• Corpus ID: 211020766

Reinhardt cardinals and non-definability

@article{Schlutzenberg2020ReinhardtCA,
  title={Reinhardt cardinals and non-definability},
  author={Farmer Schlutzenberg},
  journal={arXiv: Logic},
  year={2020}
}
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… 

Even ordinals and the Kunen inconsistency

This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of

A note on L\"owenheim-Skolem cardinals

In this note we provide some applications of Lowenheim-Skolem cardinals introduced in \cite{U}.