On a question of Silver about gap-two cardinal transfer principles

Abstract

Assuming the existence of a Mahlo cardinal, we produce a generic extension of Gödel's constructible universe $L$, in which the transfer principles $({\aleph }_2,{\aleph }_0)\to ({\aleph }_3,{\aleph }_1)$ and $({\aleph }_3,{\aleph }_1)\to ({\aleph }_2,{\aleph }_0)$ fail simultaneously. The result answers a question of Silver from 1971. We also extend our result to higher gaps.