To the language of set theory add a primitive unary predicate $\operatorname {Universe}$ and a primitive total unary function $j$. Add all axioms of $\sf ZF$ in the language of this theory, i.e. the new primitives allowed to be used in instances of Separation and Replacement, I'll refer to this simply as $\sf ZFj$.


We add the following axioms:


**Cumulative:** $\forall X: \operatorname {Universe}(X) \to \exists \lambda: X=V\_\lambda$


**Modeling:** if $\psi$ is an axiom of $\sf ZFj$, and $\psi^X$ is the $``\in X"$ bounded form of $\psi$; then: $$\forall X: \operatorname {Universe}(X) \to \psi^X$$


**Elementarity:** if $\phi(x\_1,..,x\_n)$ is a formula in signature $\{=,\in\}$, having $``x\_1,..,x\_n\! \!"$ as its sole free variables, and none of them occur bound, then: $$\forall X: \operatorname {Universe}(X) \to\forall x \in X \, (j(x) \in X) \land \\\forall x\_1,..,x\_n \in X \\ (\phi(x\_1,..,x\_n) \iff \phi(j(x\_1),..,j(x\_n))) \\ \land \exists x \in X: j(x) \neq x$$


The intention is to render the restriction of $j$ to any Universe $X$ a non-trivial elementary embedding from $X \to X$.


**Reflection:** if $\phi$ is a formula [defined functions and predicates allowed] not using the symbol $X$, and $\phi^X$ is the $``\in X"$ bounded form of $\phi$; then: $$\forall \vec{p} \, (\phi \to \exists X: \operatorname {Universe} (X) \land \phi^X)$$



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> Is this consistent? If so, what's its consistency strength?
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> In particular does it manage to interpret Reinhardt cardinals at each universe? Would we have a club of Reinhardt cardinals?
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Can we interpret Reinhardt cardinals this way?
It is inconsistent. Call an ordinal b an independent critical point if for every for every Universe X,


if c∈X then there is a function f with domain X and an ordinal α, such that "f is an elementary


embedding from X to Vα" holds, for all x∈X f(x)∈X, and b is the least ordinal with f(b)≠b.


Note that the critical point of j is an independent critical point because the restriction of j to any


Universe X has the above property of f. Let c be the least independent critical point. By the


axiom schema of Reflection, there is a universe K such that "c is the least independent critical


point" holds relativized to K. By the definition of independent critical point, there is a function


g with domain K and ordinal γ such tha g is an elementary embedding from K to Vγ, and for all x∈K g(x)∈K


and c is the least ordinal with g(c)≠c. Let F(x) be a formula expressing


"x is the least independent critical point". Define a sequence s by


s0=c and s(n+1) is the least ordinal α such that α is greater than g(s(n)) and for K, Vα reflects all


subformulas of F. Let t=U{sn|n∈}. Then g(t)=t, and F(c) holds in V(t). By elementarity,F(g(c)) holds


in V(g(t)). That is F(g(c)) holds in V(t). But this is impossible.


