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	Wilhelm Ackermann - Wikipedia, the free encyclopedia
		Wilhelm Ackermann (March 29, 1896, Herscheid municipality, Germany – December 24, 1962 Lüdenscheid, Germany) was a German mathematician best known for the Ackermann function, an important example in the theory of computation.
		Ackermann was awarded the Ph.D. by the University of Goettingen in 1925 for his thesis Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit, which was a consistency proof of arithmetic apparently without full Peano induction (although it did use e.g.
		Ackermann went on to construct consistency proofs for set theory (1937), full arithmetic (1940), type-free logic (1952), and a new axiomatization of set theory (1956).
	en.wikipedia.org /wiki/Wilhelm_Ackermann (245 words)

	Wilhelm Ackermann
		Wilhelm Ackermann (1896-1962) was a mathematician and is most famous for the Ackermann function named after him, an important example in the theory of computation.
		Ackermann was born on March 29, 1896 in Schönebecke (then Altena district, now part of Herscheid[?] municipality), Germany, and received his doctoral degree in 1925 with his thesis Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit, which was a consistency proof of arithmetic without induction.
		Wilhelm Ackermann died in Lüdenscheid, Germany on December 24, 1962.
	www.ebroadcast.com.au /lookup/encyclopedia/ac/Ackermann.html (180 words)

	Ackermann function - Wikipedia, the free encyclopedia
		Ackermann originally considered a function A(m, n, p) of three variables, the p-fold iterated exponentiation of m with n, or m → n → p as expressed using the Conway chained arrow.
		Ackermann proved that A is a recursive function, a function a computer with unbounded memory can calculate, but it is not a primitive recursive function, a class of functions including almost all familiar functions such as addition and factorial.
		The Ackermann function, due to its definition in terms of extremely deep recursion, can be used as a benchmark of a compiler's ability to optimize recursion.
	en.wikipedia.org /wiki/Ackermann_function (1857 words)

	PlanetMath: Ackermann function
		Ackermann's function is an example of a recursive function that is not primitive recursive, but is instead
		Ackermann's function wasn't actually written in this form by its namesake, Wilhelm Ackermann.
		This is version 6 of Ackermann function, born on 2002-03-23, modified 2004-03-30.
	planetmath.org /encyclopedia/AckermannFunction.html (159 words)

	Ackermann (print-only) (Site not responding. Last check: 2007-10-13)
		Ackermann received his doctoral degree in 1925 with a thesis Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit written under Hilbert and was a proof of the consistency of arithmetic without induction.
		Ackermann was also the main contributor to the development of the logical system known as the epsilon calculus, originally due to Hilbert.
		In 1928, Ackermann observed that A(x, y, z), the z-fold iterated exponentiation of x with y, is an example of a recursive function which is not primitive recursive.
	www-history.mcs.st-and.ac.uk /Printonly/Ackermann.html (255 words)

	Wilhelm Ackermann: bio and encyclopedia article (Site not responding. Last check: 2007-10-13)
		Wilhelm Ackermann (March 29, EHandler: no quick summary.
		In the theory of computation, the ackermann function or ackermann-peter function is a simple example of a recursive function that is not primitive recurs...
		Karl wilhelm feuerbach (30 may 1800-12 march 1834) was a german geometer....
	www.absoluteastronomy.com /encyclopedia/w/wi/wilhelm_ackermann.htm (934 words)

	Wilhelm Ackermann (Site not responding. Last check: 2007-10-13)
		Wilhelm Ackermann received his doctoral degree in 1925 with a thesis written under Hilbert.
		In 1928, Ackermann observed that A(x,y,z), the z-fold iterated exponentiation of x with y, is an example of a recursive function which is not primitive recursive.
		Among Ackermann's later work there are consistency proofs for set theory, full arithmetic, and type free logic.
	www.stetson.edu /~efriedma/periodictable/html/Ac.html (172 words)

	THE ACKERMANN INSTITUTE
		Wilhelm Paul Ackermann, was born in germany and started his studies in natural medicine chiropractics because of own back problems.
		Ackermann studied chiropractics in USA and in England and thereafter worked as assistant school doctor.
		Later he founded both the Ackermann Institute (1968) and College (1980), where his developments on specific diagnosis- and treatmentsystem on the basis of american chiropractic are being taught today.
	www.ackermann-institutet.se /AI-ENG.htm (167 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		Submissions ----------- Eligible for the 2005 ACKERMANN Award are PhD dissertations in topics specified by the EACSL and LICS conferences, which were formally accepted as PhD theses at a university or equivalent institution between 1.1.
		------------------------------------------------- Wilhelm Ackermann was born on March 29, 1896 and died on December 24, 1962.
		Ackermann was also the main contributor to the logical system known as the epsilon calculus, originally due to Hilbert.
	dept-info.labri.u-bordeaux.fr /~dicky/LaBRI-Hebdo/43/ackermann.txt (482 words)

	Ackermann - TheBestLinks.com - Austria, Actor, Author, Ackermann function, ... (Site not responding. Last check: 2007-10-13)
		Anton Ackermann, Minister of Foreign Affairs of the German Democratic Republic in 1953.
		Ernst Christian Wilhelm Ackermann, 1761-1835, Bohemian public servant, close friend of August von Kotzebue.
		Wilhelm Ackermann, 1896-1962, German mathematician known for the Ackermann function.
	www.thebestlinks.com /Ackermann.html (190 words)

	Dotzel: The Ackermann function (12 KB)
		The so-called Ackermann function was developed by Wilhelm Ackermann (1896 to 1962) in the year 1928.
		Also its nesting level (i.e.: not the number of calls of the Ackermann function) is at least as deep as its function value.
		The first argument x determines the complexity or kind of the function, the Ackermann function represents, whereas the second argument y is the iteration count for the function determined by the first argument.
	www.modulaware.com /mdlt08.htm (955 words)

	Franz Ackermann - Paintings - The Saatchi Gallery
		His jumbled composition is harmonious in its turmoil: concentric patterns of colour expose hints of identifiable place (a street map, a building interior, a snippet of landscape) only to dislocate them in a maze of organic generalisations.
		Franz Ackermann is a perpetual tourist: his paintings are like large trippy postcards from the edge.
		Franz Ackermann’s painting has an undertone of catastrophe: desert sunsets, rocky coves and industrial parks clash together like tectonic plates in an ethical snafu.
	www.saatchi-gallery.co.uk /artists/franz_ackermann.htm (341 words)

	Lavenz-Schwarz and Knecht-Eischeid Genealogy (Site not responding. Last check: 2007-10-13)
		Wilhelm KNECHT and Anna Maria LEŸ were married in 1858.
		Johann Wilhelm LEŸ was born on 3 Apr 1718 in Niederwennerscheid.
		Wilhelm LEŸ and Anna Maria AGGERMIENS (ACKERMANN) were married about 1791.
	members.aol.com /mtl1963/b126.htm (902 words)

	Ackermann Function (Site not responding. Last check: 2007-10-13)
		The U.S. Department of Education continues to emphasize the arts as part of a comprehensive education in order to meet the requirements of the No Child Left Behind Act.
		In the theory of computation, the Ackermann function or Ackermann-Péter function is a simple example of a recursive function that is not
		The Ackermann function is the simplest example of a well defined total function which is computable but not primitive recursive, providing a...
	www.ackermannfunction.info (1061 words)

	Ackermann's function (Site not responding. Last check: 2007-10-13)
		Note: In 1928, Wilhelm Ackermann observed that A(x,y,z), the z-fold iterated exponentiation of x with y, is an example of a recursive function which is not primitive recursive.
		Many people have given other versions of Ackermann's function, some of which are not simply a restating of this one.
		Robert Munafo's Versions of Ackermann's Function and analysis.
	www.nist.gov /dads/HTML/ackermann.html (165 words)

	Hilbert's Program
		Ackermann (1924) attempted to extend Hilbert's idea to a system of analysis.
		Shortly thereafter, von Neumann showed that Ackermann's consistency proof is flawed and provided a counterexample to the proposed ε-substitution procedure (see Zach 2003).
		Ackermann, Wilhelm, 1924, "Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit", Mathematische Annalen, 93: 1-36.
	plato.stanford.edu /entries/hilbert-program (7534 words)

	Citebase - The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program (Site not responding. Last check: 2007-10-13)
		The main innovation was the invention of the epsilon-calculus, on which Hilbert's axiom systems were based, and the development of the epsilon-substitution method as a basis for consistency proofs.
		The paper traces the development of the "simultaneous development of logic and mathematics" through the epsilon-notation and provides an analysis of Ackermann's consistency proofs for primitive recursive arithmetic and for the first comprehensive mathematical system, the latter using the substitution method.
		Ackermann, W.:1928b, `Zum Hilbertschen Aufbau der reellen Zahlen'.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0102189 (706 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		December 24 - Wilhelm Ackermann, German mathematician (b.
		Wilhelm Ackermann died in L denscheid, Germany on December 24, 1962.
		The question goes back to Gottfried Leibniz, who in the seventeenth century, after having constructed a successful mechanical calculating machine, dreamt of building a machine that could manipulate symbols in order to determine the truth values of mathematical statements.
	www.worldhistory.com /wiki/W/Wilhelm-Ackermann.htm (221 words)

	Richard Zach: Hilbert's Finitism: Historical, Philosophical and Metamathematical Perspectives
		In this respect Wilhelm Ackermann's 1924 dissertation is a milestone both in the development of the Program and in proof theory in general.
		Ackermann gives a consistency proof for a second-order version of primitive recursive arithmetic which, surprisingly, explicitly uses a finitistic version of transfinite induction up to
		The acceptance of non-primitive recursive methods in Ackermann's dissertation presented in Chapter 3, together with additional textual evidence presented in Chapter 4, shows that this identification is untenable as far as Hilbert's conception of finitism is concerned.
	www.ucalgary.ca /~rzach/papers/hilbert.html (573 words)

	Ackermann - Wikipedia, the free encyclopedia
		Ernst Christian Wilhelm Ackermann (1761–1835), Bohemian public servant
		Rudolph Ackermann (1764–1834), German book trader and inventor
		This human name article is a disambiguation page — a list of pages that might otherwise share the same title, which is a person's or persons' name.
	en.wikipedia.org /wiki/Ackermann (102 words)

	[No title]
		Ackermann Christina Schneider Maria Schiven Catharina Weber Catharina Weber Anna Maria Wertmann Maria Barbara Wertmann Barbara Wertmann Elisabeth Segler Catharina Mauser Maria Schoen ------------------------------------------------------- UNION LUTHERAN & REFORMED CHURCH Hemlock Twp.
		Wilhelm Ingold and wife Anna Maria Child - Hanna Born - Mar 7, (no year given) Bap.
		Wilhelm Ingold and wife Anna Maria Child - Elisabeth Born - Dec 11, 1810 Bap.
	ftp.rootsweb.com /pub/usgenweb/pa/columbia/church/union001.txt (3823 words)

	More on Ackermanns Function
		A few months ago, I posted an algorithm for Ackermann's function, but I did not give a very good background or explanation for it.
		In the 1920's the German logician and mathematician Wilhelm Ackermann defined a function that now bears his name.
		The special properties of the Ackermann function are a consequence of it's phenominal rate of growth.
	www.fortunecity.com /skyscraper/false/780/ack.html (151 words)

	Directorio - Historia (Site not responding. Last check: 2007-10-13)
		Bernoulli, Daniel (1700-1782) Most important work considered the basic properties of fluid flow, pressure, density and velocity, and gave their fundamental relationship now known as Bernoulli's principle.
		Bessel - Friedrich Wilhelm Bessel (1784-1846) Catalogued stars, predicted a planet beyond Uranus as well as the existence of dark stars, investigated Johann Kepler's problem of heliocentricity, and systematized the mathematical functions involved, which now bear his name.
		Leibniz - Gottfried Wilhelm Leibniz (1646-1716) Invented the differential and integral calculus (independently of Sir Isaac Newton),
	www.satd.uma.es /matap/svera/links/matnetg12.html (7133 words)

	[No title]
		Ackermann +------------------------------------------------------------ Ackermann Ackermann Wilhelm (1896-1962) +------------------------------------------------------------
		Bessel +------------------------------------------------------------ Bessel Friedrich Wilhelm, (1784-1846) calculated orbit of Halley's orbit as 20 year old.
		Blaschke +------------------------------------------------------------ Blaschke Blaschke Wilhelm (1885-1962) +------------------------------------------------------------
	www.math.harvard.edu /~knill/sofia/data/mathematicians.txt (6427 words)

	Who Can Name the Bigger Number?
		Ackermann’s idea was to create an endless procession of arithmetic operations, each more powerful than the last.
		Ackermann number is bigger than X. The inverse grows as slowly as Ackermann’s original sequence grows quickly; for all practical purposes, the inverse is at most 4.
		Ackermann numbers are pretty big, but they’re not yet big enough.
	www.scottaaronson.com /writings/bignumbers.html (6372 words)

	[No title]
		Wilhelm Ackermann}}, journal= JSL, volume = 1, number = 1, month = {Mar}, year = 1936, pages = {43-44} } @Article{re:Bernays36c, author = {Paul Bernays}, title = {{Review of: On the Independence of Hilbert and Ackermann's Postulates for the Calculus of Propositional Functions.
		Wilhelm Ackermann}}, journal= JSL, volume = 3, number = 2, month = {Jun}, year = 1938, pages = {85} } @Article{re:Bernays38b, author = {Paul Bernays}, title = {{Review of: A Purely Topological Form of Non-Aristotelian Logic.
		W. Ackermann}}, journal= JSL, volume = 22, number = 1, month = {Mar}, year = 1957, pages = {68-72} } @Article{re:Bernays57b, author = {Paul Bernays}, title = {{Review of: Creative Sets.
	www.phil.cmu.edu /projects/bernays/bernays-reviews.bib (2809 words)

	1896 - Article and Reference from OnPedia.com
		January 4 - Utah is admitted as the 45th U.S. state.
		January 5 - An Austrian newspaper reports that Wilhelm Rntgen discovered a type of radiation later known as X-rays.
		January 12 - H.L. Smith takes the first X-ray photograph.
	www.onpedia.com /encyclopedia/1896 (596 words)

	[No title]
		To show the existence of certain difficult-to-compute functions, mathematicians have invoked the Ackermann numbers (named after Wilhelm Ackermann of the Gymnasien in Luedenscheid, Germany), which compose a rapidly growing sequence that runs: 0, 1, 2^2, 3^3^^3^^^3....
		The fourth Ackermann number, involving exponentiated 3's, is approximately 10^3,638,334,640,024.
		Compared with the fifth Ackermann number the mighty googolplex is but a spit in the proverbial bucket.
	cryptome.sabotage.org /googol.txt (3200 words)

	heiner ackermann - ResearchIndex document query (Site not responding. Last check: 2007-10-13)
		of Z the ag subunits as previously suggested Ackermann and 28 Taylor, 1997 Ackermann et al.
		where ff(n) is the functional inverse of the Ackermann function.
		the primitive predicates of the theory (Hilbert &Ackermann 1999, pp.
	citeseer.ist.psu.edu /cis?q=Heiner+Ackermann (562 words)

	Carnegie Mellon Department Of Philosophy: Research
		It wasn't until almost two decades later, however, that this problem could be formulated in a clear and methodologically sound way.
		Doing so involved extensive foundational work by Hilbert and his collaborators (among them Paul Bernays and Wilhelm Ackermann), who, strongly influenced by Russell and Whitehead's Principia Mathematica, helped to lay the groundwork for modern mathematical logic.
		Around 1922, Hilbert introduced the subject of proof theory as a means of addressing the consistency problem: viewing proofs in formalized axiomatic theories as objects of investigation, the new theory was to establish - using only restricted, finitist means - that such proofs cannot lead to a contradiction.
	www.hss.cmu.edu /philosophy/research-proof.php (1488 words)

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