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For every tuple of parameters $\bar{a}\in M^n$, a downward absolute formula transfers truth from the larger structure to the smaller one:The condition concerns every parameter tuple from $M$, not merely parameter-free sentences.

A formula has set-theoretic absoluteness between these structures when both directions hold for every $\bar{a}\in M^n$:Equivalently, it is both a downward absolute formula and an upward absolute formula between them.

N. A larger model may collapse a singular cardinal number, destroying cardinalhood, so the assertion is not described by an upward absolute formula. It may instead add a short cofinal function to a regular cardinal in the smaller model, so it is not described by a downward absolute formula. This is the nonabsoluteness of singular cardinalhood.

An strongly inaccessible cardinal $\kappa$ has the Keisler extension property when there is a proper transitive set $X⊋V_{\kappa }$ such that

Suppose $\kappa$ is strongly inaccessible and has this property. Because $X$ properly extends the transitive set $V_{\kappa }$, it contains $\kappa$. Strong inaccessibility of $\kappa$ is downward absolute from the ambient universe to the transitive set $X$: any internal witness that $\kappa$ is countable, singular, or not a strong limit would also be an ambient witness. Hencewith $\kappa$ as a witness. Since $V_{\kappa }\prec X$, the same sentence holds in $V_{\kappa }$. Its witness is an ordinal $\mu <\kappa$. The set $V_{\kappa }$ contains $V_{\mu +1}$, so it computes all subsets of cardinals below $\mu$ correctly; strong inaccessibility of $\mu$ is therefore absolute between $V_{\kappa }$ and the universe. Thus there is a strongly inaccessible $\mu <\kappa$, and $\kappa$ cannot be the least strongly inaccessible cardinal.

With the same parameter $\alpha =({\omega }_1{)}^M$, letThis formula is downward absolute: if the larger transitive model has no such function, then neither can the smaller model, since any witness in the smaller model would remain a witness in the larger one. The model $M$ satisfies ${\phi }_2(\alpha )$, while the generic surjection $\bigcup G$ makes it false in $M[G]$. Therefore it is not upward absolute.