Vol. 41, No. 1, 1972

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On a question of Tarski and a maximal theorem of Kurepa

James Daniel Halpern

Vol. 41 (1972), No. 1, 111–121

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.

Mathematical Subject Classification
Primary: 02K05
Secondary: 04A25, 02K20, 02J05
Received: 5 October 1970
Revised: 9 September 1971
Published: 1 April 1972
James Daniel Halpern