Perspex machine VIII: axioms of transreal arithmetic

Transreal arithmetic is a total arithmetic that contains real arithmetic, but which has no arithmetical exceptions. It allows the specification of the Universal Perspex Machine which unifies geometry with the Turing Machine. Here we axiomatise the algebraic structure of transreal arithmetic so that it provides a total arithmetic on any appropriate set of numbers. This opens up the possibility of specifying a version of floating-point arithmetic that does not have any arithmetical exceptions and in which every number is a first-class citizen. We find that literal numbers in the axioms are distinct. In other words, the axiomatisation does not require special axioms to force non-triviality. It follows that transreal arithmetic must be defined on a set of numbers that contains{-8,-1,0,1,8,&pphi;} as a proper subset. We note that the axioms have been shown to be consistent by machine proof.

Do Different Models of Integration Affect Actual Integration? The Cases of France and Great Britain Revisited

Britain and France adapted two different integration models, namely assimilationist and multiculturalism to integrate their immigrants. These two big models of integration have distinctive characteristics to integrate immigrants. There is a general claim that multiculturalism model is the best for integrating immigrants in terms of actual integration, however, some argue the opposite, that French assimilationist model is ‘better off.’ This study examines these controversial claims by looking at the level to which immigrants are integrated in economic, social, political, cultural dimensions of integration and attitudes towards immigrants in Britain and France. Within a given theoretical framework, this study compares the overall competency level of immigrants’ integration in terms of actual integration between British multiculturalism model and French assimilationist model and validate that both these two big models of integration have reached a comparable level of integration and they do not have any decisive impact on actual integration.

Note by the Secretariat (Request by the Ministry of Foreign Affairs of France, on behalf of the Regional Council of Martinique for admission as an associate member of ECLAC)

Spanish version available