Constructive Geometry and the Parallel Postulate

Euclidean geometry consists of straightedge-and-compass constructions and reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. We consider three versions of Euclid's parallel postulate: Euclid's own formulation in his Postulate 5; Playfair's 1795 version, and a new version we call the strong parallel postulate. These differ in that Euclid's version and the new version both assert the existence of a point where two lines meet, while Playfair's version makes no existence assertion. Classically, the models of Euclidean (straightedge-and-compass) geometry are planes over Euclidean fields. We prove a similar theorem for constructive Euclidean geometry, by showing how to define addition and multiplication without a case distinction about the sign of the arguments. With intuitionistic logic, there are two possible definitions of Euclidean fields, which turn out to correspond to the different versions of the parallel axiom. In this paper, we completely settle the questions about implications between the three versions of the parallel postulate: the strong parallel postulate easily implies Euclid 5, and in fact Euclid 5 also implies the strong parallel postulate, although the proof is lengthy, depending on the verification that Euclid 5 suffices to define multiplication geometrically. We show that Playfair does not imply Euclid 5, and we also give some other independence results. Our independence proofs are given without discussing the exact choice of the other axioms of geometry; all we need is that one can interpret the geometric axioms in Euclidean field theory. The proofs use Kripke models of Euclidean field theories based on carefully constructed rings of real-valued functions.Comment: 114 pages, 39 figure

Some new results on decidability for elementary algebra and geometry

We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic. We find that the purely universal and purely existential fragments of the theory of normed spaces are decidable, as is the AE fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbert's 10th problem show that the EA fragments for metric and normed spaces and the AE fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3

From mathematical axioms to mathematical rules of proof : recent developments in proof analysis

A short text in the hand of David Hilbert, discovered in Gottingen a century after it was written, shows that Hilbert had considered adding a 24th problem to his famous list of mathematical problems of the year 1900. The problem he had in mind was to find criteria for the simplicity of proofs and to develop a general theory of methods of proof in mathematics. In this paper, it is discussed to what extent proof theory has achieved the second of these aims. This article is part of the theme issue 'The notion of 'simple proof' - Hilbert's 24th problem'.Peer reviewe

Note di Matematica 26

Abstract. We point out the geometric significance of a part of the theorem regarding the maximality of the orthogonal group in the equiaffine group proved in Keywords: Erlanger Programm, definability, Lω 1 ω -logic MSC 2000 classification: 03C40, 14L35, 51F25, 51A99 A. Schleiermacher and K. Strambach [12] proved a very interesting result regarding the maximaility of the group of orthogonal transformations and of that of Euclidean similarities inside certain groups of affine transformations. Although similar results have been proved earlier, this is the first time that the base field for the groups in question was not the field of real numbers, but an arbitrary Pythagorean field which admits only Archimedean orderings. They also state, as geometric significance of the result regarding the maximality of the group of Euclidean motions in the unimodular group over the reals, that there is "no geometry between the classical Euclidean and the affine geometry". The aim of this note is to point out the exact geometric meaning of the positive part of the 2-dimensional part their theorem, in the case in which the underlying field is an Archimedean ordered Euclidean field. In this case their theorem states that: (1) the group G 1 of Euclidean isometries is maximal in the group H 1 of equiaffinities (affine transformations that preserve non-directed area), and that (2) the group G 2 of Euclidean similarities is maximal in the group H 2 of affine transformations. The restriction to the 2-dimensional case is not essential but simplifies the presentation. The geometric counterpart of group-theoretic results in the spirit of the Erlanger Programm is given by Beth's theorem, as was emphasized by Büch