Journal of Advances in Mathematics and Computer Science

Inconsistent Countable Set in Second Order ZFC and Nonexistence of the Strongly Inaccessible Cardinals

Jaykov Foukzon *

Center for Mathematical Sciences, Israel Institute of Technology, Haifa, Israel.

*Author to whom correspondence should be addressed.

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Abstract

In this article we derived an important example of the inconsistent countable set in second order ZFC (ZFC₂) with the full second-order semantics. Main results: (i) ¬Con(ZFC₂), (ii) let k be an inaccessible cardinal and H_k is a set of all sets having hereditary size less then k, then ¬Con(ZFC + (V = H_k)).

Keywords: Gödel encoding, Completion of ZFC2, Russell′ s paradox, ω-model, Henkin semantics, full second-order semantics.

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