Suppose V is a model of Godel-Berney's set theory with the axiom of choice. A well-known result of Kunen says that there can be no elementary embedding $V$ to itself. This result further implies that there can be embedding from $V$ to $V$ which is $\Sigma\_1$-elementary. 


Is it consistent (with say Godel-Berney's set theory with the axiom of choice) that there is a $\Delta\_1$-elementary embedding from $V$ to $V$? How about a $\Delta\_0$-elementary embedding? 



Can $\Delta_1$ (or $\Delta_0$)-elementary embeddings from $V$ to $V$ exist?
The answer is yes, the existence of such embeddings is equiconsistent over ZFC with a measurable cardinal. 


If $\kappa$ is measurable, then there is a fully elementary embedding $j:V\to M$ into a transitive class $M$ with critical point $\kappa$. By composing this map with the inclusion $M\subset V$, we may view $j:V\to V$. The observation to make is that this remains $\Delta\_0$-elementary, since every transitive class is $\Delta\_0$-elementary in the universe $M\prec\_{\Sigma\_0}V$, and so we may view $j:V\to V$ as a composition of the original elementary embedding with this $\Delta\_0$-elementary inclusion. 


In fact, the embedding $j:V\to V$ is $\Delta\_1$-elementary, since $M$ is actually $\Delta\_1$-elementary in $V$, as $\Sigma\_1$ assertions go up and $\Pi\_1$ assertions go down. Since $M$ and $V$ are elementarily equivalent, they agree on the equivalence or non-equivalence of any two formulas. 


(Note that there is a subtle issue about what exactly one means by $\Delta\_1$-elementary, since there is no such thing as a $\Delta\_1$-formula, unlike $\Delta\_0$. For example, in the argument, it was important that $M$ and $V$ agreed on the equivalence of the $\Sigma\_1$ nad $\Pi\_1$ formulas in question. It can happen, however, that two models of ZFC do not necessarily agree like this, and there is a difference between $\Delta\_1$ and provably-$\Delta\_1$.)


Conversely, let us show that measurable cardinals are required. If $j:V\to V$ is a nontrivial $\Delta\_0$-elementary embedding, then I claim, first, that there is a critical point, that is, a first ordinal that is moved. Notice that $j$ must take ordinals to ordinals. Assume toward contradiction that $j$ fixes all ordinals. Let $a$ be an $\in$-minimal set with $j(a)\neq a$. So $a\subset j(a)$ since $j(x)=x$ for all $x\in a$. By the axiom of choice, we may well-order $a$ with a relation $\lhd$ in some order-type $\gamma=|a|$. It follows that $j(\lhd)$ well-orders $j(a)$ in order-type $j(\gamma)=\gamma$. Furthermore, the $\beta^{th}$ element of $a$ gets mapped to the $j(\beta)=\beta^{th}$ element of $j(a)$ with respect to $j(\lhd)$. So $j$ is onto $j(a)$, and therefore $j(a)=j"a=a$, a contradiction. So we have shown that there must be a critical point $\kappa$, the least ordinal with $\kappa<j(\kappa)$. Secondly, now, let $\mu$ be the set of $X\subseteq\kappa$ with $\kappa\in j(X)$. It follows now that $\mu$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$. This is easily seen to be a nonprincipal filter, and it is $\kappa$-complete, since if $X\_\alpha\in \mu$ for all $\alpha<\gamma$, some $\gamma<\kappa$, then $j(\langle X\_\alpha\mid\alpha<\gamma\rangle=\langle j(X\_\alpha)\mid\alpha<\gamma\rangle$, and $\kappa$ is in every element of the right hand side and hence in the intersection of the right-hand side. So $\cap\_\alpha X\_\alpha\in\mu$. 


Lastly, let me point out that if one weakens much more, below $\Delta\_0$, then one gets the embeddings in ZFC. This idea grew out of my work in: 


*Hamkins, Joel David*, [Every countable model of set theory embeds into its own constructible universe](http://jdh.hamkins.org/every-model-embeds-into-own-constructible-universe/), J. Math. Log. 13, No. 2, Article ID 1350006, 27 p. (2013). [DOI](http://dx.doi.org/10.1142/S0219061313500062), [ZBL1326.03046](https://zbmath.org/?q=an:1326.03046),
 [blog](http://jdh.hamkins.org/every-model-embeds-into-own-constructible-universe/).


In that article, I consider the $\in$-embeddings, which are embeddings $j$ for which merely $$x\in y\iff j(x)\in j(y).$$ This is a weakening of $\Delta\_0$-elementarity. These are precisely the self-embeddings of the structure $\langle V,\in\rangle$ in the model-theorists sense of the term for a relational structure. 


**Theorem.** ZFC proves that the map defined by the following recursion is an $\in$-embedding $j:V\to V$. 
$$j(x)=\{j(y)\mid y\in x\}\cup\{\{\emptyset,x\}\}$$


See theorem 19 and lemma 18 in the paper. Basically, it is clear that $y\in x\to j(y)\in j(x)$ by definition. And conversely, $j$ is injective, since we can recover $x$ from the last part of $j(x)$, and if $j(y)\in j(x)$, then it must have come from $y\in x$, since $j(y)\neq\{\emptyset,x\}$ as $j(z)\neq\emptyset$ and $\emptyset\notin j(z)$ for any $z$.


Let me also mention my question, [Can there be an embedding $j:V\to L$?](https://mathoverflow.net/questions/101821/can-there-be-an-embedding-jv-to-l-from-the-set-theoretic-universe-v-to-the-con), which asks whether one can have in GBC an $\in$-embedding from $V$ to $L$, in cases where $V\neq L$. The full question remains open, although we now know some things about partial answers, for work-in-progress. 


