= Rieger-Bernays permutation model
{c}
{title2=$x\mathrel E y\Longleftrightarrow x\in\pi(y)$}
A definable class bijection $\pi$ of a model's universe changes membership as displayed. Images and inverse images of sets under $\pi$ remain sets by the <Axiom schema of replacement>. The new extension of $y$ is the old set $\pi(y)$. Thus the representative of any desired extension $b$ is $\pi^{-1}(b)$. Extensionality follows from bijectivity; separation and replacement translate to old definable formulas; pairs, unions and power sets have explicit representatives. Infinity follows by iterating the new empty set and successor. Foundation need not survive: interchanging $\varnothing$ and $\{\varnothing\}$ makes the old empty set a <Quine atom>. A full such permutation of a model of choice still satisfies choice; failure of choice requires an additional symmetry restriction.
Back to article page
