A common axiom set for classical and intuitionistic plane geometry

Lombard, Melinda; Vesley, R. E.

doi:10.1016/s0168-0072(98)00017-7

Cited by 14 publications

(11 citation statements)

References 8 publications

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“…. This is shown in Lombard and Vesley (1998), but is merely a straightforward consequence of Kleene' s realizability notions (1952 ;with Vesley 1965). Combined with classical logic these axioms produce a system CEG equivalent to Tarski' s T, whereas combined with intuitionistic logic they give an intuitionistic geometry IEG , which is a proper subsystem of CEG .…”

Section: Richard Vesleymentioning

confidence: 97%

“…In Lombard and Vesley (1998) a prim itive predicate with this eOE ective enumerability property in the intended m odel is chosen as the sole primitive for a system of constructive geom etry, patterned after Tarski' s T. The predicate Dist(x,y,z,u,v,w) is m ore com plicated than g or ! .…”

Section: Richard Vesleymentioning

confidence: 99%

“…Suppose to the constructive geometry IEG of Lombard and Vesley (1998) is added an additional axiom : A " h C A " , where A " is the form ula which asserts the decidability of equality for points ; so A " is c xc y(x¯y h C x¯y). The ® rst view, under which a formal theory is constructive if it has the h -property, disregards all other familiar attributes of constructive theories, in particular the semantic interpretations which are often considered to be their foundations.…”

Section: Richard Vesleymentioning

confidence: 99%

“…The last derives from Brouwer and Heyting, with recent work by van Dalen (1990), von Plato (1994 and Lombard and Vesley (1998). The ® rst, urged by a referee of Lom bard and Vesley (1998), is proof-theoretic.…”

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Constructivity in Geometry

Vesley

1999

History and Philosophy of Logic

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W e review and contrast three ways to make up a forma l Euclidean geometry which one m ight call construc tive, in a computa tional sense. The starting poin t is the ® rst-orde r geom etry created by Tarski.Through much of his life Alfred Tarski worked oOEand on to achieve a neat ® rst order form ulation of Euclidean plane geom etry. The ® nal form ulations are included in Schwabhau$ ser et al. (1983) andTarski (1959), while the history of this development is reviewed in Szczerba (1986). As Tarski (1959) observed, the question whether this theory captures accurately the geom etry intended by Euclid is unrewarding. But it does capture a theory which deserves the m odern title of elem entary or ® rst order Euclidean plane geometry, as is supported by Tarski' s proof that the theory is complete.A notable feature of Tarski' s system T is its economy of language. Only one sort of variable is used, ranging over points. In addition to equality, only two prim itive predicates are introduced, B(a,b,c) for`point b is between points a and c ' and D (a,b,c,d) for`a is the same distance from b as c is from d ' . In this lim ited language, all the fam iliar results of Euclidean plane geom etry can be represented. A similarly spare axiom set allows proofs of these results, including theorems which assert the existence of points obtained by familiar Euclidean constructions, like that for ® nding the perpendicular bisector of a line segment, and also theorems which are less immediately correlated to Euclidean constructions, like the congruence theorem s for triangles.How much of this theory T is constructive ? This question did not concern Tarski, but it does suggest some interesting investigations. Evidently, any answers will depend on our ® xing sharply the meaning of the question. W e may see ourselves as continuing a centuries-old discussion concerning geometric constructions, dating from celebrated questions in which constructions are limited to those involving compass and straight edge, and generalizing now to broader views of what might be allowed in constructions.In this brief paper, three approaches to constructivity in Euclidean plane geometry are reviewed and put into context. The ® rst, urged by a referee of Lom bard and Vesley (1998), is proof-theoretic. The second has been developed by M oler and Suppes (1968) and Pam buccian (1989, 1992). The last derives from Brouwer and Heyting, with recent work by van Dalen (1990), von Plato (1994 and Lombard and Vesley (1998). There are other approaches possible and the last two of those mentioned actually subdivide into a number of related approaches.The ® rst view regards a formal geometry as constructive if there is an algorithm determining for any closed formula A whether or not A is a theorem , symbolically whether r-A or not r-A. As a consequence of com pleteness, Tarski' s formal geometry T has this property. Thus it is constructive in the sense that in principle no ingenuity is required in m aking proofs or refutations. Rather, there is a single master or universal algo...

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Section: Richard Vesleymentioning

confidence: 97%

Section: Richard Vesleymentioning

confidence: 99%

Section: Richard Vesleymentioning

confidence: 99%

mentioning

confidence: 96%

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Constructivity in Geometry

Vesley

1999

History and Philosophy of Logic

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“…Further work on affine and projective geometry was done by Heyting and van Dalen, an account of which can be found in [136]. The intuitionistic counterpart to Tarski's first-order axiomatization of Euclidean geometry was obtained in [58]. All of these use quantifiers, and thus do not qualify as constructive in the sense used in this paper.…”

Section: Intuitionistic Constructive Geometrymentioning

confidence: 99%

Axiomatizing geometric constructions

Pambuccian

2008

Journal of Applied Logic

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In this survey paper, we present several results linking quantifier-free axiomatizations of various Euclidean and hyperbolic geometries in languages without relation symbols to geometric constructibility theorems. Several fragments of Euclidean and hyperbolic geometries turn out to be naturally occurring only when we ask for the universal theory of the standard plane (Euclidean or hyperbolic), that can be expressed in a certain language containing only operation symbols standing for certain geometric constructions.

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