• Corpus ID: 211020766

Reinhardt cardinals and non-definability

  title={Reinhardt cardinals and non-definability},
  author={Farmer Schlutzenberg},
  journal={arXiv: Logic},
Work in $\mathsf{ZF}$ or $\mathsf{ZF}_2$ (second order $\mathsf{ZF}$), as appropriate. Recall that a Reinhardt cardinal is the critical point of a (non-trivial) elementary embedding $j:V\rightarrow V$. Beyond these, one has super-Reinhardt, total Reinhardt and Berkeley cardinals. We prove the following results. Let $X$ be a set and $A$ a class. Then (i) if there is a Reinhardt cardinal then $V\neq\mathrm{HOD}(X)$, and (ii) if $V$ is total Reinhardt or there is a Berkeley cardinal then $V\neq… 

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