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	Heyting algebra
		Heyting algebras arise as models of intuitionistic logic, a logic in which the law of excluded middle does not in general hold.
		A Heyting algebra H is a bounded lattice such that for all a and b in H there is a greatest element x of H such that a ^ x ≤ b.
		Arend Heyting (1898-1980) was himself interested in clarifying the foundational status of intuitionistic logic, in introducing this type of structures.
	encyclopedia.codeboy.net /wikipedia/h/he/heyting_algebra.html (948 words)

	Encyclopedia: Arend Heyting (Site not responding. Last check: 2007-11-03)
		Heyting arithmetic is the basic arithmetic of intuitionism (not to be confused with Heyting algebra).
		Heyting was not well placed to make contact with colleagues in the universities, yet he spent all his free time working on his research.
		Heyting moved away from these big problems, concentrating on trying to identify formal, intuitive, and logical concepts in the study of mathematics.
	www.nationmaster.com /encyclopedia/Arend-Heyting (518 words)

	Heyting (Site not responding. Last check: 2007-11-03)
		Although Heyting's father was a successful school teacher, the family were still in financial problems when Heyting began his studies in 1916 at the University of Amsterdam.
		Although Heyting's version of intuitionist logic differed somewhat from that of Brouwer, it is clear that one of his main aims was to make Brouwer's ideas more accessible and better known.
		Heyting was appointed as a Privatdozent at the University of Amsterdam in 1936 and in the following year he was appointed as a lecturer.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Heyting.html (1200 words)

	[No title]
		Abstract: A divisibility test of Arend Heyting, for polynomials over a field in an intuitionistic setting, may be thought of as a kind of division algorithm.
		The model that Heyting had in mind was the real numbers, with apartness being a positive form of inequality: two real numbers are apart if we can find a positive rational number that bounds them away from one another.
		Heyting does not assume that law, so being a unit may be stronger than being nonzero, which is the whole point of introducing the apartness.
	www.math.fau.edu /richman/Docs/heyting2.html (1634 words)

	Arend Heyting (Site not responding. Last check: 2007-11-03)
		Arend Heyting (May 9, 1898 – July 9, 1980) was a Dutch mathematician and logician.
		He was born in Amsterdam, Netherlands, and died in Lugano, Switzerland.
		Kolmogorov, Heyting and Gentzen on the intuitionistic logical constants *.
	hallencyclopedia.com /Arend_Heyting (136 words)

	Thematic Afternoon on Constructivism (Site not responding. Last check: 2007-11-03)
		The Arend Heyting Foundation came into being as the result of the last will of ms J.F. Heyting-van Anrooij, who died in september 1998.
		heyting stipulated in he last will that the first Arend Heyting lecture should be devoted to workconnected with Heytings own work on intuitionism.
		In "Untersuchungen über intuitionistische Algebra", Heyting reduces the question of whether a nonzero polynomial with coefficients in a field divides another polynomial, to whether a finite number of field elements are zero.
	staff.science.uva.nl /~anne/heyting.html (685 words)

	Reference.com/Encyclopedia/Heyting algebra
		A Heyting algebra H is a bounded lattice such that for all a and b in H there is a greatest element x of H such that
		Unless all elements of the Heyting algebra are regular, this Boolean algebra will not be a sublattice of the Heyting algebra, because its join operation will be different.
		A Heyting algebra, from the logical standpoint, is essentially a generalization of the usual system of truth values.
	www.reference.com /browse/wiki/Heyting_algebras (790 words)

	Philosophical Fortnights: Friday nights at the Library
		One moral to draw from Wood’s treatment of lattices (and the treatment of Heyting lattices and Boolean algebras that follows) is that not all axiomatizations of a theory are equal.
		(Arend Heyting was a student of L. Brouwer, the founder of intuitionism.
		Heyting algebras are models for intuitionistic logic, and Boolean algebras of course for classical logic, which is thus situated in a spectrum of logics.
	tlonuqbar.typepad.com /phfn/2005/12/friday_nights_a.html (7772 words)

	References for Heyting (Site not responding. Last check: 2007-11-03)
		J Alcolea Banegas, Arend Heyting (Spanish), Mathesis (México) 4 (2) (1988), 189-220.
		B A Kushner, Arend Heyting : a short sketch of his life and work (Russian), Methodological analysis of the foundations of mathematics 'Nauka' (Moscow, 1988), 121-135.
		A S Troelstra, Arend Heyting and his Contribution to Intuitionism, Nieuw archief voor wiskunde 29 (1981), 1-23.
	www-groups.dcs.st-and.ac.uk /history/References/Heyting.html (149 words)

	Luitzen Egbertus Jan Brouwer
		Brouwer's principal students were Maurits Belinfante and Arend Heyting; the latter, in turn, was the teacher of Anne Troelstra and Dirk van Dalen.
		Cooling off of his relationship with Arend Heyting, his successor at the post of director of the Mathematical Institute, as a result of disagreement over the exact role the retired Brouwer could still play there.
		A comparison of the arguments for intuitionism advanced by, respectively, Brouwer, Heyting, and Dummett, in particular with respect to the possibility of intersubjective validity of intuitionistic mathematics.
	plato.stanford.edu /entries/brouwer (4195 words)

	Citations: North-Holland - Heyting (ResearchIndex) (Site not responding. Last check: 2007-11-03)
		For example, a proof of A and B is a pair of a proof of A and a proof of B, so the logical notion, conjunction, can be represented by the type theoretic notion of cartesian product.
		and related citations there) However, it was argued by Heyting [56] that, without negation, there would be no calculus of propositions because only true propositions make sense, and thus the logic of negationless mathematics would be difficult to formalise.
		Heyting, Intuitionism, an Introduction, 3rd rev. ed., North-Holland, Amsterdam, 1971.
	citeseer.ist.psu.edu /context/318/0 (1526 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Although Arend Heyting showed in 1930 that one could formally redefine negation to that end, most mathematicians, including many otherwise sympathetic to the program of avoiding completed infinities, sensed incoherence.
		Unsurprisingly, Kurt G”del's demonstration in 1932 that Heyting's calculus could be reinterpreted "in terms of the concepts of the usual sentential logic and of the concept p is provable'" was read by most mathematicians as having given sense to a system previously unintelligible.
		Heyting's Intuitionism: An Introduction (Amsterdam: North-Holland, 1956) remains to this day the best introduction to the subject in English.
	www.hanover.edu /philos/film/vol_02/cameron.htm (6432 words)

	Is Mathematics a Scientific Discipline?
		Arend Heyting, "The Intuitionist Foundations of Mathematics," Philosophy of Mathematics, ed.
		Arend Heyting, Intuitionism: An Introduction (3rd rev. ed., 1971), p.
		Arend Heyting, "After Thirty Years," Logic, Methodology, and Philosophy of Science, ed.
	www.henryflynt.org /studies_sci/mathsci.html (9178 words)

	Constructive Mathematics
		Secondly, there is a model theory (Kripke models) in which it can be shown that LPO is not derivable in Heyting arithmetic — that is, Peano arithmetic using the computational interpretation of the connectives and quantifiers that we state in more detail in the next section (Bridges and Richman 1987, Chapter 7).
		The desire to retain the possibility of a computational interpretation is one motivation for using the constructive reinterpretations of the logical connective and quantifiers that we gave above; but it is not exactly the motivation of the pioneers of constructivism in mathematics.
		In 1930, Brouwer's most famous pupil, Arend Heyting, published a set of formal axioms which so clearly characterise the logic used by the intuitionist that they have become universally known as the axioms for intuitionistic logic (Heyting 1930).
	plato.stanford.edu /entries/mathematics-constructive (6375 words)

	Luitzen Egbertus Jan Brouwer
		Among Brouwer's assistants were Heyting, Hans Freudenthal, Karl Menger, and Witold Hurewicz, the latter two of whom were not intuitionistically inclined.
		In a style that is more down-to-earth and oecumenical than Brouwer's, Heyting presents the intuitionistic versions of various basic subjects in everyday mathematics.
		Brouwer and Heyting have some philosophical disagreements that make a difference in their appreciation of some aspects of intuitionistic mathematics.
	www.science.uva.nl /~seop/archives/fall2005/entries/brouwer (3679 words)

	I P M - Homepage (Site not responding. Last check: 2007-11-03)
		One of the most important consequences of this limitation is the restriction of formal proofs to direct proofs.
		In 1930, Brouwer's student Arend Heyting gave the first axiomatization of intuitionistic logic.
		The first comprehensive study of Kripke models of Heyting Arithmetic $HA$ (the intuitionistic counterpart of first order Peano Arithmetic $PA$) was done by Craig Smorynski in his PhD thesis appeared in 1973.
	www.ipm.ac.ir /ipm/activities/ViewProgramInfo.jsp?PTID=206 (507 words)

	Citations: Mathematische Grundlagenforschung - Heyting (ResearchIndex) (Site not responding. Last check: 2007-11-03)
		Since irreducible proofs explicitly represent mathematical constructions, reduction rules for intuitionistic logic turn out to have a computational meaning.
		, in the form stated by Heyting [40] states that a proof of an implication P Q is a construction which transforms any proof of P into a proof of Q. This idea was formalized independently by Kleene s realizability interpretation [46,47] in which proofs of intuitionistic number theory are....
		The Brower Heyting Kolmogorov (BHK) interpretation [15,51,40] in the form stated by Heyting
	citeseer.ist.psu.edu /context/2865/0 (524 words)

	Nat' Academies Press, Biographical Memoirs V.82 (2003)
		Back at Yale one of the first courses Robinson taught was “Chapters in the History of Mathematics,” offered in the spring of 1968.
		Subsequently, in a Festschrift to honor Arend Heyting he chose to consider in partly historical terms the “ultimate foundation” for mathematics.
		Just as non-Euclidean geometry destroyed faith in Euclidean geometry as the one true geometry of space, so too, Robinson held, did the results of Gödel and Cohen destroy any faith one might have had in the existence of a single, absolutely true set theory.
	www.nap.edu /books/0309086981/html/271.html (463 words)

	AIP International Catalog of Sources
		Consists of papers of Gödel relating to all periods of his life, including scientific correspondence, notebooks, drafts, unpublished manuscrfipts, academic, legal, and financial records, and additional loose notes and memoranda.
		Correspondents include Paul Bernays, William Boone, Rudolf Carnap, Paul J. Cohen, Gotthard Gunther, Jacques Herbrand, Arend Heyting, Georg Kreisel, Karl Memger, Oskar Morgenstern, Abraham Robinson, Paul A. Schilpp, Dana Scott, Gaisi Takeuti, Jean van Heijenoort, Oswald, Veblen, John von Neumann, and Hao Wang.
		Prior to its arrangement in 1983-84, the collection was stored in filing cabinets and moving cartons in the basement of the Historical Studies Library of the Institute for Advanced Study (Princeton, N.J.).
	www.aip.org /history/catalog/icos/156.html (297 words)

	bibast.html (Site not responding. Last check: 2007-11-03)
		1978A A. Heyting on the formalization of intuitionistic mathematics.
		1981 Arend Heyting and his contribution to intuitionism, Nieuw Archief voor Wiskunde (3) 29, 1--23.
		(Postscript to: A. Heyting, Continuum en Keuzerij bij Brouwer, 125-139.)
	staff.science.uva.nl /~anne/bibast.html (1505 words)

	Letter from Paul La Montagne - September 12, 2002
		It is certainly not the only kind of logic that there is. Mathematicians have devised three valued, multivalued, and even infinite valued logics (see the work of Jan Lukasiewicz and E. Post).
		Moreover, even within the realm of two valued logic, there are formulations in which the law of the excluded middle does not hold, at least when dealing with infinite objects (See the work of L. Brouwer and Arend Heyting’s axiomatization of Brouwer’s position).
		Second of all, it is one of the key limitations of strict rigorous two valued logic that it can only be properly applied to purely formal systems, such as propositional logic or mathematics.
	www.presbyweb.com /2002/Letters/091201.htm (1174 words)

	TAG Meetings: Spring 2000
		A logic programming representation of the blocks world due to Ilkka Niemelae is discussed as an example.
		Vladimir Lifschitz talked about the "logic of here-and-there." This is a 3-valued (monotonic propositional) logic invented 70 years ago by Arend Heyting.
		David Pearce showed how this logic can be used to prove the "monotonic" (or "strong") equivalence of logic programs.
	www.cs.utexas.edu /users/tag/meetings-S00.html (988 words)

	Find in a Library: Logic and foundations of mathematics.
		Find in a Library: Logic and foundations of mathematics.
		Subjects: Heyting, A. -- 1898- -- (Arend),
		To find this item in a library, enter a postal code, state, province, or country in the field above.
	www.worldcatlibraries.org /wcpa/ow/36d3c990f8f36c82.html (52 words)

	CS 486: Applied Logic (Site not responding. Last check: 2007-11-03)
		Formal Number Theory (Peano Arithmetic (PA) and Heyting Arithmetic (HA)): Suppes book
		Arend Heyting showed that these propositions form the core of first-order logic and arithmetic, and that Peano arithmetic (PA) can be factored into Heyting arithmetic (HA) and Aristotle's axiom of excluded middle.
		N.G. deBruijn generalized this computational meaning by defining a proposition as the type of its proofs.
	www.cs.cornell.edu /courses/cs486/2001sp/Summary/summary.html (992 words)

	Godel 5
		H-Z; includes James Halpern, Arend Heyting, Richard Jeffrey, Stephen C. Kleene, Georg Kreisel, Akira Nakamura, J. Robert Oppenheimer, Robert W. Robinson, Dana Scott, William W. Tait, Gaisi Takeuti, and Ernst Zermelo
		Notes or items inserted in books: These items (including both AMs notes and printed material) were found inserted in the books of Gödel's library; they are here filed by author of book.
		Originals are stored at the Heyting Archief, Entschede.
	libweb.princeton.edu /libraries/firestone/rbsc/aids/godel/godel4.html (2113 words)

	[No title]
		Speech, Association for Symbolic Logic, Amsterdam, Holland 16 General Correspondence 17 Heyting, Arend, 1954 18 Schrodinger, Erwin, 1954 19 Notes 20-22 Holograph 23 Typescript "Towards a Liberal Theory of Public Opinion," 1954 September 6-11.
		Includes correspondence with Paul Bernays, Arend Heyting, Stephen C. Kleene, D. MacKinnon, Hao Wang 13 Silvey, S. 14-15 Typescript and holograph.
		Includes correspondence neither to nor from Karl Popper 3 Church, Alonzo, 1964 4 Findlay, Allan F., 1967 5 Freudenthal, H., 1964 6 Godel, Kurt, 1964 7 Henkin, Leon, 1964 8 Heyting, Arend, 1964 80.
	www.eeng.dcu.ie /~tkpw/hoover/hoover.txt (9901 words)

	The Philosophy Family Tree
		Brouwer was in turn the advisor of Maurtis Belinfante and Arend Heyting.
		Heyting was the adivsor of Anne Troelstra and Dirk Van Dalen.
		It seems that Ernst Cassirer was, while at Yale, the main influence on Arthur Pap and Susanne Langer, though I don't know if he was either's official advisor.
	philtree.blogspot.com (10179 words)

	[No title]
		Brouwer's disciple, Arend Heyting, took on the challenge of explaining to the mathematical community what intutionism is all about.
		Contrary to Brouwer's wishes, Heyting formalized intuitionistic logic and intuitionistic number theory.
		Brouwer was furious, but in the end, Heyting's approach won; the intuitionism dealt with today is largely that which Heyting formalized.
	badros.com /greg/doc/philmath.htm (12971 words)

	Practical Foundations of Mathematics
		Dependent Choice does not imply excluded middle or vice versa.
		Logic in a topos For Jan Brouwer and his student Arend Heyting, Intuitionism was a profound philosophy of mathematics [Hey56, Dum77,Man98], but like increasingly many logicians, we shall use the word intuitionistic simply to mean that we do not use excluded middle.
		Joachim Lambek and Philip Scott [LS86] show that the so-called ``term model'' of the language of mathematics, also known as the ``free topos,'' may be viewed as a preferred world, but Gödel's incompleteness theorem shows that the term model of classical mathematics won't do.
	www.geocities.com /yury_bendersky/b/f/s18.html (2566 words)

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