Hacker News Reader

Comment by bmacho | Hacker News Reader

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?

View on HN

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.

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?

How would that work?

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

Reply on news.ycombinator.com