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Zuhair Al-Johar
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Can we iteratively reflect on self elementary embeddable stages of the cumulative hierarchy?

Is it possible to iterate elementary embeddability and reflect on those stages that are elementary embeddable to themselves?

The following is a formal capture of that idea:

To the language of $\sf ZF$ (i.e., mono-sorted $\sf FOL(=,\in)$) add primitive partial unary functions $W$ and $j$.

To the axioms of $\sf ZF$, add the following axioms:

Restriction: $\forall \alpha: W_\alpha \lor j_\alpha \to \operatorname {ordinal}(\alpha)$

Injectivity: $W_\alpha \land W_\beta \land \alpha \neq \beta \to W_\alpha \neq W_\beta$

Cumulation: $\forall \operatorname {ordinal} \alpha \exists \lambda: W_\alpha=V_\lambda$

Elementarity: $\forall \operatorname {ordinal} \alpha \, (j_\alpha: W_\alpha \to W_\alpha \land \\ \forall \vec{x} \in W_\alpha [ \phi(\vec x) \leftrightarrow \phi(j_\alpha[\vec x ])] \land \\ \exists x: j_\alpha (x) \neq x) \\\text {where } \phi \text { is purely set theoretic }$

Reflection: $\forall \vec{x} (\phi \to \exists \alpha: \phi^{W_\alpha}); \text { if } W_\alpha \text { doesn't occur in } \phi$

Where $V_\lambda$ stands for the $\lambda^{th}$ stage of the cumulative hierarchy, defined in the customary manner. $\phi^X$ stands for relativising all quantifiers in $\phi$ to $``\in X\!"$.

Is the above theory consistent relative to some large cardinal property? If so, Which one?

Zuhair Al-Johar
  • 11.3k
  • 1
  • 13
  • 47