[FOM] Query: references on intuitionistic Euclidean geometry
vladik at utep.edu
Sun Nov 22 13:05:11 EST 2009
One more related reference:
O. M. Kosheleva, "Hilbert Problems (Almost) 100 Years Later
(From the Viewpoint of Interval Computations)",
Reliable Computing, 1998, Vol. 4, No. 4, pp. 399-403.
give an example of a problem where an algorithm was not expected: 3rd,
the axiomatization of volume in elementary geometry. Traditional
a volume requires not only additivity but also the so-called method of
* In the plane, every two polygons of equal area are equi-decomposable,
therefore, any additive function on polygons is either an area or a
of an area.
* In 3D case, in 1900, it was not known whether any two polytopes of
volume are equi-decomposable.
Dehn has proved that some are not, and moreover, he proved the existence
an additive function that is neither a volume nor a function of a
function was strongly non-constructive (used axiom of choice), and it
believed that no constructive function of this type is actually
possible. This was
proven in ."
 O. M. Kosheleva, "Axiomatization of volume in elementary geometry",
Siberian Mathematical Journal, 1980, Vol. 21, No. 1, pp. 106-114 (in
sian); English translation: Siberian Mathematical Journal, 1980, Vol.
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf
Of Arnon Avron
Sent: Sunday, November 22, 2009 5:34 AM
To: fom at cs.nyu.edu
Subject: Re: [FOM] Query: references on intuitionistic Euclidean
Another work that should perhaps be mentioned is a series
of papers by Victor Pambuccian about constructive geometry.
The last published one is:
Constructive Axiomatizations of Plane Absolute, Euclidean and
Hyperbolic Geometry. Math. Log. Q. 47(1): 129-136 (2001)
(see there for references to other papers on the subject by the same
> Dear FOMers,
> a student of mine plans to write a thesis on Euclidean geometry
> developed over intuitionistic logic. I would appreciate any reference
> books, papers, or any other source on this topic.
> Thank you very much in advance for collaboration
> Giovanni Sambin
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