NE PEUT-ON PAS CONTOURNER L'IMPASSE IDENTIFIÉE PAR GÖDEL ET PROUVER LA CONSISTANCE DES MATHÉMATIQUES USUELLES ?

Francis Bessière (1)

1 Retraité

The departure point of this study is the fact that the well known Gödel's theorems apply specifically to "hypothetico-deductive systems" - which I will rather call "deductive theories". Then it may be possible to turn around the dead-end identified by Gödel, by trying to find other kinds of theories (with the condition that they shall not be equivalent to a deductive theory). So I will define "translative theories" and "asymptotic theories". Chapter 1 and the thirteen following "Memos" lay down their foundations and present (Memo 4) the theory M : the Annexes A, B and C will prove that this theory is equivalent to "usual Mathematics", i.e. [ZF] plus the axiom of foundation (AF) - but with no necessity to take into account the axiom of choice (AC). Using always the intuitive notion of "dictives" (Memo 10), the chapters 2 and 3 will introduce some new concepts: the "superblocks" of bounded letters, and the "useful" and "sub-transitive" dictives. The chapter 4 deals with a family of translative theories which are all consistent and leading to M. The unwritten chapter 5 hoped to finish the present work, but my age and other uncontrollable factors prevented me from a real success: I hope that other researchers will take over ...

Mots clés: Gödel; Littème; Littéral; Transcription; Logique; Dédductive; Traductive; Dictif; Asymptotique; Transitif; Denexion; Prenexion; Bloc; Libération; Duplication; Fondation; Sélection

Domaines: Logique [math.LO]

Francis Bessière. NE PEUT-ON PAS CONTOURNER L'IMPASSE IDENTIFIÉE PAR GÖDEL ET PROUVER LA CONSISTANCE DES MATHÉMATIQUES USUELLES ?. 2013. ⟨hal-00812912⟩