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

TL;DR

This paper constructs a model of set theory where certain transfer principles between different infinite cardinals fail simultaneously, answering a question posed by Silver in 1971, under the assumption of a Mahlo cardinal.

Contribution

It demonstrates the consistency of the failure of specific gap-two transfer principles in set theory, extending previous results and answering a longstanding open question.

Findings

Failure of both transfer principles in the constructed model

Extension of results to higher gaps

Answer to Silver's 1971 question

Abstract

Assuming the existence of a Mahlo cardinal, we produce a generic extension of G\"{o}del'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.