# Sets definable by $\Sigma_2$ parameters

Say that a formula $\phi$ defines a set $x$ from parameters $a_1, \dots, a_n$ if $\phi(a_1, \dots, a_n, x)$ holds (in $V$) but for $y \neq x$ $\phi(a_1, \dots, a_n, y)$ does not hold.

Is it true that: If $x$ is definable from ordinal parameters, then it is definable from ordinal parameters by a $\Sigma_2$ formula.

Note that in In the model $L$, everything is definable by ordinal parameters.

• Do you mean $\Sigma_2$? If so, then this is a consequence of the reflection theorem. – Andrés E. Caicedo Sep 6 '18 at 12:52
• We can do slightly better with respect to your last remark. Every $x \in L$ is $\Sigma_1$ definable in ordinal parameters: "$x$ is the $\beta$th element of the canonical well-order $L_\alpha$ and $x \not \in L_{< \alpha}$." (Recall that the constructible hierarchy as well as the sequence of their canonical well-orders are $\Sigma_1$ definable.) – Stefan Mesken Sep 10 '18 at 21:17

Any set definable from ordinal parameters is definable from a slightly different set of ordinal parameters by a $\Sigma_2$ formula; this is a consequence of the reflection theorem, which - given a definition $\varphi$ of $x$ with ordinal parameters $\gamma_1,...,\gamma_n$ - states that there is an ordinal $\alpha$ such that $x,\gamma_1,...,\gamma_n\in V_\alpha$ and $$V_\alpha\models\forall z(\varphi(\gamma_1,...,\gamma_n,z)\iff z=x).$$ We can now define $x$ from the parameters $\gamma_1,...,\gamma_n,\alpha$ via the above condition; saying "$V_\alpha\models ...$" is $\Sigma_2$ (in fact, $\Sigma_1$) in $V$: saying that an appropriate Skolem function exists is an $\exists$ ranging over $V$ on top of a bunch of quantifiers bounded by $V_\alpha$.
• @OP: Note that $V_{\alpha}$ is $\Sigma_1$ definable from $\alpha$: $x \in V_{\alpha}$ iff $\exists y \exists f \colon y \text{ is transitive } \wedge x \subseteq y \wedge f \colon (y; \in) \to (\alpha;\in) \text{ is order preserving}$. – Stefan Mesken Sep 10 '18 at 21:24