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### The axioms of constructive geometry

Jan von Plato
1995 Annals of Pure and Applied Logic
I first give the axioms of a general plane geometry of apartness and convergence.  ...  Elementary geometry can be axiomatized constructively by taking as primitive the concepts of the apartness of a point from a line and the convergence of two lines, instead of incidence and parallelism  ...  Acknowledgements I am much indebted to Per Martin-Liif for his active interest in the development of constructive geometry. The type logic used in Section 11 stems from joint efforts with  ...

### Logic of Ruler and Compass Constructions [chapter]

Michael Beeson
2012 Lecture Notes in Computer Science
We cannot go the other way by defining multiplication and addition geometrically without the strong parallel axiom.  ...  We also take as an axiom ¬¬B(x, y, z) → B(x, y, z), or "Markov's principle for betweenness", enabling us to drop double negations on atomic sentences. Here betweenness is strict.  ...  for the Parallel Axioms Models of the Elementary Constructions ECG proves to exist, can be constructed with ruler and compass Three versions of the parallel postulate Constructive Geometry and Euclidean  ...

### Page 3071 of Mathematical Reviews Vol. , Issue 86g [page]

1986 Mathematical Reviews
From a given model of axioms 1-5 is constructed an orthogonal geometry, satisfying the point-hyperplane axioms of the previous paper.  ...  geometry, the construction of a geometry from the group is facilitated if we can construct an orthogonal geometry from its points and orthocomplemented hyperplanes.  ...

### Herbrand's theorem and non-Euclidean geometry [article]

Michael Beeson, Pierre Boutry, Julien Narboux
2015 arXiv   pre-print
We use Herbrand's theorem to give a new proof that Euclid's parallel axiom is not derivable from the other axioms of first-order Euclidean geometry.  ...  Previous proofs involve constructing models of non-Euclidean geometry.  ...  Here k = 2 and the constructed points are indicated by the open circles. Table 1 . 1 Tarski's axioms for geometry There is no "standard" name for this axiom.  ...

### A constructive version of Tarski's geometry

Michael Beeson
2015 Annals of Pure and Applied Logic
Constructivity, in this context, refers to a theory of geometry whose axioms and language are closely related to ruler and compass constructions.  ...  It may also refer to the use of intuitionistic (or constructive) logic, but the reader who is interested in ruler and compass geometry but not in constructive logic, will still find this work of interest  ...  They connect the axioms of geometry with ruler and compass constructions and, in the case of Pasch's axiom, with its intuitive justification.  ...

### A Coq-Based Library for Interactive and Automated Theorem Proving in Plane Geometry [chapter]

Tuan-Minh Pham, Yves Bertot, Julien Narboux
2011 Lecture Notes in Computer Science
Guilhot for interactive theorem proving at the level of high-school geometry [7], where we eliminate redundant axioms and give formalizations for the geometric concepts using a vector approach.  ...  This library includes the basic geometric notions to state theorems and provides a database of theorems to construct interactive proofs more easily. It is an extension of the library of F.  ...  She first constructs a vector space attached to the affine geometry, then she construct an Euclidean space by adding the notion of scalar product to the affine geometry.  ...

### A Constructive Version of Tarski's Geometry [article]

Michael Beeson
2015 arXiv   pre-print
Tarski's elegant and concise first-order theory of Euclidean geometry, on the other hand, is essentially non-constructive, even if we restrict attention (as we do here) to the theory with line-circle and  ...  The third version has a function symbol for the intersection point of two non-parallel, non-coincident lines, instead of only for intersection points produced by Pasch's axiom and the parallel axiom; this  ...  They connect the axioms of geometry with ruler and compass constructions and, in the case of Pasch's axiom, with its intuitive justification.  ...

### HERBRAND'S THEOREM AND NON-EUCLIDEAN GEOMETRY

MICHAEL BEESON, PIERRE BOUTRY, JULIEN NARBOUX
2015 Bulletin of Symbolic Logic
We use Herbrand's theorem to give a new proof that Euclid's parallel axiom is not derivable from the other axioms of first-order Euclidean geometry.  ...  Previous proofs involve constructing models of non-Euclidean geometry.  ...  Here k = 2 and the constructed points are indicated by the open circles. Table 1 . 1 Tarski's axioms for geometry There is no "standard" name for this axiom.  ...

### On the equivalence of Playfair's axiom to the parallel postulate [article]

Elizabeth T. Brown, Emily Castner, Stephen Davis, Edwin O'Shea, Edouard Seryozhenkov, AJ Vargas
2019 arXiv   pre-print
In particular, we construct a non-SAS geometry that models the Playfair axiom but not the parallel postulate.  ...  We show that the classical equivalence of Euclid's parallel postulate and Playfair's axiom collapses in the absence of triangle congruence.  ...  If so, does this provision form an intermediate axiom between non-SAS and SAS in the presence of Hilbert's other axioms?  ...

### Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries

Marvin Jay Greenberg
2010 The American mathematical monthly
By "elementary" plane geometry I mean the geometry of lines and circles-straightedge and compass constructions-in both Euclidean and non-Euclidean planes.  ...  Hilbert not only made Euclid's geometry rigorous, he investigated the minimal assumptions needed to prove Euclid's results, he showed the independence of some of his own axioms from the others, he presented  ...  I also thank the referees for their helpful suggestions.  ...

### Degree of Negation of an Axiom [article]

Florentin Smarandache
2010 arXiv   pre-print
(or a degree of negation) of an axiom in geometry.  ...  The most important contribution of this article is the introduction of the degree of negation (or partial negation) of an axiom and, more general, of a scientific or humanistic proposition (theorem, lemma  ...  The most important contribution of Smarandache geometries was the introduction of the degree of negation of an axiom (and more general the degree of negation of a theorem, lemma, scientific or humanistic  ...

### Page 337 of Christian Remembrancer. A Quarterly Review Vol. 10, Issue 50 [page]

1845 Christian Remembrancer. A Quarterly Review
‘To construct such a System requires labour and thought of quite a different kind from that which is requisite in the discussion of the questions, whether Geometry rests upon Axioms?  ...  The construction of the Elements of Geometry, be- sides being the creation of a precious and imperishable body of Scientific Truth, was the first step in the Philosophy of Geometry.  ...

### Poincaré's review of Hilbert's " foundations of geometry."

H. Poincaré
1903 Bulletin of the American Mathematical Society
THE LIST OF AXIOMS. -The first thing to do was to enumerate all the axioms of geometry.  ...  We wish to construct our geometry without making use of the metrical axioms ; the word length has then for us no meaning ; we have no right to use the compass ; on the other hand, we may use the ruler,  ...  Lobachevsky and Riemann rejected the postulate of Euclid, but they preserved the metrical axioms ; in the majority of his geometries, Professor Hilbert does the opposite.  ...

### Tarski's System of Geometry

Alfred Tarski, Steven Givant
1999 Bulletin of Symbolic Logic
of independence of axioms and primitive notions, and versions of the system suitable for the development of 1-dimensional geometry.  ...  It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the history of specific axioms, the questions  ...  of betweenness, and hence is useful in the construction of an axiom set for affine geometry.  ...

### Axiomatizing Changing Conceptions of the Geometric Continuum II: Archimedes-Descartes-Hilbert-Tarski†

John T Baldwin
2017 Philosophia Mathematica
the Grundlagen is an immodest axiomatization of any of these geometries.  ...  In this paper we argue: 1) Tarski's axiom set E 2 is a modest complete descriptive axiomatization of Cartesian geometry (Section 2; 2) the theories EG π,C,A and E 2 π,C,A are modest complete descriptive  ...  Since, in the Elements the limit always has a simple description, the construction of the sequence can be done within the bounds of elementary geometry; and the question of constructing a sequence for  ...
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