Let PI be the statement,
“Every Boolean algebra has a prime ideal.”; let SPI be the statement, “Every infinite
set algebra has a nonprincipal prime ideal.”; let K be the statement, “Every family of
sets includes a maximal subfamily of pairwise incomparable (by the inclusion
relation) sets.”. A model is exhibited of set theory without regularity in
which SPI and K hold, PI fails. Furthermore all finitary versions of the
axiom of choice fail in this model and the model contains a set which is
infinite Dedekind finite in the sense of the model. A proof is given that the
local version of SPI implies the axiom of choice for families of finite sets in
ZF.
