2019/11/21 by Karagila, Asaf
#FOS: Mathematics #Logic (math.LO) #Primary 03E25 #Secondary 03E35
paper · doi:10.48550/arxiv.1911.09285
We combine several folklore observations to provide a working framework for iterating constructions which contradict the axiom of choice. We use this to define a model in which any kind of structural failure must fail with a proper class of counterexamples. For example, the rational numbers have a proper class of non-isomorphic algebraic closures, every partial order embeds into the cardinals of the model, every set is the image of a Dedekind-finite set, every weak choice axiom of the form ACXY fails with a proper class of counterexamples, every field has a vector space with two linearly independent vectors but without endomorphisms that are not scalar multiplication, etc.