Nontrivial automorphisms of
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DOI: 10.4153/s0008439526102653 ISSN: 0008-4395

Nontrivial automorphisms of Will Brian, Alan Dow

We show that if

κ < ℵ ω

$\kappa < \aleph _\omega $

kappa less than normal first transfinite cardinal Subscript omega

Cohen reals are added to a model of

CH

$\textsf {CH}$

sans serif CH

, then there are nontrivial automorphisms of

P ( ω ) / Fin

$\mathcal{P}(\omega )/\mathrm {Fin}$

script upper P left parenthesis omega right parenthesis divided by upper F i n

in the extension. Under some further hypotheses on the ground model, namely, the existence of long enough sage Davies trees (which follows from

SCH

$\textsf {SCH}$

sans serif SCH

plus

◻ λ

$\square _\lambda $

white medium square Subscript lamda Baseline

for every

λ

$\lambda $

lamda

with

cf ( λ ) = ω

$\mathrm {cf}(\lambda ) = \omega $

cf left parenthesis lamda right parenthesis equals omega

), we prove the same result for cardinals

κ ≥ ℵ ω

$\kappa \geq \aleph _\omega $

kappa greater than or equals normal first transfinite cardinal Subscript omega

as well. This extends a result by Shelah and Steprāns, who proved the result for

κ = ℵ 2

$\kappa = \aleph _2$

kappa equals normal first transfinite cardinal 2

.