### Abstract

We provide a universal axiom system for plane hyperbolic geometry in a first-order language with two sorts of individual variables, 'points' (upper-case) and 'lines' (lower-case), containing three individual constants, A_{0}, A_{1}, A_{2}, standing for three non-collinear points, two binary operation symbols, φ and ι, with φ(A, B) = ι to be interpreted as 'ι is the line joining A and B' (provided that A ≠ B, an arbitrary line, otherwise), and ι(g, h) = P to be interpreted as 'P is the point of intersection of g and h (provided that g and h are distinct and have a point of intersection, an arbitrary point, otherwise), and two binary operation symbols, π_{1}(P, ι) and π_{2}(P, ι), with π_{i}(P, ι) = g (for i = 1, 2) to be interpreted as 'g is one of the two limiting parallel lines from P to ι (provided that P is not on ι, an arbitrary line, otherwise).

Original language | English (US) |
---|---|

Pages (from-to) | 475-488 |

Number of pages | 14 |

Journal | Mathematical Logic Quarterly |

Volume | 47 |

Issue number | 4 |

DOIs | |

State | Published - 2001 |

### Fingerprint

### Keywords

- Constructive axiomatization
- Hyperbolic geometry

### ASJC Scopus subject areas

- Logic

### Cite this

**Constructive axiomatization of plane hyperbolic geometry.** / Pambuccian, Victor.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Constructive axiomatization of plane hyperbolic geometry

AU - Pambuccian, Victor

PY - 2001

Y1 - 2001

N2 - We provide a universal axiom system for plane hyperbolic geometry in a first-order language with two sorts of individual variables, 'points' (upper-case) and 'lines' (lower-case), containing three individual constants, A0, A1, A2, standing for three non-collinear points, two binary operation symbols, φ and ι, with φ(A, B) = ι to be interpreted as 'ι is the line joining A and B' (provided that A ≠ B, an arbitrary line, otherwise), and ι(g, h) = P to be interpreted as 'P is the point of intersection of g and h (provided that g and h are distinct and have a point of intersection, an arbitrary point, otherwise), and two binary operation symbols, π1(P, ι) and π2(P, ι), with πi(P, ι) = g (for i = 1, 2) to be interpreted as 'g is one of the two limiting parallel lines from P to ι (provided that P is not on ι, an arbitrary line, otherwise).

AB - We provide a universal axiom system for plane hyperbolic geometry in a first-order language with two sorts of individual variables, 'points' (upper-case) and 'lines' (lower-case), containing three individual constants, A0, A1, A2, standing for three non-collinear points, two binary operation symbols, φ and ι, with φ(A, B) = ι to be interpreted as 'ι is the line joining A and B' (provided that A ≠ B, an arbitrary line, otherwise), and ι(g, h) = P to be interpreted as 'P is the point of intersection of g and h (provided that g and h are distinct and have a point of intersection, an arbitrary point, otherwise), and two binary operation symbols, π1(P, ι) and π2(P, ι), with πi(P, ι) = g (for i = 1, 2) to be interpreted as 'g is one of the two limiting parallel lines from P to ι (provided that P is not on ι, an arbitrary line, otherwise).

KW - Constructive axiomatization

KW - Hyperbolic geometry

UR - http://www.scopus.com/inward/record.url?scp=0035541188&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035541188&partnerID=8YFLogxK

U2 - 10.1002/1521-3870(200111)47:4<475::AID-MALQ475>3.0.CO;2-S

DO - 10.1002/1521-3870(200111)47:4<475::AID-MALQ475>3.0.CO;2-S

M3 - Article

VL - 47

SP - 475

EP - 488

JO - Mathematical Logic Quarterly

JF - Mathematical Logic Quarterly

SN - 0942-5616

IS - 4

ER -