It’s a nice follow up. I think it is worthwhile to point out some counter intuitive results that happen in the absence of choice. Check out this post: https:

It’s a nice follow up. I think it is worthwhile to point out some counter intuitive results that happen in the absence of choice. Check out this post: https://mathoverflow.net/questions/20882/most-unintuitive-application-of-the-axiom-of-choice/70435#70435 https://mathoverflow.net/questions/20882/most-unintuitive-ap... To me the essence of the issue is with dealing with uncountable sets. Our intuition fails in some cases when dealing with uncountable sets. I suspect that whatever axioms you have that seem reasonable for a foundation for set theory then those axioms will lead to bizarre results when dealing with uncountable sets. EDIT: When I wrote worthwhile to point out above, I was referring to people who may read the comments here and not the actual blogpost. I can’t construct the function requested in the blogpost but this isn’t a problem for me philosophically. Probably because I’ve been brainwashed to deal comfortably with such situations.