4 ms·
This is not exactly a list when AC is dropped but if one takes as an axiom 'Axiom of Choice is false'
by ithinkso 4y ago
This is not exactly a list when AC is dropped but if one takes as an axiom 'Axiom of Choice is false'
- syzarian 4y agoIn a given model the axiom is either true or false. When I say “dropped” I mean “false” in that model. It isn’t true that in all models where choice fails we have fields with no algebraic closure.
- gizmo686 4y agoWhat about ZF; which is ZFC without the axiom of choice, and notably not ZF¬C, which is ZF with the negation of choice.
- syzarian 4y agoIn a given model of ZF either choice is true or it is false. In the absence of the Axiom of Choice you can’t say that not all fields have algebraic closures. What you can say is that there is a model in which there are fields with no algebraic closure.