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Comment by syzarian | Hacker News Reader

There are a lot of seemingly bizarre consequences of the Axiom of Choice. This is well known. Sometimes people haven’t seen a list of bizarre consequences when the Axiom of Choice is dropped. Here’s a nice write up of this:

https://mathoverflow.net/questions/20882/most-unintuitive-ap...

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“The Axiom of Choice is obviously true, the Well–ordering theorem is obviously false; and who can tell about Zorn’s Lemma?" -- Jerry Bona

This is not exactly a list when AC is dropped but if one takes as an axiom 'Axiom of Choice is false'

In 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.

What about ZF; which is ZFC without the axiom of choice, and notably not ZF¬C, which is ZF with the negation of choice.

In 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.

Imagine having countable infinite many prisoners, put hats on them colored with 0-1 real numbers, and let them to guess their own colors. And then all but finite many prisoners guess their color correctly.

Would you believe in axiom of choice now, wouldn't you?

I don’t understand your point. There are seemingly bizarre consequences of choice and of the negation of choice. They revolve around uncountable sets. For me, the bizarre thing is uncountable sets. I’m fine with them existing in ZFC and use them when doing mathematics but uncountability is bizarre.

I had no real point. Just imagine it, and laugh, because it is a funny scenario. It doesn't need uncountable sets btw, countable infinite many prisoners, and 2 colors do the trick.

I don't know what you mean by uncountable sets are bizarre. Newtonian physics uses a lot of uncountable sets, and it is consistent witch our everyday experiences, isn't it? Or things like Banach-Tarski would appear anyway? Do you have some examples?

Most of the bizarre things that happen in set theory come from uncountability. Of course there are lots of examples where uncountable sets aren’t bizarre. Physics uses real manifolds and other continuous mathematics to model reality but this doesn’t mean reality contains any actual uncountable sets.

Can you give examples?

Without the Axiom of Choice it’s possible for a vector space to not have a basis. Examples of this phenomena require uncountable direct sums of vector spaces.

How would that work?

https://www.startpage.com/sp/search?query=axiom+of+choice+in...

This seems very weird. Where does this come from?

I'd assume none will guess correctly because you have countably infinite guesses from uncountably infinite set (the colors) so the set of guesses is very sparse, meaning pretty much none will hit it.

How on earth can almost all hit a point in a continuum exactly with random chance?

my favorite joke about this is:

"the axiom of choice is obviously true, and the well ordering principle is obviously false, but Zorn's lemma? who can say?"

Are all bizarre consequences related to infinity or infinite sets?

Yes. I’m not an expert in this area but the intuition for finite sets is well modeled by the standard axioms of set theory and I’ve never heard of an issue with finite sets. Things like Russell’s Paradox had to do with constructing sets when the foundations for doing this weren’t well established. Countably infinite sets are fine too, for the most part. The truly weird results come from uncountably infinite sets.

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