Set truth and the Tarski interface

Set satisfaction and relativization must be distinguished. Set satisfaction is uniform in the code of a formula and the data of a set structure. Relativization to a definable proper class supplies an ambient formula separately for each fixed input formula. Writing $V\models \varphi (\vec{a})$ in this latter sense merely abbreviates $\varphi (\vec{a})$.

For the truth-with-parameters interface, a proposed pure membership formula $T(w,x,y,z)$ and a set parameter $t$ would have to satisfy, for every pure membership formula $\psi$ with free variables among $x,y$ and all sets $r,s$, the biconditional $\psi (r,s)\leftrightarrow T(<mi\; mathvariant="normal">⌜</mi>\psi <mi\; mathvariant="normal">⌝</mi>,r,s,t)$. Here $<mi\; mathvariant="normal">⌜</mi>\psi <mi\; mathvariant="normal">⌝</mi>$ is its finite set code. This is a scheme of requirements, not a single first-order assertion quantifying over ambient truths. The companion refutation tests this exact scheme by one formula built from the proposed $T$. No sentence-only arithmetized diagonal lemma or representability theorem is asserted here.

Conventions and prerequisites: Relativization agrees with induced set satisfaction.