IJPAM: Volume 52, No. 4 (2009)

AN IMPLICATION OF GÖDEL'S
INCOMPLETENESS THEOREM

Hitoshi Kitada
Graduate School of Mathematical Sciences
University of Tokyo
Komaba, Meguro, Tokyo, 153-8914, JAPAN
e-mail: kitada@ms.u-tokyo.ac.jp


Abstract.A proof of Gödel's incompleteness theorem is given. With this new proof a transfinite extension of Gödel's theorem is considered. It is shown that if one assumes the set theory ZFC on the meta level as well as on the object level, a contradiction arises. The cause is shown to be the implicit identification of the meta level and the object level hidden behind the Gödel numbering. An implication of these considerations is stated.

Received: March 22, 2009

AMS Subject Classification: 03F40, 03F15, 03B25, 03E99

Key Words and Phrases: consistency, incompleteness, Gödel, transfinite extension

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 52
Issue: 4