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**Permutations and stratified formulae - a preservation theorem.**
*(English)*
Zbl 0717.03019

The permutation construction that concerns us here is familiar to students of ZF as the technique used in the standard proof of the independence of the axiom of foundation. The technique is originally due to L. Rieger [Czech. Math. J. 7(82), 323-357 (1957; Zbl 0089.244)] and P. Bernays [J. Symb. Logic 19, 81-96 (1954; Zbl 0055.046)].

If we have a model \(<V,\in >\) of ZF and let \(\sigma\) be the transposition exchanging the empty set and its singleton. Then we define \(x\in_{\sigma}y\) by \(x\in \sigma \text{`}y\). It turns out that in the model \(V^{\sigma}\) consisting of the old universe and the new membership relation \(\in_{\sigma}\) the “old” empty set has become an object identical to its own singleton, and foundation has failed. As it happens all the other axioms of ZF are preserved.

The purpose of this note is to prove a preservation theorem saying that the sentences of the language of set theory preserved by this construction are precisely those that obey a certain typing discipline.

If we have a model \(<V,\in >\) of ZF and let \(\sigma\) be the transposition exchanging the empty set and its singleton. Then we define \(x\in_{\sigma}y\) by \(x\in \sigma \text{`}y\). It turns out that in the model \(V^{\sigma}\) consisting of the old universe and the new membership relation \(\in_{\sigma}\) the “old” empty set has become an object identical to its own singleton, and foundation has failed. As it happens all the other axioms of ZF are preserved.

The purpose of this note is to prove a preservation theorem saying that the sentences of the language of set theory preserved by this construction are precisely those that obey a certain typing discipline.