Criterion-referenced test - Revision history

en>Rjwilmsi: Added 1 doi to a journal cite using AWB (10222)

2014-06-08T13:32:55Z

Added 1 doi to a journal cite using AWB (10222)

← Older revision		Revision as of 15:32, 8 June 2014
Line 1:		Line 1:
	~~{{Expert-subject\|Mathematics\|date=November 2008}}~~		Hi there, I am Sophia. Her family members life in Ohio. What I love doing is soccer but I don't have the time recently. My working day occupation is an information officer but I've currently applied for an additional one.<br><br>My weblog :: [http://black7.mireene.com/aqw/5741 psychic solutions by lynne]

	~~'''Gentzen's consistency proof''' is a result of [[proof theory]]~~ in ~~[[mathematical logic]]~~. It "reduces" the consistency of a simplified part of mathematics, not to something that could be proved in that same simplified part of mathematics (which would contradict the basic results of [[Kurt Gödel]]), but ~~rather to a simpler logical principle.~~

	~~==Gentzen~~'~~s theorem==~~
	~~In 1936 [[Gerhard Gentzen]] proved~~ the consistency of [[first-order arithmetic]] using combinatorial methods. Gentzen's proof shows much more than merely that first-order arithmetic is [[consistent]]. Gentzen showed that the consistency of first-order arithmetic is provable, over the base theory of [[primitive recursive arithmetic]] with the additional principle of quantifier-free [[transfinite induction]] up to the [[ordinal number\|ordinal]] [[epsilon zero\|ε<sub>0</sub>]]. Informally, this additional principle means that there is a [[well-ordering]] on the set of finite rooted [[tree (mathematics)\|tree]]s.

	~~The principle of quantifier-free transfinite induction up to ε<sub>0</sub> says that for any formula A(x) with no bound variables transfinite induction up to ε<sub>0</sub> holds~~. ~~ε<sub>0</sub>~~ is ~~the first ordinal <math>\alpha</math>, such that <math>\omega^\alpha = \alpha</math>, i.e. the limit of the sequence:~~
	~~:<math>\omega,\ \omega^\omega,\ \omega^{\omega^\omega},\ \ldots</math>~~
	~~To express ordinals in the language of arithmetic~~ an ~~[[ordinal notation]] is needed, i.e. a way to assign natural numbers to ordinals less than ε<sub>0</sub>. This can be done in various ways, one example provided by Cantor~~'~~s normal form theorem. That transfinite induction holds~~ for ~~a formula A(x) means that A does not define~~ an ~~infinite descending sequence of ordinals smaller than ε<sub>0</sub> (in which case ε<sub>0</sub> would not be well-ordered)~~. Gentzen assigned ordinals smaller than ε<sub>0</sub> to proofs in first-order arithmetic and showed that if there is a proof of contradiction, then there is an infinite descending sequence of ordinals < ~~ε<sub~~>0<~~/sub~~> ~~produced by a [[primitive recursive]] operation on proofs corresponding to a quantifier-free formula.~~

	~~==Relation to Gödel's theorem==~~
	Gentzen's proof also highlights one commonly missed aspect of [[Gödel's second incompleteness theorem]]. It is sometimes claimed that the consistency of a theory can only be proved in a stronger theory. The theory obtained by adding quantifier-free transfinite induction to primitive recursive arithmetic proves the consistency of first-order arithmetic but is not stronger than first-order arithmetic. For example, it does not prove ordinary mathematical induction for all formulae, while first-order arithmetic does (it has this as an axiom schema). The resulting theory is not weaker than first-order arithmetic either, since it can prove a number-theoretical fact - the consistency of first-order arithmetic - that first-order arithmetic cannot. The two theories are simply incomparable.

	Gentzen's proof is the first example of what is called proof-theoretical [[ordinal analysis]]. In ordinal analysis one gauges the strength of theories by measuring how large the (constructive) ordinals are that can be proven to be well-ordered, or equivalently for how large a (constructive) ordinal can transfinite induction be proven. A constructive ordinal is the order type of a [[recursive set\|recursive]] well-ordering of natural numbers.

	~~[[Laurence Kirby]] and [[Jeff Paris]] proved in 1982 that [[Goodstein's theorem]] cannot be proven in Peano arithmetic based on Gentzen's theorem.~~

	~~==References==~~
	*{{Citation\|last1=Gentzen\|doi=10.1007/BF01565428\|first1=Gerhard\|author-link=Gerhard Gentzen\|title=Die Widerspruchsfreiheit der reinen Zahlentheorie\|journal=Mathematische Annalen\|volume=112\|year=1936\|publisher=\|pages=493–565\|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002278391}} - Translated as 'The consistency of arithmetic', in {{Harv\|Gentzen\|Szabo\|1969}}.
	*{{Citation\|last1=Gentzen\|first1=Gerhard\|author-link=Gerhard Gentzen\|title=Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie\|journal=Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften\|volume=4\|year=1938\|publisher=\|pages=19–44}} - Translated as 'New version of the consistency proof for elementary number theory', in {{Harv\|Gentzen\|Szabo\|1969}}.
	*{{Citation\|last=Gentzen\|first=Gerhard\|editor-last=M. E.\|editor-first=Szabo\|editor-link=M. E. Szabo\|year=1969\|title=Collected Papers of Gerhard Gentzen\|publisher=North-Holland\|location=Amsterdam\|edition=Hardcover\|series=Studies in logic and the foundations of mathematics\|isbn=0-7204-2254-X\|ref={{Harvid\|Gentzen\|Szabo\|1969}}}} - an English translation of papers.
	*{{Citation\|last=Gödel\|first=K.\|author-link=Kurt Gödel\|editor1-last=Feferman\|editor1-first=Solomon\|editor1-link=Solomon Feferman\|origyear=1938\|year=2001\|title=Kurt Gödel: Collected Works\|chapter=Lecture at Zilsel’s\|publisher=Oxford University Press Inc.\|pages=87–113\|edition=Paperback\|volume=vol.III Unpublished Essays and Lectures\|isbn=0-19-514722-7\|ref={{Harvid\|Gödel\|Feferman et al.\|2001}}}}
	*{{Citation\|last=Jervell\|first=Herman Ruge\|author-link=Herman Ruge Jervell\|year=1999\|title=A course in proof theory\|edition=textbook draft\|url=http:~~//folk.uio.no/herman/bevisteori.ps}}~~
	*{{Citation\|last1=Kirby\|first1=L.\|author1-link=Laurie Kirby\|last2=Paris\|first2=J.\|author2-link=Jeff Paris\|year=1982\|title=Accessible independence results for Peano arithmetic\|journal=[~~[Bull. London Math. Soc.]]\|publisher=[[London Mathematical Society\|LMS]]\|pages=285–293\|volume=14\|url=~~http://~~blms~~.~~oxfordjournals~~.~~org~~/~~content~~/~~14/4/285.full.pdf\|format=PDF}}~~
	*{{Citation\|last=Tait\|first=W. W.\|author-link=William W. Tait\|year=2005\|title=Gödel's reformulation of Gentzen's first consistency proof for arithmetic: the no-counterexample interpretation\|journal=The Bulletin of Symbolic Logic\|volume=11\|issue=2\|publisher=[[Association for Symbolic Logic\|ASL]]\|pages=225–238\|url=http://home.uchicago.edu/~wwtx/GoedelandNCInew1.pdf\|format=PDF\|issn=1079-8986}}

	~~[[Category:Metatheorems]]~~
	~~[[Category:Proof theory]~~]

65.255.37.171: /* Comparison of criterion-referenced and norm-referenced tests */

2013-12-22T14:39:36Z

Comparison of criterion-referenced and norm-referenced tests

← Older revision		Revision as of 16:39, 22 December 2013
Line 1:		Line 1:
	~~The writer~~'s ~~name~~ is ~~Andera and she believes it seems fairly good~~. ~~What me and my family adore is bungee jumping~~ but I'~~ve been~~ using ~~on new things lately~~. ~~My day occupation~~ is ~~an information officer but I've currently utilized for an~~ additional [~~http~~://~~modenpeople~~.co.kr/~~modn~~/~~qna~~/~~292291 accurate psychic predictions~~] 1. For ~~many years~~ are ~~psychics real~~ ([http://~~www~~.~~chk~~.~~woobi~~.~~co.kr~~/xe/?~~document_srl~~=~~346069 www~~.~~chk~~.~~woobi~~.co.~~kr]) she's been living~~ in ~~Kentucky but her spouse desires them to transfer~~.~~<br><br>Feel free to visit my weblog~~ ... ~~online psychics (~~[http://~~netwk~~.~~hannam~~.~~ac.kr~~/xe/~~data_2~~/~~85669~~ http://~~netwk~~.~~hannam~~.ac.~~kr/~~])		{{Expert-subject\|Mathematics\|date=November 2008}}

			'''Gentzen's consistency proof''' is a result of [[proof theory]] in [[mathematical logic]]. It "reduces" the consistency of a simplified part of mathematics, not to something that could be proved in that same simplified part of mathematics (which would contradict the basic results of [[Kurt Gödel]]), but rather to a simpler logical principle.

			==Gentzen's theorem==
			In 1936 [[Gerhard Gentzen]] proved the consistency of [[first-order arithmetic]] using combinatorial methods. Gentzen's proof shows much more than merely that first-order arithmetic is [[consistent]]. Gentzen showed that the consistency of first-order arithmetic is provable, over the base theory of [[primitive recursive arithmetic]] with the additional principle of quantifier-free [[transfinite induction]] up to the [[ordinal number\|ordinal]] [[epsilon zero\|ε<sub>0</sub>]]. Informally, this additional principle means that there is a [[well-ordering]] on the set of finite rooted [[tree (mathematics)\|tree]]s.

			The principle of quantifier-free transfinite induction up to ε<sub>0</sub> says that for any formula A(x) with no bound variables transfinite induction up to ε<sub>0</sub> holds. ε<sub>0</sub> is the first ordinal <math>\alpha</math>, such that <math>\omega^\alpha = \alpha</math>, i.e. the limit of the sequence:
			:<math>\omega,\ \omega^\omega,\ \omega^{\omega^\omega},\ \ldots</math>
			To express ordinals in the language of arithmetic an [[ordinal notation]] is needed, i.e. a way to assign natural numbers to ordinals less than ε<sub>0</sub>. This can be done in various ways, one example provided by Cantor's normal form theorem. That transfinite induction holds for a formula A(x) means that A does not define an infinite descending sequence of ordinals smaller than ε<sub>0</sub> (in which case ε<sub>0</sub> would not be well-ordered). Gentzen assigned ordinals smaller than ε<sub>0</sub> to proofs in first-order arithmetic and showed that if there is a proof of contradiction, then there is an infinite descending sequence of ordinals < ε<sub>0</sub> produced by a [[primitive recursive]] operation on proofs corresponding to a quantifier-free formula.

			==Relation to Gödel's theorem==
			Gentzen's proof also highlights one commonly missed aspect of [[Gödel's second incompleteness theorem]]. It is sometimes claimed that the consistency of a theory can only be proved in a stronger theory. The theory obtained by adding quantifier-free transfinite induction to primitive recursive arithmetic proves the consistency of first-order arithmetic but is not stronger than first-order arithmetic. For example, it does not prove ordinary mathematical induction for all formulae, while first-order arithmetic does (it has this as an axiom schema). The resulting theory is not weaker than first-order arithmetic either, since it can prove a number-theoretical fact - the consistency of first-order arithmetic - that first-order arithmetic cannot. The two theories are simply incomparable.

			Gentzen's proof is the first example of what is called proof-theoretical [[ordinal analysis]]. In ordinal analysis one gauges the strength of theories by measuring how large the (constructive) ordinals are that can be proven to be well-ordered, or equivalently for how large a (constructive) ordinal can transfinite induction be proven. A constructive ordinal is the order type of a [[recursive set\|recursive]] well-ordering of natural numbers.

			[[Laurence Kirby]] and [[Jeff Paris]] proved in 1982 that [[Goodstein's theorem]] cannot be proven in Peano arithmetic based on Gentzen's theorem.

			==References==
			*{{Citation\|last1=Gentzen\|doi=10.1007/BF01565428\|first1=Gerhard\|author-link=Gerhard Gentzen\|title=Die Widerspruchsfreiheit der reinen Zahlentheorie\|journal=Mathematische Annalen\|volume=112\|year=1936\|publisher=\|pages=493–565\|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002278391}} - Translated as 'The consistency of arithmetic', in {{Harv\|Gentzen\|Szabo\|1969}}.
			*{{Citation\|last1=Gentzen\|first1=Gerhard\|author-link=Gerhard Gentzen\|title=Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie\|journal=Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften\|volume=4\|year=1938\|publisher=\|pages=19–44}} - Translated as 'New version of the consistency proof for elementary number theory', in {{Harv\|Gentzen\|Szabo\|1969}}.
			*{{Citation\|last=Gentzen\|first=Gerhard\|editor-last=M. E.\|editor-first=Szabo\|editor-link=M. E. Szabo\|year=1969\|title=Collected Papers of Gerhard Gentzen\|publisher=North-Holland\|location=Amsterdam\|edition=Hardcover\|series=Studies in logic and the foundations of mathematics\|isbn=0-7204-2254-X\|ref={{Harvid\|Gentzen\|Szabo\|1969}}}} - an English translation of papers.
			*{{Citation\|last=Gödel\|first=K.\|author-link=Kurt Gödel\|editor1-last=Feferman\|editor1-first=Solomon\|editor1-link=Solomon Feferman\|origyear=1938\|year=2001\|title=Kurt Gödel: Collected Works\|chapter=Lecture at Zilsel’s\|publisher=Oxford University Press Inc.\|pages=87–113\|edition=Paperback\|volume=vol.III Unpublished Essays and Lectures\|isbn=0-19-514722-7\|ref={{Harvid\|Gödel\|Feferman et al.\|2001}}}}
			*{{Citation\|last=Jervell\|first=Herman Ruge\|author-link=Herman Ruge Jervell\|year=1999\|title=A course in proof theory\|edition=textbook draft\|url=http://folk.uio.no/herman/bevisteori.ps}}
			*{{Citation\|last1=Kirby\|first1=L.\|author1-link=Laurie Kirby\|last2=Paris\|first2=J.\|author2-link=Jeff Paris\|year=1982\|title=Accessible independence results for Peano arithmetic\|journal=[[Bull. London Math. Soc.]]\|publisher=[[London Mathematical Society\|LMS]]\|pages=285–293\|volume=14\|url=http://blms.oxfordjournals.org/content/14/4/285.full.pdf\|format=PDF}}
			*{{Citation\|last=Tait\|first=W. W.\|author-link=William W. Tait\|year=2005\|title=Gödel's reformulation of Gentzen's first consistency proof for arithmetic: the no-counterexample interpretation\|journal=The Bulletin of Symbolic Logic\|volume=11\|issue=2\|publisher=[[Association for Symbolic Logic\|ASL]]\|pages=225–238\|url=http://home.uchicago.edu/~wwtx/GoedelandNCInew1.pdf\|format=PDF\|issn=1079-8986}}

			[[Category:Metatheorems]]
			[[Category:Proof theory]]

en>Senjuto: removed extra spaces between sentences.

2012-01-02T15:22:10Z

removed extra spaces between sentences.

New page

The writer's name is Andera and she believes it seems fairly good. What me and my family adore is bungee jumping but I've been using on new things lately. My day occupation is an information officer but I've currently utilized for an additional [http://modenpeople.co.kr/modn/qna/292291 accurate psychic predictions] 1. For many years are psychics real ([http://www.chk.woobi.co.kr/xe/?document_srl=346069 www.chk.woobi.co.kr]) she's been living in Kentucky but her spouse desires them to transfer.<br><br>Feel free to visit my weblog ... online psychics ([http://netwk.hannam.ac.kr/xe/data_2/85669 http://netwk.hannam.ac.kr/])