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Let $\lambda$ be a cardinal. Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. If $f:V_{\lambda}\rightarrow V_{\lambda}$ is a function and $\gamma<\lambda$, then let $f\upharpoonright_{V_{\gamma}}:V_{\gamma}\rightarrow V_{\gamma+1}$ be the function where $f\upharpoonright_{V_{\gamma}}(x)=f(x)\cap V_{\gamma}$ for each $x\in V_{\gamma}$. We shall say that a function $T:V_{\lambda}^{n}\rightarrow V_{\lambda}$ is definable in $(V_{\lambda},\in)$ with small parameters if $T$ is definable in $(V_{\lambda},\in)$ with parameters $A_{1},...,A_{n}$ but such that if $j\in\mathcal{E}_{\lambda}\setminus\{1_{V_{\lambda}}\}$, then $A_{1},...,A_{n}\in V_{\text{crit}(j)}$.

Do there exist elementary embeddings $j_{1},...,j_{n},j\in\mathcal{E}_{\lambda}$, and a limit ordinal $\gamma$ along with a function $T$ definable in $(V_{\lambda},\in)$ from small parameters such that

  1. $T(j_{1}\upharpoonright_{V_{\gamma}},...,j_{n}\upharpoonright_{V_{\gamma}})=j\upharpoonright_{V_{\gamma}}$ and

  2. there does not exist an $n$-ary term $t$ in the language of self-distributive algebras such that $j\upharpoonright_{V_{\gamma}}=t(j_{1},...,j_{n})\upharpoonright_{V_{\gamma}}$?

I am also interested in the answer an analogous version of this question for stronger large cardinal axioms such as $I2$ and $I1$ or more generally $E_{n}(\lambda),E_{\omega}(\lambda)$. I allow for small parameters since these small parameters are invariant under rank-into-rank embeddings.

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  • $\begingroup$ Could you clarify: when you say $T$ is definable, which structure and language do you intend to run the definition? In $\langle V,\in\rangle$? In $\langle V_\lambda,\in\rangle$? $\endgroup$ – Joel David Hamkins Feb 1 '16 at 3:41
  • $\begingroup$ I have clarified what structure $T$ is definable in. $\endgroup$ – Joseph Van Name Feb 1 '16 at 4:53

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