28,483 research outputs found
The weak choice principle WISC may fail in the category of sets
The set-theoretic axiom WISC states that for every set there is a set of
surjections to it cofinal in all such surjections. By constructing an unbounded
topos over the category of sets and using an extension of the internal logic of
a topos due to Shulman, we show that WISC is independent of the rest of the
axioms of the set theory given by a well-pointed topos. This also gives an
example of a topos that is not a predicative topos as defined by van den Berg.Comment: v2 Change of title and abstract; v3 Almost completely rewritten after
referee pointed out critical mistake. v4 Final version. Will be published in
Studia Logica. License is CC-B
Polynomials with prescribed bad primes
We tabulate polynomials in Z[t] with a given factorization partition, bad
reduction entirely within a given set of primes, and satisfying auxiliary
conditions associated to 0, 1, and infinity. We explain how these sets of
polynomials are of particular interest because of their role in the
construction of nonsolvable number fields of arbitrarily large degree and
bounded ramification. Finally we discuss the similar but technically more
complicated tabulation problem corresponding to removing the auxiliary
conditions.Comment: 26 pages, 3 figure
- …