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+∃M^ZFC_st),

(ii) ¬Con(N F+∃M^NF_st),

(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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