A theorem of A. Levy says that, if $\kappa$ is an inaccessible cardinal, then $V\_\kappa\prec\_{\Sigma\_1}V$ namely $V\_\kappa$ is an elementary submodel when considering only $\Sigma\_1$ formulas.


Where can I find a proof of this theorem ?


Is this property true also for some other (non inaccessible) cardinals ?



Inaccessible cardinal and $\Sigma_1$ reflection
Here's one way to prove it. Let $H\_\kappa = \{x: |tc(\{x\})|<\kappa\}$. Then we have:


**Theorem 1** If $\kappa$ is an uncountable cardinal, then $H\_\kappa\prec\_1 V$. 


*Proof.* Let $\phi$ be $\Delta\_0$ with free variables among $y,x$. Since $\Sigma\_1$ formulas are upward absolute for transitive models, if $H\_\kappa\vDash \exists y\phi$, then $\exists y\phi$. So suppose $\exists y\phi$ for $x\in H\_\kappa$. Now let $M\prec\_1 V$ with $tc(\{x\})\subseteq M$ and $|M|<\kappa$ (such an $M$ exists because $\kappa$ is uncountable and $|tc(\{x\})|<\kappa$ by definition of $H\_\kappa$). By the Mostowski collapse lemma, there is an isomorphism $j:M \to M'$ for some transitive $M'$. Since $tc(\{x\})\subseteq M$, $j(x) = x$ and thus $M'\vDash \exists y\phi$. Finally, we note that because $M'$ is transitive and has cardinality less than $\kappa$, $M'\in H\_\kappa$ and so $H\_\kappa\vDash \exists y\phi$ (again by upward absoluteness). $\Box$


Our result then follows from the fact that $H\_\kappa = V\_\kappa$, for $\kappa$ inaccessible. It is furthermore optimal for inaccessibles since ``$\kappa$ is inaccessible" is $\Pi\_1$. But stronger results hold for other large cardinals. For example:


**Theorem 2** If $\kappa$ is supercompact (or even strong), then $V\_\kappa\prec\_2 V$.


*Proof.* See Kanamori The Higher Infinite (2003) p. 299 and p. 359. $\Box$


**Theorem 3** If $\kappa$ is extendible, then $V\_\kappa\prec\_3 V$.


*Proof.* See Kanamori The Higher Infinite (2003) p. 318. $\Box$


This result is also optimal for extendibles since ``$\kappa$ is an extendible" is $\Pi\_3$. 


