4 ms·

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 finit

by bmacho 4y ago

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?

syzarian 4y ago
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.
- bmacho 4y ago
  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?
  - syzarian 4y ago
    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.
    - bmacho 4y ago
      Can you give examples?
      - syzarian 4y ago
        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.
Brigand 4y ago
How would that work?
- bmacho 4y ago
  https://www.startpage.com/sp/search?query=axiom+of+choice+infinite+prisoners https://www.startpage.com/sp/search?query=axiom+of+choice+in...
sharpneli 4y ago
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?

by bmacho 4y ago

syzarian 4y ago

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.

bmacho 4y ago
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?
- syzarian 4y ago
  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.
  - bmacho 4y ago
    Can you give examples?
    - syzarian 4y ago
      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.

Brigand 4y ago

How would that work?

bmacho 4y ago
https://www.startpage.com/sp/search?query=axiom+of+choice+infinite+prisoners https://www.startpage.com/sp/search?query=axiom+of+choice+in...

sharpneli 4y ago

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?