Martin's axiom is consistent with the existence of nowhere trivial automorphisms (Q2782641)

A trivial automorphism of \(\mathcal P(\omega)/\)fin is induced by a 1-1 map from some cofinite set onto another cofinite set. A somewhere trivial automorphism of \(\mathcal P(\omega)/\)fin is trivial on some \(\mathcal P(A)/\)fin. It has been known for some time that ``all automorphisms of \(\mathcal P(\omega)/\)fin are trivial'' is consistent, as is its negation. This paper explores somewhere trivial automorphisms, and (a) shows that there's a model of MA in which some automorphism fails to be somewhere trivial, and (b) develops a technique avoiding oracle cc forcing for constructing models in which every automorphism is somewhere trivial; this technique has the advantage of allowing \(\mathfrak d = \aleph_1\). The reader is warned that key definitions are not stated, e.g. trivial, somewhere trivial, Sacks forcing, and Silver forcing.