
The Axiom of Choice Is Wrong (2007) - jaybosamiya
https://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/
======
fmap
The problem solution in the article uses the axiom of choice to construct a
"nonprincipal ultrafilter" on the natural numbers. This is actually weaker
than the full axiom of choice, but you can still show that no such object is
computable. It's a nice exercise to show that with the same assumptions as in
the article you can decide the halting problem. (hint: consider the boolean
sequence where the nth element is true iff the Turing machine halts within n
steps)

As for the axiom of choice, the real problem is trying to claim that it is
right or wrong in the first place. Mathematics as a whole has never quite
recovered from the failure of Hilbert's program... The bottom line is that
there is no complete and consistent notion of "truth". There is no objective
mathematical reality, because it cannot include a statements about its own
consistency (and it's easy to translate this into "statements about certain
hard problems", by exactly the same process we use to show that some problems
are NP complete by reduction from another NP complete problem).

On the other hand, this is not actually detrimental to mathematical practice.
It only means that you have a lot more freedom in modeling your problem
domain. For instance, it turns out that set theory with the axiom of choice is
a horrible place to do probability theory in (non-measurable sets and
functions are a direct consequence, and you have to go to a lot of trouble to
exclude them everywhere). If ZFC was part of some objective mathematical
reality, then this would in some sense be unavoidable, since ultimately you
want to make statements describing reality. On the other hand, once we realize
that this assumption is just plainly false, we can start looking for more
refined models.

~~~
cygx
_There is no objective mathematical reality, because it cannot include a
statements about its own consistency_

I'm assuming we're talking about Gödel's incompleteness theorems?

Don't they just say that if there's such a thing as objective mathematical
reality, it can't be effectively axiomatized?

~~~
candiodari
> Don't they just say that if there's such a thing as objective mathematical
> reality, it can't be effectively axiomatized?

No they don't. Real space doesn't appear to be infinite, and Zn is not subject
to Godel's incompleteness theorem.

If you drop the requirement of infinite numbers and "recursive" infinites
(e.g. real numbers), as reality appears to do, there is no problem.

~~~
Jweb_Guru
> Real space doesn't appear to be infinite

Citation needed. People keep saying things like this ("real space isn't
infinite" or "real spacetime must be discrete") but never show their work.

~~~
warrenpj
There was a post here two weeks ago discussing this exact problem. [1] It
starts by explaining Democritus' reasoning that matter must be made of
discrete atoms. The main point of the article is that a similar logic applies
to the future theory of quantum gravity; space and time too must be discrete.

From the article:

> Importing the atomic philosophy of Democritus into modern physics might be
> essential for reconciling general relativity (which assumes a continuous
> reality) with quantum mechanics (which very much does not).

As far as infinite vs finite space: because there is a fixed rate of
information propagation, we only need to show that time is finite in extent,
and it follows that space is also finite. We know that the visible universe is
finite (by looking in the sky) and we can explain why: the big bang happened
at a finite time in the past.

We don't know if the universe will have a finite future, but it might be the
case. One possible cause of a finite future could be accelerating expansion of
the universe due to dark energy. [2]

Of course there are other interpretations of "real space", but this one
(causally connected) seems parsimonious to me.

[1]
[https://news.ycombinator.com/item?id=13466254](https://news.ycombinator.com/item?id=13466254)

[2]
[https://en.wikipedia.org/wiki/Ultimate_fate_of_the_universe](https://en.wikipedia.org/wiki/Ultimate_fate_of_the_universe)

~~~
Jweb_Guru
To the best of my knowledge, mainstream quantum mechanics assumes continuous
spacetime, so the argument that the primary reason it can't be reconciled with
general relativity is _continuity_ seems pretty specious to me. And it's
possible to come up with a model where the universe is infinite but the
visible universe is finite. In fact, I believe FLRW is one such a model, and
is supported with high accuracy by the latest experimental observations. So
again, your argument doesn't seem that compelling to me.

~~~
warrenpj
> To the best of my knowledge, mainstream quantum mechanics assumes continuous
> spacetime, so the argument that the primary reason it can't be reconciled
> with general relativity is continuity seems pretty specious to me.

Your criticism is right and that is a bad argument. The real reasons they are
hard to reconcile are technical (see for example [1]), and not mentioned in
the source I referred to.

The article only says that a discrete theory of spacetime "might be essential"
for the reconciliation; the arguments for this are philosophical - it would be
beautiful and historically completing. Read the article if you're interested
in the arguments, which aren't included in my comments.

> And it's possible to come up with a model where the universe is infinite but
> the visible universe is finite. In fact, I believe FLRW is one such a model,
> and is supported with high accuracy by the latest experimental observations.

I agree with you that the universe could be infinite, but that doesn't
contradict my previous comment. I explained that I was talking about local
(causally connected to us) space-time, only. I admit this is a bit of a trick,
but the parent comment used the term "real space" which I interpreted in such
a way to favour an argument for finite space.

> So again, your argument doesn't seem that compelling to me.

I've given reasons but I think it might be helpful to restate what I am
actually claiming. I admit I don't know that space and time are discrete. The
source I gave is a poetic and philosophical argument that it might be. I think
it's established fact that energy and matter are, and that time is finite in
the past, so the visible universe is finite. I think it's possible that the
future is finite too.

I don't know what FLRW is. My best guess is
[https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–W...](https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric)

[1]
[http://www.theory.caltech.edu/people/jhs/strings/str115.html](http://www.theory.caltech.edu/people/jhs/strings/str115.html)

~~~
cygx
> I think it's established fact that energy and matter are, and that time is
> finite in the past, so the visible universe is finite.

None of that is established, actually:

Metric expansion of space comes with a cosmic event horizon where redshift
goes to infinity. While the energy contained within that region is finite,
there's no reason to assume that the universe outside the horizon looks any
different than inside the horizon, and it might very well be spatially
infinite.

As to time being finite in the past, it is possible to generalize Einstein's
equations so that they can be extended past the big bang singularity, and what
you get is a mirror universe on the other side. The issue here is that as we
approach Planck scale, quantum gravity becomes relevant, and we have no
predictive theory of that.

------
CJefferson
As time goes by, I increasingly view things like uncountable infinities and
the axiom of choice as "a fun maths game", rather than having any intrinsic
truth or falsity. Other's view may differ.

Here is an interesting thing I've never seen anyone write down (I should do it
myself) -- we don't need uncountable infinities.

* How many natural numbers are there? Countable

* How many rational numbers are there? Countable

* How many numbers are solution to a polynomial? Countable

* How many numbers are the output of any turing machine (including programs that run forever, producing an infinite decimal)? Countable

* How many numbers are the answer to any maths problem anyone can write down (where the problem has at most a countable number of answers)? Countable.

If the set of all numbers any can express in any sensible way, and the
solution to any problem any could ever have, is countable, why do we need the
other uncountables?

~~~
heinrichhartman
Are you aware of the concept of periods ? This is a quite fascinating,
countable, ring of numbers, that captures all of the above:

Kontsevich and Zagier introduce and develop this concept quite far:

[http://www.maths.ed.ac.uk/~aar/papers/kontzagi.pdf](http://www.maths.ed.ac.uk/~aar/papers/kontzagi.pdf)

~~~
wolfgke
Just to give a little bit more attention to this link: Wikipedia link to one
of the authors:

>
> [https://en.wikipedia.org/wiki/Maxim_Kontsevich](https://en.wikipedia.org/wiki/Maxim_Kontsevich)

Proof that one of the authors really is no other "M. Kontsevich" (I openly
admit that I wanted to be sure since I associate Maxim Kontsevich mostly with
other mathematical areas):

>
> [http://www.ihes.fr/~maxim/publicationsanglais.html](http://www.ihes.fr/~maxim/publicationsanglais.html)

------
Grue3
There are many problems with this puzzle that go against the intuition.

\- the number of prisoners is infinite, so they will never finish answering
the question. At any point in time, only a finite number of prisoners will be
freed.

\- a single prisoner must process an infinite amount of information to reach
the decision. In fact, by observing only a finite number of hats he cannot
possibly choose the answer.

\- the number of equivalence classes is uncountable. Not even a countably
infinite number of prisoners can possibly have enough time to pick out a
single element from every equivalence class.

~~~
Tloewald
As soon as you start treating the axiom of choice as a superpower for a
conscious being you are in conceptual la la land.

How do I guess the color of my hat if there's an uncountable number of
possible colors? Doesn't that mean that communicating the value of that color
involves transmitting an infinite amount of informtion?

~~~
waqf
Quick, tell me your favourite number between 0 and 1.

How did you do that? Weren't there an uncountable number of possible numbers?

Ah, you may say, but I obviously wasn't going to choose one with an infinite
information content, so all but countably many possible numbers had
probability 0.

Which is true. But in fact it's true that _whenever_ you have a probability
measure on an uncountable space then all but countably many elements have
probability 0, so that escape clause applies equally well to the hat colours.

~~~
TheCoelacanth
But the axiom of choice applies to all sets, not just ones that you can easily
choose a number from. For instance, what is your favorite non-computable
number between 0 and 1?

~~~
red75prime
Inverse of Kolmogorov complexity of thirteenth bit-string with uncomputable
Kolmogorov complexity, of course.

------
ramblenode
What an interesting thought experiment. Lying in bed last night, the best I
could come up with (before seeing the optimal solution this morning) was an
average of 83 with a minimum of 66.

The prisoners agree that every third prisoner, beginning with the first, uses
"white" to convey that the subsequent two prisoners are wearing the same color
and "black" to convey different colors. Since the second prisoner in each
triple knows the color of the third, he can deduce his own color, leaving the
third prisoner to also deduce his own color. And of course there's a 50/50
chance the sacrificial first prisoner in the triple still gets out. I suppose
this could be improved upon by later prisoners having a longer memory, but
I've already seen the optimal solution. ;)

Interested to hear others' attempts.

------
Arnavion
Mathologer recently did a video on the puzzles mentioned in this article:
[https://www.youtube.com/watch?v=aDOP0XynAzA](https://www.youtube.com/watch?v=aDOP0XynAzA)

------
EGreg
The axiom of choice is used to demonstrate the _existence_ of something, in
this case a perfect strategy.

However, if said strategy's implementation requires actual infinities, eg each
prisoner having an infinite memory, then that is why you find it intuitively
objectionable.

It is useful here to think of computer algorithms and not just math. While
mathematical arguments have no problem supposing infinite amounts of actors,
the next question is whether each actor can have infinite memory.

In mathematics, infinity can be thought of as a property of a _set_. It can
also be thought of as some limit of an infinite sequence of operations on
sets, which is a statement that is simultaneously true about each member of
that sequence.

This is useful because it can tie constructions we observe in the real world
into patterns that approximate and converge to the limit of this infinite
sequence. And then the question is how the computational complexity grows.

So in your example here, each FINITE set of prisoners can't coordinate a
strategy. So there is no "approaching a limit" \- the thing only starts
working with an infinite set of prisoners, each of whom has infinite memory
etc. And that is why you get your intuition alarm bells go off :)

But it is even more than that. Your construction requires each prisoner to
_use the axiom of choice in order to take an action based on the NAME of the
chosen member_ which is used to demonstrate the existence of a sequence of
_actions_ that satisfies a certain property. However, when the axiom of choice
is used normally, it is not used to actually NAME the chosen element, but
merely work with it like a black box. By NAME, I mean an id that distinguishes
it from all other elementa, and lets you pick it out and examine is properties
THAT ARE DIFFERENT than all other elements in that set.

In other words, Sure, you can assume that the chosen "representative" sequence
has the same property as any other in the equivalence class -- namely that all
but finitely many terms are equal. BUT the part where you "cheat" is having
the prisoner "find out" more than that about the representative sequence, in
particular its initial values up to an arbitrary depth.

------
twic
The axiom of choice always seemed intuitively wrong to me. You can't just take
a set and arbitrarily pick something out of it! Making a choice requires
information, and you can't pluck information out of thin air at whim; applying
the axiom amounts to creating information out of nothing.

I suppose this is because i'm not a mathematician, but have a natural sciences
background. In the physical universe, memorably, "the law that entropy always
increases holds, I think, the supreme position among the laws of Nature" [1],
and so we do not accept the mathematicians' fake information.

More specifically related to choice from a set, Curie's principle that "when
certain causes produce certain effects, it is the elements of symmetry of the
causes that may be found in the effects produced" [2] forbids something
uniform from becoming arbitrarily non-uniform; whenever that appears to
happen, there must be some hidden cause which already carries that non-
uniformity.

[1]
[https://en.wikipedia.org/wiki/Second_law_of_thermodynamics](https://en.wikipedia.org/wiki/Second_law_of_thermodynamics)

[2] [https://hal.archives-ouvertes.fr/jpa-00239814](https://hal.archives-
ouvertes.fr/jpa-00239814) \- "Enfin, lorsque certaines causes produisent
certains effets, les éléments de symétrie des causes doivent se retrouver
dans les effets produits."; please excuse the not-so-literal translation

~~~
hackinthebochs
> You can't just take a set and arbitrarily pick something out of it! Making a
> choice requires information

True, but how can you "have a set", i.e. reference a set in any way, without
having information about that set? There seems to be a requirement of some
bare minimum of information enough to specify the set, and so enough to pick
out a member of the set.

~~~
twic
Having enough information to specify the set isn't enough to pick out one
particular member.

For example, if i say "the colours teal, maroon, and taupe", you have enough
information to know what's in the set, but no extra information that would let
you pick one element out of it.

~~~
hackinthebochs
Well the issue is whether its possible in principle to define such a function
for all possible sets. But if you can enumerate the elements of the set, such
a function is simply to pick the first element.

So the issue is how much information is necessary to guarantee that its
possible in principle to construct such a function. It should be clear that
enough information to enumerate each set is enough to create a decision
function. The remaining question seems to be whether one can specify a
countable set of countable sets without specifying enough info about each
member set to enumerate the set's members. It seems trivial that a set with
countable elements is enumerable, but I might be missing some subtlety.

------
im3w1l
I'd assume that even if in every case the number of incorrect guesses is
finite, the _expected number_ of people that fail to guess their color is
infinite.

Am I right about this?

~~~
baddox
Im not sure what you mean by "expected number." Do you mean if you try to
guess how many inmates the warden has managed to guarantee will fail? Per the
article, the warden can guarantee that an "arbitrarily large finite number of
them" will fail. But it's still always finite despite being unbounded. If you
want to predict a lower bound on the number of failures the warden has
guaranteed, you just have to guess a larger natural number than the warden. :)

~~~
im3w1l
I mean if the warden picks a sequence randomly.

------
rtpg
I have a vague understanding of the Axiom of Choice, but I've always had
trouble with some of the analogies people use to explain it.

Two things that have bugged me for a while:

\- why is it usually talked about only in the context of infinite sets? Is
there a general trick to building a choice function if all you have are finite
sets?

\- There's a saying like "you can choose from an infinite set of shoes, but
not from an infinite set of socks without AC". Why exactly?

~~~
petteris
The axiom of choice is about making an _infinite_ number of _arbitrary_
choices simultaneously. If you can specify some rule, this rule is just one
choice. Finitely many choices are always fine and don't need the axiom of
choice.

If you have a finite set, you can number its elements and make rules by saying
"let's take the element with the smallest number having this or that
property", so you don't need the axiom of choice when dealing with finite
sets.

The point of Russell's shoes versus socks analogy is that shoes are
distinguishable while socks aren't: To choose one shoe from each of an
infinite set of pairs of shoes, you can always choose the left shoe, or
specify some pattern (so you don't need the axiom of choice), whereas when
choosing socks, you have to make an arbitrary choice to select one from each
pair (so you do need the axiom of choice).

------
wolfgke
An interesting alternative to the axiom of choice (AC) is the axiom of
determinacy (AD):
[https://en.wikipedia.org/w/index.php?title=Axiom_of_determin...](https://en.wikipedia.org/w/index.php?title=Axiom_of_determinacy&oldid=738341699)

------
DigitalPhysics
If you're interested in the Axiom of Choice, Godel, finitism, pseudo-
randomness, complexity, information, and other foundational topics, check out
the indie film "Digital Physics" on iTunes, Amazon, or Vimeo. Free packs of
trading cards (with gum!) are available too! Check the website.

------
bananabiscuit
This problem has exactly the wrong setup for using th the axiom of choice:

First, the axiom of choice requires that you have a countable number of sets
that you are choosing elements from, but there are undoubtably many of the
equivalence classes that he described [0]. So the author is using something
stronger than the axiom of choice to arrive at his paradox.

Second, if you actually are in a situation where you have to choose from a
countably infinite number of sets, you only need the axiom of choice if there
is no selection rule for choosing an element available. In this case there is
a rule you can use, namely: select the sequence in which the "finite prefix"
is all zeros.

[0]: the number of equivalence classes is uncountable because there is a 1:1
relation between the equivelance class and an the infinite sequence that is
common to all the sequences in the equivalence class once theirs uncommon
prefixes have been truncated.

~~~
n4r9
> the axiom of choice requires that you have a countable number of sets that
> you are choosing elements from

You're referring to the "axiom of countable choice", which is a different
axiom. The Wikipedia entry for the axiom of choice makes it clear that the
number of sets can be uncountable.

> select the sequence in which the "finite prefix" is all zeros

This doesn't really make sense as a selection rule. The size of the prefix can
vary between pairs of members from the same equivalence class.

------
mrob
Infinities aren't real, so you shouldn't be surprised if unrealistic things
happen when you invoke infinities. That you can duplicate a sphere by cutting
it into a finite number of pieces and reassembling it is a "fact" in the same
sense as "Luke Skywalker destroyed the Death Star". It might be interesting
and culturally important, but it's talking about fictional entities. Both
spheres and arbitrarily detailed pieces of spheres only exist in the
imaginations of mathematicians. The Axiom of Choice is obviously true, and the
fact that it lets you invent weird sounding stories from weird components is
no evidence against it. Don't confuse mathematical tools with reality.

~~~
llamaz
Mathematics is formalised is to avoid this sort of philosophizing.

I used to think, for example, that the dirac delta function was mathematical
fiction - a mathematical "hack". But then in an engineering control systems
class, we did an experiment where we used a step function to approximate a
dirac delta function. I could see the results both on the computer screen and
in physical reality through a mass-spring-damper system. From that moment on I
saw the dirac delta function in the same way that I see cosine/sine: The
reason it works in math is because it has a basis in physical reality.

The lesson to be learned here, is that you don't know in advance whether
something is obvious or not. To me, it doesn't make sense to decide whether
the axiom of choice can be justified by looking at the axiom itself. You have
to look at where it's used and required, and whether the proofs convey
something that matches your physical intuition.

~~~
vostok
> I used to think, for example, that the dirac delta function was mathematical
> fiction - a mathematical "hack".

Can you expand on this? The Dirac delta function is not a function from R to R
for example.

~~~
monochromatic
Treating it as a function is the fiction (and the reason is probably that the
average college freshman doesn't know what a distribution is). But it's
"enough" like a function that unless you look too closely, no real problems
arise from glossing over the function/distribution distinction.

~~~
vostok
Right. That was kind of my point. It is a "mathematical trick" in the sense
that it's not a function, but a distribution.

~~~
llamaz
You're right, but what I should have emphasized is that if you accept that
cos/sin are based in physical reality because they model a circle, then you
have to have just as much faith in the dirac delta.

Cos/sin are infinitely precise, and can't be reduced to algebra (as opposed to
calculus/analysis). This in itself is an idealisation, a fiction.Sin/cos seem
natural because they model the ideal unit circle, while the dirac delta is
natural because it models an ideal impulse. The latter seems more abstract
than the former because we all have been exposed to unit circles, but usually
we are not exposed to unit impulses.

You can apply a step function to a mass-spring-damper system, then observer
the response. Thereafter, you can make the step function narrower but taller
and observer the response. As you continue this process, the response
approaches something simple. Maybe this way of thinking about the dirac delta
in obvious to everyone, but to me it was a major breakthrough because I
couldn't imagine how something infinitely tall and infinitely narrow could
model anything in the real world.

So my ultimate takeaway message is that cos/sin are to an n-gon what the dirac
delta is to a step function, and that they are just as "real" as one another.
Alternately, the dirac delta doesn't work for "symbolic" or algebraic reasons.
It's an idealisation of reality, not an abstraction _away_ from reality for
the sake of convenience (e.g. how we sometimes artificiallyl define 0^0 (zero
to the zero) to be 1 in some cases or 0 in other cases)).

~~~
vostok
I was under the impression that we were talking about mathematics and not
physical reality.

In mathematics, cos/sin are just functions cos:R -> R and sin:R -> R. As far
as functions go, Dirac delta is kind of a fiction because we don't have a
function δ: R -> R. That's why I think that your initial characterization was
correct.

Of course if we define it with distribution or measure then we don't have this
issue.

> Maybe this way of thinking about the dirac delta in obvious to everyone, but
> to me it was a major breakthrough because I couldn't imagine how something
> infinitely tall and infinitely narrow could model anything in the real
> world.

By the way, if you're looking for intuition on the Dirac delta I think a good
way to think of it is just the identity element under convolution.
Equivalently, you can think of it as a thing whose Fourier transform is 1.

------
alsadi
> Sure, this is based primarily on my intuition for finite things and a naive
> hope that they should extend to infinities.

but it's known that it does not extend!

------
pron
So what bothered the author was the axiom of choice and not the part where a
prisoner with a finite brain needs to memorize an infinite amount of
information and then perform a computation on infinite information to compute
the equivalence class? For this strategy to work you must assume that the
prisoners are capable of carrying out noncomputable computations, too. _That
's_ the more problematic assumption.

------
eru
You can generalize their prisoner example to using a countable number of
colours, not just two or finite.

------
jiiam
In the comments to the linked article there's a nice explanation by Terry Tao
of why this is not so spectacular, in the sense that our intuition with
probabilities here relies on Fubini's theorem, which in this case do not apply
due to measure theoretic obstacles. But, again, our intuition of probability
fails with much easier examples.

Here follows Terence Tao's comment (refer to the original to have proper
rendering of math symbols):

"This paradox is actually very similar to Banach-Tarski, but involves a
violation of additivity of probability rather than additivity of volume.

Consider the case of a finite number N of prisoners, with each hat being
assigned independently at random. Your intuition in this case is correct: each
prisoner has only a 50% chance of going free. If we sum this probability over
all the prisoners and use Fubini’s theorem, we conclude that the expected
number of prisoners that go free is N/2\. So we cannot pull off a trick of the
sort described above.

If we have an infinite number of prisoners, with the hats assigned randomly
(thus, we are working on the Bernoulli space {\Bbb Z}_2^{\Bbb N}), and one
uses the strategy coming from the axiom of choice, then the event E_j that the
j^th prisoner does not go free is not measurable, but formally has probability
1/2 in the sense that E_j and its translate E_j + e_j partition {\Bbb
Z}_2^{\Bbb N} where e_j is the j^th basis element, or in more prosaic
language, if the j^th prisoner’s hat gets switched, this flips whether the
prisoner gets to go free or not. The “paradox” is the fact that while the E_j
all seem to have probability 1/2, each element of the event space lies in only
finitely many of the E_j. This can be seen to violate Fubini’s theorem – if
the E_j are all measurable. Of course, the E_j are not measurable, and so
one’s intuition on probability should not be trusted here.

There is a way to rephrase the paradox in which the axiom of choice is
eliminated, and the difficulty is then shifted to the construction of product
measure. Suppose the warden can only assign a finite number of black hats, but
is otherwise unconstrained. The warden therefore picks a configuration
“uniformly at random” among all the configurations with finitely many black
hats (I’ll come back to this later). Then, one can again argue that each
prisoner has only a 50% chance of guessing his or her own hat correctly, even
if the prisoner gets to see all other hats, since both remaining
configurations are possible and thus “equally likely”. But, of course, if
everybody guesses white, then all but finitely many go free. Here, the
difficulty is that the group \lim_{n \to \infty} {\Bbb Z}_2^n is not compact
and so does not support a normalised Haar measure. (The problem here is
similar to the two envelopes problem, which is again caused by a lack of a
normalised Haar measure.)"

------
IshKebab
How can an axiom be 'wrong'?

~~~
dvt
An axiom is simply a statement that is _defined_ as being true. Often times on
grounds of being self-evident. For example, one of Euler's axioms is "Things
that are equal to the same thing are also equal to one another".

Whereas Euler's may seem pretty tame, the Axiom of Choice is not. That's why
some people think it might be wrong.

~~~
IshKebab
Exactly, so unless it leads to contradictions it may be weird and unintuitive
but it's not _wrong_.

------
Ceezy
YOUR MATH ARE WRONG

If prisonners follow the last strategy, the dude number 0 as a probability of
finding his hat of 50% and the dude number 1 000 000 50%. Why? because nobody
knows how many times they will lose. Those who are sure to win are those close
to infinity... This have nothing to do with Axiom of Choice. But only because
almost all the mass of your distribution is near infinty. With or without
Axiom of Choice this things exist.

