A model for Smarandache's Anti-Geometry
by
Roberto Torretti
Universidad de Chile, Casilla 20017, Correo 20, Santiago, Chile

Anti-Geometry rests on a system of nineteen axioms, each one of which is the negation of one of Hilbert's nineteen axioms. 3 Such wholesale negation brings about a complete collapse of the constraints imposed by Hilbert's axioms on its conceiv able models. The immediate consequence of this is that models of Anti-Geometry can be readily found in all walks of life. 4 On the other hand, and for the same reason, the truths concerning these models that can be obtained from Smarandache's axi oms by deductive inference are somewhat uninteresting, to say the least.

I shall now state my interpretation of the undefined terms in Smarandache's (and Hilbert's) axioms and show, thereupon, that Smarandache's nineteen axioms come out true under this interpretation. Following Chimienti and Bencze,2 I say 'line' where Hilbert says 'straight' ( gerade).5 Points lying on one and the same line are said to be collinear; points or lines lying on one and the same plane are said to be coplanar. Two lines are said to meet or intersect each other if they have a point in common. In my interpretation the geometrical terms employed in the axioms are made to stand for ordinary, non-geometric objects and relations, with which I assume the reader is familiar. As a matter of fact, Smarandache's system, despite its vaunted vanguardistic libertarianism, still imposes a few existential constraints on admissible models; for example, his Axiom III presupposes the existence of infinitely many of the objects called 'lines'. This has forced me to introduce three existence postulates which my model is required to comply with, at least one of which is plainly unnatural (EP3).

A model for Smarandache's Anti-Geometry

Date received: October 1, 2000