>[!note] Notation >If $M$ is a transitive class, we consider it as the model $(M,\in)$ and write, for a formula $\varphi(x_{1},\dots,x_{n})$ and $a_{1},\dots,a_{n}$: $M\vDash\varphi(a_{1},\dots,a_{n})$ for "$(M,\in)$ satisfies $\varphi$ under the assignment $x_{i}\mapsto a_{i}quot; >[!info] Definition > Let $M,N$ be transitive classes, $M \subseteq N$. We say that $\varphi$ is > - *upward absolute between $M$ and $N$* if for every $a_{1},\dots,a_{n}\in M$, $M\vDash\varphi(a_{1},\dots,a_{n}) \implies N\vDash\varphi(a_{1},\dots,a_{n})$ > - *downward absolute between $M$ and $N$* if for every $a_{1},\dots,a_{n}\in M$, $N\vDash\varphi(a_{1},\dots,a_{n}) \implies M\vDash\varphi(a_{1},\dots,a_{n})$ > - *absolute between $M$ and $N$* if for every $a_{1},\dots,a_{n}\in M$, $M\vDash\varphi(a_{1},\dots,a_{n}) \iff N\vDash\varphi(a_{1},\dots,a_{n})$ > >If we mention only one model, we mean the other is $V$. >If we say that $\varphi$ is (upward/downward)-absolute, we mean it is (upward/downward)-absolute for any two transitive models. ## Facts - In [[The Lévy hierarchy]] ^18da90 - $\Sigma_{0}=\Pi_{0}=\Delta_{0}$ formulas are absolute. - $\Sigma_{1}$ formulas are upward absolute. - $\Pi_{1}$ formulas are downward absolute. - (Hence) $\Delta_{1}$ formulas are absolute. # Generic absoluteness