Title: "Paradoxical" sets with no well-ordering of the reals

Abstract: 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 with a Luzin set, a Sierpiński set, a Burstin basis, and a Mazurkiewicz set,

but with no well-ordering of R. In joint work with V. Kanovei, we constructed such a model in which even all those "paradoxical" sets are projective.

Abstract: 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 with a Luzin set, a Sierpiński set, a Burstin basis, and a Mazurkiewicz set,

but with no well-ordering of R. In joint work with V. Kanovei, we constructed such a model in which even all those "paradoxical" sets are projective.

## Date:

Wed, 27/03/2019 - 14:00 to 15:30

## Location:

Ross 63