable in the language of arithmetic is called '''''[[analytical hierarchy\|analytical]]'''''.

			===Definability with ordinal parameters===
			Sometimes it is of interest to consider definability ''with parameters''; that is, to give a definition relative to another object that remains undefined. For example, a real ''a'' (or for that matter, any set ''a'') is called '''''ordinal definable''''' if there is a first-order formula ''φ'' in the language of set theory, with ''two'' free variables, and an [[Ordinal number\|ordinal]] γ, such that ''a'' is the unique object such that ''φ''(''a'',γ) holds (in V).

			The other sorts of definability thus far considered have only countably many defining formulas, and therefore allow only countably many definable reals. This is not true for ordinal definability, because an ordinal definable real is defined not only by the formula ''φ'', but also by the ordinal γ. In fact it is consistent with [[ZFC]] that ''all'' reals are ordinal-definable, and therefore that there are uncountably many ordinal-definable reals. However it is also consistent with ZFC that there are only countably many ordinal-definable reals.

			== See also ==
			* [[Berry paradox]]
			* [[Computable number]]
			* [[Constructible number]]
			* [[Constructible universe]]
			* [[Entscheidungsproblem]]

			==References==

			* {{Citation \| last1=Kunen \| first1=Kenneth \| author1-link=Kenneth Kunen \| title=[[Set Theory: An Introduction to Independence Proofs]] \| publisher=North-Holland \| location=Amsterdam \| isbn=978-0-444-85401-8 \| year=1980}}

			* [http://rothkamm.com/download.cfm/turing/Turing.pdf Alan Turing, "On Computable Numbers, With An Application to the Entscheidungsproblem", ''Proceedings of the London Mathematical Society'', 1936] ([[Alan Turing\|Turing's]] original paper distinguishing computable and definable numbers)

			[[Category:Set theory]]

en>Ohconfucius: General formatting, inc. removal of superscripted ordinals

2011-09-23T07:53:28Z

General formatting, inc. removal of superscripted ordinals

New page

I am 28 years old and my name is Phillipp Bussau. I life in Neuchatel (Switzerland).<br><br>Feel free to surf to my webpage [http://Perysa.com/noticias/noticia-1.html fifa 15 coin Hack]