Аннотация:
By a Hamel basis we mean a basis for the reals, R, construed as a
vecor space over the field of rationals. In 1905, G. Hamel constructed such a basis from a
well-ordering of R. In 1975, D. Pincus and K. Prikry asked "whether a Hamel basis exists
in any model in which R cannot be well ordered." About two years ago, we answered
this positively in a joint paper with M. Beriashvili, L. Wu, and L. Yu. In more recent joint
work, additionally with J. Brendle and F. Castiblanco, we constructed a
model of ZF plus DC with a Luzin set, a Sierpiński set, a Burstin basis, and a
Mazurkiewicz set, but with no well-ordering of R. We will discuss the methods which get
exploited and give an outline of the constructions.
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