

The Axiom of Choice is Wrong (2007) - rw
http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/

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limmeau
So this guy is against using the axiom of choice because a countably infinite
number of prisoners in a row who have agreed on an infinite set of equivalence
classes just between breakfast and hat game time and remember them and each of
them can recognize infinitely many hat colors in finite time (takes deep
breath) suddenly becomes unintuitive using the axiom of choice?

I can't help him there.

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bdr
Be sure to check out the comments. Lots of good discussion in there, on both
the paradox at hand and mathematical Platonism in general, by people who know
what they're talking about.

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bitdiddle
More troubling to me has been the continuum hypothesis. The notion that you
can toss a dart (whose tip is the width of a point) at the real number line
and hit an integer or rational number with probability zero is very
unintuitive.

I can kind of grasp the continuum hypothesis, there does seem to be a
distinction between countable and uncountable infinity. Continuing the game
beyond that to this area of large cardinals strikes me as just a language
game.

Most mind blowing, IMHO, is Cantor's middle third's set[1]. Uncountable
nowhere dense totally disconnected, ... it's an amazing set and so easy to
construct.

[1] <http://en.wikipedia.org/wiki/Cantor_set>

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stanleydrew
You're not talking about the continuum hypothesis. The continuum hypothesis
just states that there isn't any set with cardinality larger than the natural
numbers (aleph_0) but smaller than the reals (2^aleph_0). Interestingly this
can't be proven or disproven within set theory if you assume the axiom of
choice.

Comparing the cardinality of the natural and real numbers doesn't have much to
do with the continuum hypothesis. It's simply true that there are many many
many more reals than rationals. It's not a hypothesis.

But I do agree that the Cantor set is pretty awesome.

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bitdiddle
Actually I was talking about it. My mind wanders. Sorry to be so vague (it's
been years since I read this stuff). I didn't mean to imply sets of measure
zero were what CH is about. The reals can be essentially equated with the
powerset of the naturals. It's very intriguing that there is no set with
cardinality in between that of the naturals and that of the reals.

The Axiom of Choice is also independent of ZF set theory. Have you read the
more modern expositions of these independence proofs based on various Topoi?

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jules
> Its interesting to notice that a larger number of hat colors poses no
> problem here. For any set of hat colors , the prisoners can pick an abelian
> group structure on . Then, the first prisoner guesses the ’sum’ of all the
> hat colors he can see. The next guy can then subtract the sum of the hat
> colors he sees from the hat color the first guy said to find his own hat
> color. Again, this argument repeats, and so everyone except the first guy
> gets out. For the case of black and white, the previous argument used black
> = 0 (mod 2) and white = 1 (mod 2).

So the first guy says the sum, not a color? If he's allowed to say arbitrary
things he can as well tell everyone their color "next guy is white, next
green, etc.".

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jibiki
No, he translates the sum back to colors. E.g., there is a map f:colors->H
with an inverse g:H->colors and he says the color g(sum(f(hat))) where the
index hat ranges over all of the hats.

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yummyfajitas
This is still only an argument against the uncountable axiom of choice.

The set of equivalence classes of infinite strings is uncountable, so using
the axiom of choice here is invalid.

As far as I know, the countable one is still reasonable.

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jganetsk
I feel like the axiom of choice can't be used here.

Let's say we have a infinitely long bit string, b. This equivalence class has
an infinite number of elements in it! That is, there is an infinite number of
strings with suffix b.

b, 0b, 1b, 00b, 01b, 10b, 11b, etc.

Can you use the axiom of choice in this case? I think it is required that each
bin has a finite number of objects, even though there are infinite bins.

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ggchappell
> Can you use the axiom of choice in this case?

Yes, you can.

> I think it is required that each bin has a finite number of objects, even
> though there are infinite bins.

No, the axiom says only that the product of nonempty sets is nonempty. It
doesn't say anything about the number of sets or their sizes.

For those who don't see the connection: Each element of a product of sets
consists of a choice of one item from each set. Saying the product is nonempty
is saying there _is_ a way to make such choices. For example, the product of
{1,2} and {a,b} contains, among other things, the pair (1, b), which can be
thought of as choosing 1 from the first set and b from the second.

There _are_ restricted versions of the Axiom of Choice (e.g., Countable
Choice, Dependent Choice); however, none of these are called "the Axiom of
Choice".

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jganetsk
Ok. My other gripe with his argument is that these aren't equivalence classes.
Let the string be 01b, where b is an infinitely long bit string. Then the
string 01b belongs to equivalence class b, and also equivalence class 1b...
and an element cannot belong to multiple equivalence classes at once.

The only way out of this is for 1b and b to actually be the same equivalence
class, that is, that of b. But, can we do that? I feel like that is wrong.

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thisrod
It's like saying 128x and 28x, considered as finite sequences of digits
followed by a letter, are in the class of sequences that end in x. Except in
the infinite case, there are a lot more than 26 letters.

~~~
jganetsk
They also happen to be in the class of sequences that end in 8x and 28x...
also definitely their own classes. So 128x is in at least 4 classes, 28x is in
at least 3. Therefore it's not an equivalence relation.

~~~
ggchappell
Ah, but 8x and 28x are not different classes; they are all the same class.

I think it's clearer if you describe things the way the article does. Don't
say "these are the classes"; say "here is the relation".

As the article says:

> Call two such sequences ‘equivalent’ if they are equal after a finite number
> of entries.

To see whether this is an equivalence relation, you only need to check whether
it is reflexive, symmetric, and transitive. Figuring out what the classes are,
might be interesting, but it is not necessary.

~~~
jganetsk
If an item belongs to multiple equivalence classes, then transitivity does not
hold. In other words, if you show that the classes are not disjoint, then it's
not an equivalence relation.

18x belongs in the 8x class, but 28x belongs in both the 28x and 8x classes.

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jibiki
And 8x belongs in 28x's class, since 8x and 28x share a tail.

I think the author meant for you to interpret "equal after a finite number of
entries" in the more normal fashion (e.g., 123x = 456x but 1x != 111x, the
prefixes have to have equal length.) But your way seems to work fine too.

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ShardPhoenix
Looks like another example of infinity leading to nonsensical (or at least
unintuitive) results.

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Create
If you choose not to decide, you still have made a choice. -- unsourced René
Descartes

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danbmil99
Sounds like Bridge. I hate Bridge!

