There is No Standard Model of ZFC and ZFC2
Jaykov Foukzon *
Israel Institute of Technology, Haifa, Israel.
Elena Men'kova
All-Russian Research Institute for Optical and Physical Measurements, Moscow, Russia.
*Author to whom correspondence should be addressed.
Abstract
In this paper we view the rst order set theory ZFC under the canonical rst order semantics and the second order set theory ZFC2 under the Henkin semantics. Main results are: (i) Let MZFCst be a standard model of ZFC, then ¬Con(ZFC +∃MZFCst ).
(ii) Let MZFC2st be a standard model of ZFC2 with Henkin semantics, then ¬Con(ZFC2 +∃MZFC2st ).
(iii) Let k be inaccessible cardinal then ¬Con(ZFC + ∃κ).In order to obtain the statements (i) and (ii) examples of the inconsistent countable set in a settheory ZFC + ∃MZFCst and in a set theory ZFC2 + ∃MZFC2st were derived.
It is widely believed that ZFC + ∃MZFCst and ZFC2 + ∃MZFC2st are inconsistent, i.e. ZFC andZFC2 have a standard models. Unfortunately this belief is wrong.
Keywords: Gödel encoding, Russell's paradox, standard model, Henkin semantics, inaccessible cardinal.