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Ultrafilters Resulting from the Axiom of Determinateness
Author(s) -
Mig Robert
Publication year - 1981
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-43.3.582
Subject(s) - gratitude , citation , state (computer science) , axiom , computer science , doctoral dissertation , library science , mathematics , law , psychology , algorithm , political science , higher education , social psychology , geometry
The axiom of determinateness (AD), first studied by Mycielski and Steinhaus (see [11] and [15]), possesses some desirable consequences which support its position as an alternative to the axiom of choice (AC). For instance, AD implies that all sets of reals are Lebesgue measurable. Also, the paradoxical decomposition of the sphere, obtained as a consequence of AC, is eliminated when AC is replaced by AD. AD is inconsistent with AC and most of the weaker forms of choice (see [7]). However, AD implies countable choice (ACW) and, by a recent result of Solovay, AD is consistent with the axiom of dependent choice (DC) (see [14]). Nevertheless, there are nagging peculiarities which counter the pleasant consequences of AD. For example, AD implies that all u)n, with 3 ^ n < to, are singular cardinals. Because of these peculiarities and the lack of intuitive appeal in the statement of AD, the axiom has not been added to the list of axioms of set theory as being inherently true in the hierarchy of sets. However, over the past two decades AD has spawned a great deal of interesting research, particularly the relation between AD and the theory of large cardinals (see [6]). And since the consistency of AD with ZF is still a very open question, the intrigue with AD continues. The consequences of AD which strike the keynote for this paper are as follows.