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DE ZOLT’S POSTULATE: AN ABSTRACT APPROACH

Published online by Cambridge University Press:  02 September 2019

EDUARDO N. GIOVANNINI
Affiliation:
CONICET AND UNIVERSIDAD NACIONAL DEL LITORAL IHUCSO LITORAL BV. PELLEGRINI 2750 SANTA FE, S3000, ARGENTINA E-mail: engiovannini@conicet.gov.ar
EDWARD H. HAEUSLER
Affiliation:
CNPQ AND PONTIFICAL CATHOLIC UNIVERSITY OF RIO DE JANEIRO DEPT DE INFORMÁTICA RUA MARQUES DE SÃO VICENTE, 225, GÁVEA RIO DE JANEIRO, CEP22451-300, BRAZIL E-mail: hermann@inf.puc-rio.br
ABEL LASSALLE-CASANAVE
Affiliation:
CNPQ AND FEDERAL UNIVERSITY OF BAHIA RUA PROFESSOR ARISTÍDES NOVIS – FEDERAÇÃO SALVADOR, BA 40226-365, BRAZIL E-mail: abel.lassalle@gmail.com
PAULO A. S. VELOSO
Affiliation:
CNPQ AND COPPE, FEDERAL UNIVERSITY OF RIO DE JANEIRO COPPE-PESC, AVENIDA HORÁCIO MACEDO 2030 CENTRO DE TECNOLOGIA, BLOCO H, SALA 319 CIDADE UNIVERSITÁRIA, CEP: 21941-914, BRAZIL E-mail: veloso@cos.ufrj.br

Abstract

A theory of magnitudes involves criteria for their equivalence, comparison and addition. In this article we examine these aspects from an abstract viewpoint, by focusing on the so-called De Zolt’s postulate in the theory of equivalence of plane polygons (“If a polygon is divided into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon”). We formulate an abstract version of this postulate and derive it from some selected principles for magnitudes. We also formulate and derive an abstract version of Euclid’s Common Notion 5 (“The whole is greater than the part”), and analyze its logical relation to the former proposition. These results prove to be relevant for the clarification of some key conceptual aspects of Hilbert’s proof of De Zolt’s postulate, in his classical Foundations of Geometry (1899). Furthermore, our abstract treatment of this central proposition provides interesting insights for the development of a well-behaved theory of compatible magnitudes.

Type
Research Article
Copyright
© Association for Symbolic Logic, 2019

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