Von Neumann, J. (1925) An Axiomatization of Set Theory. In: Heijenoort, V., Ed., From Frege to Gödel, Harvard University Press, Cambridge, CA, 412. Here is an especially interesting remark by Von Neumann: “[...] no categorical axiomatization of set theory seems to exist at all; [...] And since there is no axiom system for mathematics, geometry, and so forth that does not presuppose set theory, there probably cannot be any categorically axiomatized infinite systems at all”.

has been cited by the following article:

• TITLE: Consequences of a Godel’s Misjudgment

  AUTHORS: Giuseppe Raguní

  KEYWORDS: Semantic Completeness, Syntactic Incompleteness, Categoricity, Arithmetic, Second-Order Languages, Paradoxes

  JOURNAL NAME: Open Access Library Journal, Vol.2 No.9, September 15, 2015

  ABSTRACT: The fundamental aim of the paper is to correct a harmful way to interpret a Godel’s erroneous remark at the Congress of Konigsberg in 1930. Although the Godel’s fault is rather venial, its misreading has produced and continues to produce dangerous fruits, so as to apply the incompleteness Theorems to the full second-order Arithmetic and to deduce the semantic incompleteness of its language by these same Theorems. The first three paragraphs are introductory and serve to define the languages inherently semantic and its properties, to discuss the consequences of the expression order used in a language and some questions about the semantic completeness. In particular, it is highlighted that a non-formal theory may be semantically complete despite using a language semantically incomplete. Finally, an alternative interpretation for the Godel’s unfortunate comment is proposed.