There is No Standard Model of ZFC and ZFC2

In this Chapter we obtain a contradictions in formal set theories under assumption that these theories have omega-models or nonstandard model with standard part. An possible generalization of Lob’s theorem is considered. Main results are:

(i) ¬Con(ZF C+MZFCst),

(ii) ¬Con(N F+MNFst),

(iii) ¬Con(ZF C2),

(iv) let k be an inaccessible cardinal then ¬Con(ZF C+κ),

(v) ¬Con(ZF C+ (V=L)),

(vi) ¬Con(ZF+ (V=L)).

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