Axiom of Choice

[summary: The axiom of choice states that given an infinite collection of non-empty sets, there is a function that picks out one element from each set. ]

"The Axiom of Choice is necessary to select a set from an infinite number of pairs of socks, but not an infinite number of pairs of shoes." — *Bertrand Russell,  Introduction to mathematical philosophy*

"Tarski told me the following story. He tried to publish his theorem \[the equivalence between the Axiom of Choice and the statement 'every infinite set A has the same cardinality as AxA'\] in the Comptes Rendus Acad. Sci. Paris but Fréchet and Lebesgue refused to present it. Fréchet wrote that an implication between two well known propositions is not a new result. Lebesgue wrote that an implication between two false propositions is of no interest. And Tarski said that after this misadventure he never again tried to publish in the Comptes Rendus."
- *Jan Mycielski, A System of Axioms of Set Theory for the Rationalists*

[toc:]

#Obligatory Introduction#
The Axiom of Choice, the most controversial axiom of the 20th Century.

The axiom states that a certain kind of function, called a `choice' function, always exists. It is called a choice function, because, given a collection of non-empty sets, the function 'chooses' a single element from each of the sets. It is a powerful and useful axiom, asserting the existence of useful mathematical structures (such as bases for [-3w0 vector spaces] of arbitrary [-dimension_mathematics dimension], and [-ultraproduct ultraproducts]). It is a generally accepted axiom, and is in wide use by mathematicians. In fact, according to Elliott Mendelson in Introduction to Mathematical Logic (1964) "The status of the Axiom of Choice has become less controversial in recent years. To most mathematicians it seems quite plausible and it has so many important applications in practically all branches of mathematics that not to accept it would seem to be a wilful hobbling of the practicing mathematician. "

Neverless, being an [-axiom_mathematics axiom], it cannot be proven and must instead be assumed.
In particular, it is an axiom of [-set_theory set theory] and it is not provable from the other axioms (the Zermelo-Fraenkel axioms of Set Theory). In fact many mathematicians, in particular [-constructive_mathematics constructive] mathematicians, reject the axiom, stating that it does not capture a 'real' or 'physical' property, but is instead just a mathematical oddity, an artefact of the mathematics used to approximate reality, rather than reality itself. In the words of the LessWrong community: the constructive mathematicians would claim it is a statement about [-https://wiki.lesswrong.com/wiki/Map_and_Territory_(sequence) the map, and not the territory].

Historically, the axiom has experienced much controversy. Before it was shown to be independent of the other axioms, it was believed either to follow from them (i.e., be 'True') or lead to a contradiction (i.e., be 'False'). Its independence from the other axioms was, in fact, a very surprising result at the time.

#Getting the Heavy Maths out the Way: Definitions#
Intuitively, the [-axiom_mathematics axiom] of choice states that, given a collection of *[-5zc non-empty]* [-3jz sets], there is a [-3jy function] which selects a single element from each of the sets.

More formally, given a set $X$ whose [-5xy elements] are only non-empty sets, there is a function
$$
f: X \rightarrow \bigcup_{Y \in X} Y
$$
from $X$ to the [-5s8 union] of all the elements of $X$ such that, for each $Y \in X$, the [-3lh image] of $Y$ under $f$ is an element of $Y$, i.e., $f(Y) \in Y$.

In [-logical_notation logical notation],
$$
\forall_X
\left(
\left[\forall_{Y \in X} Y \not=  \emptyset \right]
\Rightarrow
\left[\exists
\left( f: X \rightarrow \bigcup_{Y \in X} Y \right)
\left(\forall_{Y \in X}
\exists_{y \in Y} f(Y) = y \right) \right]
\right)
$$

#Axiom Unnecessary for Finite Collections of Sets#
For a [-5zy finite set]  $X$ containing only [-5zy finite] non-empty sets, the axiom is actually provable (from the [-zermelo_fraenkel_axioms Zermelo-Fraenkel axioms] of set theory ZF), and hence does not need to be given as an [-axiom_mathematics axiom]. In fact, even for a finite collection of possibly infinite non-empty sets, the axiom of choice is provable (from ZF), using the [-axiom_of_induction axiom of induction]. In this case, the function can be explicitly described. For example, if the set $X$ contains only three, potentially infinite, non-empty sets $Y_1, Y_2, Y_3$, then the fact that they are non-empty means they each contain at least one element, say $y_1 \in Y_1, y_2 \in Y_2, y_3 \in Y_3$. Then define $f$ by $f(Y_1) = y_1$, $f(Y_2) = y_2$ and $f(Y_3) = y_3$. This construction is permitted by the axioms ZF.

The problem comes in if $X$ contains an infinite number of non-empty sets. Let's assume $X$ contains a [-2w0 countable] number of sets $Y_1, Y_2, Y_3, \ldots$. Then, again intuitively speaking, we can explicitly describe how $f$ might act on finitely many of the $Y$s (say the first $n$ for any natural number $n$), but we cannot describe it on all of them at once.

To understand this properly, one must understand what it means to be able to 'describe' or 'construct' a function $f$. This is described in more detail in the sections which follow. But first, a bit of background on why the axiom of choice is interesting to mathematicians.

#Controversy: Mathematicians Divided! Counter-Intuitive Results, and The History of the Axiom of Choice#
Mathematicians have been using an intuitive concept of a set for probably as long as mathematics has been practiced.
At first, mathematicians assumed that the axiom of choice was simply true (as indeed it is for finite collections of sets).

[-https://en.wikipedia.org/wiki/Georg_Cantor Georg Cantor] introduced the concept of [-transfinite_number transfinite numbers]
and different [-4w5 cardinalities of infinity] in a 1874
[https://en.wikipedia.org/wiki/Georg_Cantor%27s_first_set_theory_article paper] (which contains his infamous
[-https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument Diagonalization Argument])
 and along with this sparked the introduction of [-set_theory set theory].
 In 1883, Cantor introduced a principle called the 'Well-Ordering Princple'
(discussed further in a section below) which he called a 'law of thought' (i.e., intuitively true).
He attempted to prove this principle from his other principles, but found that he was unable to do so.

[-https://en.wikipedia.org/wiki/Ernst_Zermelo Ernst Zermelo] attempted to
develop an [-axiom_system axiomatic] treatment of set theory. He
 managed to prove the Well-Ordering Principle in 1904 by introducing a new principle: The Principle of Choice.
This sparked much discussion amongst mathematicians. In 1908 published a paper containing responses to this debate,
as well as a new formulation of the Axiom of Choice. In this year, he also published his first version of
the set theoretic axioms, known as the [-https://en.wikipedia.org/wiki/Zermelo_set_theory Zermelo Axioms of Set Theory].
Mathematicians, [-https://en.wikipedia.org/wiki/Abraham_Fraenkel Abraham Fraenkel] and
[-https://en.wikipedia.org/wiki/Thoralf_Skolem Thoralf Skolem]  improved this system (independently of each other)
into its modern version, the [-https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
 Zermelo Fraenkel Axioms of Set Theory].

In 1914, [https://en.wikipedia.org/wiki/Felix_Hausdorff Felix Hausdorff] proved
[https://en.wikipedia.org/wiki/Hausdorff_paradox Hausdorff's paradox]. The ideas
behind this proof were used in 1924 by [-https://en.wikipedia.org/wiki/Stefan_Banach
Stefan Banach] and [-https://en.wikipedia.org/wiki/Alfred_Tarski Alfred Tarski]
to prove the more famous Banach-Tarski paradox (discussed in more detail below).
This latter theorem is often quoted as evidence of the falsehood of the axiom
of choice.

Between 1935 and 1938, [-https://en.wikipedia.org/wiki/Kurt_G%C3%B6del Kurt Gödel] proved that
the Axiom of Choice is consistent with the rest of the ZF axioms.

Finally, in 1963, [-https://en.wikipedia.org/wiki/Paul_Cohen Paul Cohen] developed a revolutionary
mathematical technique called [-forcing_mathematics forcing], with which he proved that the
axiom of choice could not be proven from the ZF axioms (in particular, that the negation of AC
is consistent with ZF). For this, and his proof of the consistency of the negation of the
[-continuum_hypothesis Generalized Continuum Hypothesis] from ZF, he was awarded a fields medal
in 1966.

This axiom came to be accepted in the general mathematical community, but was rejected by the
[-constructive_mathematics constructive] mathematicians as being fundamentally non-constructive.
However, it should be noted that in many forms of constructive mathematics,
there are *provable* versions of the axiom of choice.
The difference is that in general in constructive mathematics, exhibiting a set of non-empty sets
(technically, in constructive set-theory, these should be 'inhabited' sets) also amounts to
exhibiting a proof that they are all non-empty, which amounts to exhibiting an element for all
of them, which amounts to exhibiting a function choosing an element in each. So in constructive
mathematics, to even state that you have a set of inhabited sets requires stating that you have a choice
function to these sets proving they are all inhabited.

Some explanation of the history of the axiom of choice (as well as some of its issues)
can be found in the
paper "100 years of Zermelo's axiom of choice: what was the problem with it?"
by the constructive mathematician
[-https://en.wikipedia.org/wiki/Per_Martin-L%C3%B6f Per Martin-Löf]
at [-http://comjnl.oxfordjournals.org/content/49/3/345.full this webpage].

(Martin-Löf studied under [-https://en.wikipedia.org/wiki/Andrey_Kolmogorov Andrey Kolmogorov] of
 [-5v Kolmogorov complexity] and has made contributions to [-3qq information theory],
[-statistics mathematical_statistics], and [-mathematical_logic mathematical_logic], including developing a form of
intuitionistic [-3sz]).

A nice timeline is also summarised on [-http://plato.stanford.edu/entries/axiom-choice/index.html#note-6
The Stanford Encyclopaedia of Logic].

#So, What is this Choice Thing Good for Anyways?#
The Axiom of Choice is in common use by mathematicians in practice. Amongst its many applications are the following:

###Non-Empty Products###
This is the statment that taking the mathematical [-product_of_sets product] of non-empty sets will always yield a non-empty set.
Consider infinitely many sets $X_1, X_2, X_3, \ldots$ indexed by the natural numbers.
 Then an element of the product $\prod_{i \in \mathbb{N}} X_i$ is a family (essentially an infinite tuple)
of the form $(x_1, x_2, x_3, \ldots )$ where $x_1 \in X_1$, $x_2 \in X_2$ and so on. However, to select such a tuple in
the product amounts to selecting a single element from each of the sets. Hence, without the axiom of choice, it is
not provable that there are any elements in this product.

Note, that without the axiom, it is not necessarily the case that the product is empty. It just isn't provable that
there are any elements. However, it is consistent that there exists some product that is empty. Also, bear in mind
that even without the axiom of choice, there are products of this form which are non-empty. It just can't be shown
that they are non-empty in general.

However, it does seem somewhat counterintuitive that such a product be non-empty. In general, given two non-empty sets,
$X_1$ and $X_2$, the product is at least as large as either of the sets. Adding another non-empty set $X_3$ usually makes
the product larger still. It may seem strange, then, that taking the limit of this procedure could result in an empty
structure.

###Existence of a basis for any (esp. infinite dimensional) vector space###
This example is discussed in more detail in the next section.

A [-3w0] can be intuitively thought of as a collection of [-vector vectors] which themselves can be thought of as arrows (or the information of a 'direction' and a 'magnitude').

A vector space has a number of 'directions' in which the vectors can point, referred to as its [-vector_space_dimension]. For a finite-dimensional vector space, it is possible to find a [-vector_space_basis basis] consisting of vectors. The property of a basis is that:

 - any vector in the space can be built up as a combination of the
   vectors in the basis
 - none of the vectors in the basis can be built up as a combination of the rest of the vectors in the basis.

For finite-dimensional vector spaces, such a basis is finite and can be found. However, for infinite-dimensional vector spaces, a way of finding a basis does not always exist. In fact, if the axiom of choice is false, then there is an infinite-dimensional vector space for which it is impossible to find a basis.

###Brouwer's Fixed-Point Theorem: the existence of a fixed point for a function from a "nice" shape to itself###
Any [-continuous_function continuous functions] $f: C \rightarrow C$ from a [-closed_disk] $C$ onto itself has a [-fixed_point] $x_0$. (In full generality, $C$ may be any [-5xr] [-compact_mathematics compact] [-3jz]).

In other words, there is at least one point $x_0 \in C$ such that $f(x_0) = x_0$.

This works also for rectangles, and even in multiple dimensions. Hence, if true for real world objects, this theorem has the following somewhat surprising consequences:

 - Take two identical sheets of graph with marked coordinates. Crumple up sheet B and place the crumpled ball on top of sheet A. Then, there is some coordinate $(x , y)$ (where $x$ and $y$ are [-4bc real numbers], not necessarily [-48l integers]), such that this coordinate on sheet B is directly above that coordinate on sheet A.
 - Take a cup of tea. Stir it and let it settle. There is some point of the tea which ends up in the same place it started.
 - Take a map of France. Place it on the ground in France. Take a pin. There is a point on the map through which you could stick the pin and the pin will also stick into the ground at the point represented on the map.

###Existence of ultrafilters and hence ultraproducts###
The following example is somewhat technical. An attempt is made to describe it very roughly.

Given an indexing set $I$, and a collection of mathematical structures
$(A_i)_{i \in I}$
(of a certain type) indexed by $I$. (For example, let $I$ be the [-45h natural numbers] $\mathbb{N}$ and let each of the mathematical structures $A_n$ be numbered).

An [-ultrafilter] $\mathcal{U}$ on $I$ is a collection of subsets of $I$ of a special type. Intuitively it should be thought of as a collection of `big subsets' of $I$. It is possible to form the set of all [-cofinite] subsets
of $I$ without the axiom of choice, and $\mathcal{U}$ should contain at least these. However, for mathematical reasons, $\mathcal{U}$ should also contain 'as many sets as possible'. However, in order to do so, there are some 'arbitrary choices' that have to be made. This is where the axiom of choice comes in.

One of the applications of ultrafilters is [-ultraproduct ultraproducts]. For each subset $X \subseteq I$ such that $X \in \mathcal{U}$ there is a subcollection
$(A_i)_{i \in X}$ of $(A_i)_{i \in I}$. Call such a subcollection a "large collection". The ultraproduct $A$ is a structure that captures the properties of large collections of the $A_i$s, in the sense that a statement of [-first_order_logic] is true of the ultraproduct $A$ if and only if it is true of some large collection of the $A_i$s.

Now, any statement that is either true for cofinitely many $A_i$s or false for cofinitely many $A_i$s will be true or false respectively for $A$. But what about the other statements? This is where the arbitrary choices come in. Each statement needs to be either true or false of $A$, and we use the axiom of choice to form an ultrafilter that does that for us.

One basic example of an application of ultrafilters is forming the [-nonstandard_real_numbers].

Further examples of applications of the axiom of choice may be found on
the Wikipedia page [-https://en.wikipedia.org/wiki/Axiom_of_choice#Equivalents here]
and [-https://en.wikipedia.org/wiki/Axiom_of_choice#Results_requiring_AC_.28or_weaker_forms.29_but_weaker_than_it here].

#Physicists Hate Them! Find out How Banach and Tarski Make Infinity Dollars with this One Simple Trick! #

One of the most counter-intuitive consequences of the Axiom of Choice is the Banach-Tarski Paradox. It is a theorem provable using the Zermelo-Fraenkel axioms along with the axiom of choice.

This theorem was proven in a 1924 paper by [-https://en.wikipedia.org/wiki/Stefan_Banach Stefan Banach] and
[-https://en.wikipedia.org/wiki/Alfred_Tarski Alfred Tarski].

Intuitively, what the theorem says is that it is possible to take a ball, cut it into five pieces, rotate and shift these pieces and end up with two balls.

Now, there are some complications, including the fact the pieces themselves are infinitely complex, and the have to pass through each other when they are being shifted. There is no way a practical implementation of this theorem could be developed. Nevertheless, that the volume of a ball could be changed just by cutting, rotating and shifting seems highly counter-intuitive.

A suprisingly good video explanation in laymans terms by Vsauce can be found [-https://www.youtube.com/watch?v=s86-Z-CbaHA Here].

The video [-https://www.youtube.com/watch?v=ZUHWNfqzPJ8
Infinity shapeshifter vs. Banach-Tarski paradox] by Mathologor advertises itself as a
 prequel to the above video, which puts you 'in the mindset' of a mathematician, so-to-speak, and makes the
result a bit less surprising.

This theorem is the main counterexample used as evidence of the falsehood of the Axiom of Choice. If not taken as evidence of its falsehood, thsi is at least used as evidence of the counter-intuitiveness of AC.

Q: What's an Anagram of Banach-Tarski?
A: Banach-Tarski Banach Tarski

#How Something Can Exist Without Actually Existing: The Zermelo Fraenkel Axioms and the Existence of a Choice Function#
The [-https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory Zermelo-Fraenkel axioms of Set Theory]
(ZF)
introduce a fundamental concept, called a [-3jz set], and a [-3nt relation] between
sets, called
the [-5xy element of] relation (usally written $\in$) where $x \in X$ should be interpreted as
'$x$ is contained inside $X$ in the way that an item is contained inside a box'. There are then a number
of [-axiom_mathematics axioms] imposed on these fundamental objects and this relation.

What one must remember is that theorems derived from these axioms are merely statements of the form that
'if one has a system which satisfies these laws (even if, for example $\in$ is interpreted as something entirely
different from being contained inside something), then it must also satisfy the statements of the theorems'.
However, its general use is to imagine sets as being something like boxes which contain mathematical objects.
Moreover, almost any statement of mathematics can be stated in terms of sets, where the mathematical objects
in question become sets of a certain kind. In this way, since the mathematical objects in question are set up
to satisfy the axioms, then anything which can be derived from these axioms will also hold for the mathematical
objects.

In particular, a [-function_mathematics function] can be interpreted as a specific kind of set. In particular,
it is a set of [-ordered_pair ordered pairs] (more generally, [-tuple ordered n-tuples], each of which can
itself be interpreted as a specific kind of set) satisfying a specific property.

There are different ways of stating the same axioms (by separating or combining axioms, giving different
formulations of the same axioms, or giving different axioms that are equivalent given the other axioms) hence
what follows is only a specific formulation, namely, the
[-https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory Zermelo-Fraenkel one from Wikipedia].

The axioms begin by stating that two sets are the same if they have the same elements.
Then the axiom regularity states sets are well-behaved in a certain way that's not so
important to us right now.

Now comes the part that is important for our purposes:
The axiom schema of specification (actually a schema specifying
infinitely many axioms, but we can pretend it is
just one axiom for now). This is an axiom asserting the *existence* of certain sets.
In a sense, it allows one to 'create' a new set out of an existing one. Namely,
given a set $X$ and a statement $\phi$ of [-first_order_logic first order logic]
(a statement about sets of a specific, very formal form,
and which uses only the $\in$ symbol and the reserved symbols of logic), it is
possible to create a set $\{x \in X : \phi(x) \}$ of all of the elements of
$X$ for which the formula $\phi$ is true.

For example, if we know (or assume) the set of all numbers $\mathbb{N}$ exists, and
we have some way of formalising the statement '$x$ is an even number' as a first-order
statement $\phi(x)$, then the set of all even numbers exists.

Additionally the axioms of pairing and union, axiom schema of replacement,
and axiom of power set are all of the form "Given that some sets $A, B, C, \ldots$
exist, then some other set $X$ also exists. The axiom of infinity simply states
that an infinite set with certain properties exists.

Notice that all of the above are axioms. It is not expected that any of them be proven.
They are simply assumptions that you make about whichever system you
want to reason. Any theorems that you can prove from these axioms will then be true
about your system. However, mathematicians generally agree that these axioms capture
our intuitive notion about how "sets" of objects (or even concepts)
 should behave, and about which sets we are allowed to reason (which sets 'exist').
Most of these (except maybe the axiom of infinity, and even that one possibly)
seem to apply to our world and seem to work fine.

Now, the last axiom, the axiom of choice (or the well-ordering principle) asserts
that a certain kind of function exists. It
 cannot
be proven from the above. In other words, given a system that satisfies all of
the above, it cannot be assumed that the system also satisfies this axiom
(nor in fact that it does *not* satisfy this axiom). That's all
there is to it, really.

Yet, mathematicians do disagree about this axiom, and whether it applies to our
world as well. Some mathematicians take a [-https://en.wikipedia.org/wiki/Platonism Platonic]
view of mathematics, in which mathematical objects such as sets actually exist in some
abstract realm, and for which the axiom of choice is either true or false, but we do
not know which. Others take a highly constructive view (in many cases motivated
by realism and the ability of the mathematics to model the world) in which
either the axiom of choice is false, or infinite sets do not exist in which
case the axiom of choice is provable and hence superfluous. Others take the view
that the axiom is not true or false, but merely useless, and that anything provable
from it is meaningless. Many seem not to care: the axiom is convenient for the mathematics
they wish to do (whose application they are not much concerned about in any case) and
hence they assume it without qualm.

#How Something can be Neither True nor False:
How Could we Possibly Know that AC is Independent of ZF?#
It has been stated above multiple times that the Axiom of Choice is independent from the other
Zermelo-Fraenkel axioms. In other words, it can neither be proven nor disproven from those axioms.
But how can we possibly know this fact?

The answer lies in [-model_theory models]. However, these are not physical or even computational models,
but models of a more abstract kind.

For example, a model of [-3g8 group theory] is a specific [-3gd group],
which itself can be characterized as a specific set. Now, notice that the axioms of group theory
say nothing about whether a given group is abelian (commutative) or not. It does not follow
from the axioms that for groups it is always true that $xy = yx$, nor does it follow that for
groups there are always some $x$ and $y$ for which $xy \not= yx$. In other words, the
"abelian axiom" is independent of the axioms of group theory. How do we know this fact? We
need simply exhibit two models, two groups, one of which is abelian and the other not.
For these groups, I pick, oh say the [-47y cyclic group on 3 elements]
and the [-497 symmetric group] $S_3$. The first is abelian, the second, not.

In order to reason about such models of set theory, one assumes the existence
of "meta-sets" in some meta-theory. The entire "universe" of set theory is then a certain "meta-set"
behaving in a certain way. In case this feels too much like cheating,
[-http://math.stackexchange.com/questions/531516/meta-theory-when-studying-set-theory this
StackExchange answer] should help clear things up. In particular, the following quote from
VanLiere's thesis:

"
Since these questions all have to do with first-order provability, we could take as our metatheory some very weak theory (such as Peano arithmetic) which is sufficient for formalizing first-order logic. However, as is customary in treatises about set theory, we take as our metatheory ZF plus the Axiom of choice in order to have at our disposal the infinitary tools of model theory. We will also use locutions such as ... which are only really justifiable in some even stronger metatheory with the understanding that they could be eliminated through the use of Boolean-valued models or some other device.
"

In other words, it is possible to form these models in some weaker theory on which we have "more of a grip". The entirity
of set theory are then special objects satisfying the axioms of this weaker theory. Is it possible to repeat this process
ad infinitum? No. But if we want, we could even deal with just a finite fragment of set theory: We assume that any mathematics
we want to do only needs a finite number of the (infinitely many) axioms of set theory. We then prove what we want about this
finite fragment. But we may as well have proved it about the whole theory.

Now, pick (or construct) two specific objects of the meta-theory such
 that in one of them, the axiom of choice is true, and that
in the other, the axiom of choice is false.

To obtain these two models requires vastly different approaches which will not be
described in detail here.
More detail can be found online
 in [-http://vanilla47.com/PDFs/Cryptography/Mathematics/Set%20Theory%20PDFs/SET%20THEORY.pdf
Kunen's text]. The consistency of choice is the easier direction, and involves constructing
something called [-constructible_universe Gödel's constructible universe of sets] (or just
Gödel's universe or the constructible universe). The consistency of the negation of choice
is more difficult, and involves a technique developed by
[-https://en.wikipedia.org/wiki/Paul_Cohen_(mathematician) Paul Cohen]
called [-forcing_mathematics forcing].

#A Rose by Any Other Name: Alternative Characterizations of AC#
There are a few similar ways to state the axiom of choice. For example:

Given any set $C$ containing non-empty sets which are pairwise [-disjoint_set disjoint]
(any two sets in $C$ do not [-5sb intersect]),
then there is some set $X$ that intersects each of the sets in $C$ in exactly one element.

There are also many alternative theorems which at first glance appear to be very different from the
axiom of choice, but which are actually equivalent to it, in the sense that each of the theorems
can both be proven from the axiom of choice and be used to prove it (in conjunction
with the other ZF axioms).

A few examples:
 - Zorn's Lemma (Described in more detail below).
 - Well-Ordering Theorem (Also described in more detail below).
 - The [-5zs product] of non-empty sets is non-empty.
 - Tarski's Theorem for Binary [-product_mathematics Products]
 (from the quote at the start of this article) that $A$ is [-bijection bijective] to $A \times A$.
 - Every [-4bg surjective function] has a [-inverse_of_function right inverse].
 - Given two sets, either
 - Every [-3w0 vector space] has a [-vector_space_basis basis]
 - Every [-connected_graph connected graph] has a [-spanning_tree spanning tree].
 - For any pair of sets have comparable[-4w5 cardinality]: for any pair of sets
$X$ and $Y$, either they are the same size, or one is smaller than the other.

More examples can be found on the [-https://en.wikipedia.org/wiki/Axiom_of_choice#Equivalents Wikipedia page].

Because of the intuitive nature of some of these statements (especially that products are non-empty,
that vector spaces have bases and that cardinalities are comparable), they are often used as evidence
for the truth, or motivation for the use, of the Axiom of Choice.

#Zorn's Lemma? I hardly Know her!#
The following is a specific example of a very common way in which the axiom of choice is used, called Zorn's Lemma.
It is a statement equivalent to the axiom of choice, but easier to use in many mathematical applications.

The statement is as follows:

Every [-3rb partially ordered set] (poset) for which every [-chain_order_theory chain] has an [-upper_bound_mathematics upper bound]
has a [-maximal_mathematics maximal element].

In other words, if you have an ordered set $X$ and you consider any linearly ordered subset $C$, and there is some
element $u \in X$ which is at least as large as any element in $C$, i.e., $u \geq c$ for any $c \in C$, then there
is an element $m \in X$ which is maximal, in the sense that it is at least as big as any comparable
element of $X$, i.e., for any $x \in X$, it holds that $m \not< x$.
(It is maximal, but not necessarily a global maximum. There is nothing in $X$
lying above $m$, but there may be incomparable elements).

Again, this is provable for a finite poset X, and for some infinite posets X, but not provable in general.

Now, why is this rather arcane statement useful?

Well, often for some type of mathematical structure we are interested in,
all the structures of that type form a poset under inclusion, and if a maximal
such structure existed, it would have a particularly nice property.
Furthermore, for many structures, a union of all the structures in a
upwards-increasing chain of such structures (under inclusion) is itself
a structure of the right type, as well as an upper-bound for the chain.
Then Zorn's lemma gives us the maximal structure we are looking for.

As an example, consider a (esp. infinite-dimensional) vector space $V$, and trying to find a basis
for $V$.

Choose any element $v_1 \in V$. Now consider all possible [-linear_independence linearly independent] sets
containing $v$. These form a poset (which contains at least the set $\{v_1\}$ since that is linearly independent).
Now consider any chain, possibly infinite, of such sets. It looks like
 $\{v_1\} \subseteq \{v, v_2\} \subseteq \{v_1, v_2, v_3 \} \subseteq \cdots$. Then take the
[-union union] of all the sets in the chain
$ \{v_1\} \cup \{v_1, v_2\} \cup \{v_1, v_2, v_3 \} \cdots = \{v_1, v_2, v_3, \ldots \}$. Call it $B$.
Then $B$
contains all the elements in any of the sets in the chain.

It can be shown to be linearly independent, since if some element $v_i$ could be formed
as a linear combination of finitely many other elements in $B$, then this could be done
already in one of the sets in the chain.

Then every chain of such linearly indpendent sets has an upper bound, so
the hypothesis of Zorn's Lemma holds. Then by Zorn's Lemma, there is a maximal element
$M$. By definition, this maximal element has no superset that is itself linearly independent.
This set of vectors also spans $V$ (every element of $V$ can be written as a linear
combination of vectors in $M$), since if it did not, then there would be a vector
$v \in V$ which is linearly independent of $M$ (cannot be written as a linear
combination of vectors in $M$) and then the set $M \cup \{v\}$, which is $M$ adjoined
with $v$, strictly contains $M$, contradicting the maximality of $M$.

Since the definition of a basis is a maximal linearly independent set spanning $V$,
 the proof is done.
QED.

One might wonder why Zorn's lemma is even necessary. Why could we not just have picked the
union of the chain as our basis? In a sense, we could have, provided we use the correct chain.
For example, the chain of two elements $\{v_1\}$ and $\{v_1, v_2\}$ is not sufficient. We need
an infinite chain (and, in fact, a [-transfinite large enough infinite] chain at that) .
But there is a difference between being able to
prove that any chain has an upper bound, and being able to actually choose a specific chain that works.
In some sense, without Zorns lemma, we can reason in a very general vague way that, yes, the chains all
have upper bounds, and there *might* be a long enough chain, and if there is then its upper bound will
be the maximal element we need. Zorn's lemma formalizes this intuition, and without it,
lemma we can't always pin down a specific chain which works.

Note how Zorn's Lemma allows us to make infinitely many arbitrary choices as far as selecting
elements of the infinite basis for the vector space is concerned. In general, this is where
Zorn's lemma comes in useful.

The upper bounds are necessary. For example, in the [-45h natural numbers] $\mathbb{N}$, the
entirety of the natural numbers forms a chain. This chain has no upper bound in $\mathbb{N}$. Also,
the natural numbers do not have a maximal element.

Note that it may also be possible for a maximal element to exist without there being an upper bound.
Consider for example the natural numbers with an 'extra' element which is not comparable to any of the
numbers. Then this is a perfectly acceptable poset. Since this extra element is incomparable, then in
particular there is no

#Getting Your Ducks in a Row, or, Rather, Getting Your Real Numbers in a Row: The Well-Ordering Principle#
A [540 linearly ordered set] is called well-ordered if any of its non-empty subsets has a least element.

For example, the [-45h natural numbers] $\mathbb{N}$ are well-ordered. Consider any non-empty subset of the natural
numbers (e.g., $\{42, 48, 64, \ldots\}$. It has a least element (e.g., $42$).

The positive [-4bc real numbers] (and in fact the positive [-4zq rational numbers])
are not well-ordered.
There is no least element, since for any number bigger than zero (e.g. 1/3) it is possible to find a smaller number
(e.g. 1/4) which is also bigger than zero.

The well-ordering principle states that any set is [-bijection bijective] to some well-ordered one.
This basically states that you can have a well-ordered set of any size.

To see why this is surprising, try imaging a different linear order on the reals such that any subset
you may choose - *Any* subset - has a least element.

Again, the Axiom of Choice allows us to do this.

In fact if we are always able to well order sets, then
we are able to use it to make choice functions: imagine you needed to choose an element from each set
in a set of sets, then you can just choose the least element from each set.

#AC On a Budget: Weaker Versions of the Axiom#
There are also theorems which do not follow from ZF, and which do follow from AC, but are not strong
enough to use to prove AC. What this is equivalent to saying is that there are models of set theory
in which these theorems are true, but for which the axiom of choice does not hold in full generality.

A few examples of such theorems:

 - The Hausdorff paradox and Banach-Tarski paradox (mentioned above).
 - A [-union_mathematics union] of [-countble_infinity countably many] countable sets is countable.
 - The [-axiom_of_dependent_choice axiom of dependent choice] (given a non-empty set $X$ and
([-entire_relation entire]) [-binary_relation binary relation] $R$, there exists a sequence
$(x_n)_{n \in \mathbb{N}}$ such that $x_n$ is $R$-related to $x_{n+1}$) .
 - The [-axiom_of_countable_choice axiom of countable choice] (every countable set of sets
has a choice function).
 - Every [-field_mathematics field] has an [-algebraic_closure algebraic closure]
 - Existence of non-principal [-ultrafilter ultrafilters].
 - Gödel's [-completeness_theorem completeness theorem] for first-order logic.
 - [-booelan_prime_ideal_theorem Boolean Prime Ideal Theorem] (useful for proving existence
of non-principal ultrafilters and Gödel's completenes theorem).
 - The [-excluded_middle Law of excluded middle] for logic.

More examples may be found on
[-https://en.wikipedia.org/wiki/Axiom_of_choice#Results_requiring_AC_.28or_weaker_forms.29_but_weaker_than_it
the Wikipedia page].

#And In Related News: The Continuum Hypothesis#
Intuitively, the [-continuum_hypothesis Continuum Hypothesis] (CH) states that there is no
set strictly bigger than the set of all [-45h natural numbers],
but strictly smaller than the set of all [-4bc real numbers]. (These are two
[-infnity infinite] sets, but they are [-2w0 different infinities]).

The formal statement concerns [-4w5 cardinality] of sets. In particular,
it states that there is no set which has cardinality strictly larger than the
set of natural numbers, but strictly smaller than the set of real numbers.

It is called the 'Continuum Hypothesis' because it concerns the size of the
continuum (the set of real numbers) and was hypothesized to be true by Georg
Cantor in 1878.

It is again independent of the Zermelo-Fraenkel axioms, and this was proven in the same manner and
at the same time as the proof of the independence of AC from ZF (described in more detail above).

In fact, the continuum hypothesis was shown to be independent even from ZFC, (ZF with the Axiom of Choice).
However, the continuum hypothesis *implies* the Axiom of Choice under ZF. In other words, given the
ZF axioms, if you know that the Axiom of Choice is true then you do not yet know anything about the truth
of Continuum Hypothesis. However, if you know that the Continuum Hypothesis is true, then you know the
Axiom of Choice must also be true.

The [-generalized_continuum_hypthesis Generalized Continuum Hypthosis] (GCH) is, well, a generalized
version of the Continuum Hypothesis, stating that not only are there no sets of size lying strictly
between the natural numbers $\mathbb{N}$ and the reals $\mathbb{R}$,
but that for any set $X$, there is no set of size
lying strictly between the sizes of $X$ and the [-6gl power set] $P(X)$. In particular, note
that the reals $\mathbb{R}$ is of the same size as the power set $P(\mathbb{N})$ of the naturals,
so that GCH implies CH. It is also strictly stronger than CH (it is not implied by CH).

#Axiom of Choice Considered Harmful: Constructive Mathematics and the Potential Pitfalls of AC#
Why does the axiom of choice have such a bad reputation with [-constructive_mathematics constructive]
mathematicians?

It is important to realise that some of the reasons mathematicians had for doubting AC are no longer
relevant. Because of some of its counter-intuitive results, mathematicians

To understand this view, it is necessary to understand more about the constructive view in general. %TODO

(Posting a list of links here until I have a better idea of what to write, in approx. reverse order of relevance / usefulness)
See also

 - [-https://en.wikipedia.org/wiki/Axiom_of_choice#Criticism_and_acceptance this section of the Wikipedia page on AC]
 - [-http://mathoverflow.net/questions/22927/why-worry-about-the-axiom-of-choice this StackOverflow question],
 - [-http://comjnl.oxfordjournals.org/content/49/3/345.full this paper by Per Martin-Löf on the history and problems with the Axiom of Choice]
 - [-https://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/ this post by Greg Muller on why the Axiom of Choice is wrong]
 - [-http://www.iep.utm.edu/con-math/ this post on constructive mathematics on the Internet Encyclopaedia of Philosophy]
 - [-http://scienceblogs.com/goodmath/2007/05/27/the-axiom-of-choice/ this post
on Good Math Bad Math by Mark C. Chu-Carroll],
 - [-https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/ this
post by Terrance Tao on the usefulness of the axiom in "concrete" mathematics],
 - [-http://www.math.canterbury.ac.nz/php/groups/cm/faq/ this post regarding constructive mathematics on the University of Canterbury website]
 - [-https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=8&ved=0ahUKEwiPnovngczPAhVM1hoKHflFDtMQFghYMAc&url=http%3A%2F%2Fmath.fau.edu%2Frichman%2FDocs%2FIntrview.tex&usg=AFQjCNFJZO1QGPzG1Cqe_8XcnlzMCrlvsA&sig2=R2goExeTv4EPJ6TF9ks5Yw
this "interview with a constructivist"]
 - [-http://plato.stanford.edu/entries/mathematics-constructive/ this page on Standford Enclyclopaedia on constructive mathematics]
 - [-http://plato.stanford.edu/entries/intuitionism/ this page on Stanford Encyclopaedia on intuitionistic mathematics]

#Choosing Not to Choose: Set-Theoretic Axioms Which Contradict Choice#
It is also possible to assume axioms which contradict the axiom of choice. For example, there is the
[-axiom_of_determinancy Axiom of Determinancy]. This axiom states that for any two-person [-game_mathematics game]
of a certain type, one player has a winning strategy.
%TODO

#I Want to Play a Game: Counterintuitive Strategies Using AC#
There are a few examples of very counter-intuitive solutions to (seemingly) impossible challanges
which work only in the presence of the Axiom of Choice.

Examples may be found in the following places:

 - [-https://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/ Countably many prisoners have to guess hats with no information transfer, yet all but a finite number may go free.]
 - [-http://mathoverflow.net/questions/20882/most-unintuitive-application-of-the-axiom-of-choice Countably many people wear hats with
real numbers. They can only see everyone else's hats. Shouting at the same time, all but finitely many are guaranteed to guess correctly.]
 - [-http://math.stackexchange.com/questions/613506/real-guessing-puzzle Countably many mathematicians go into countably many identical rooms each containing countably many boxes containing real numbers. Each mathematician opens all but one box, and guesses the number in the unopened box. All but one mathematician are correct.]


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Rice's Theorem: Intro (Math 1)

Rice's Theorem says that you can't in general figure out anything about what a computer program outputs just by looking at the program. Specifically, we think of programs as taking inputs, doing some kind of computation on the input, and possibly eventually stopping and giving an output. Only "possibly" because sometimes programs get in an infinite loop and go forever. The assignment that takes an input to the output of a program on that input (if it exists) is called the [-5p2] computed by that program %%note:"Partial" functions because the output doesn't exist if the program runs forever.%%.

Rice's Theorem says that for any property of partial functions, you can't write a program that looks at some other program's source code, and tells you whether that program computes a function with the property %%note:This is true except for "trivial" properties like "all partial functions" or "no partial functions," which are easy to write programs to test.%%.

For example, say we want to know whether a program computes the [Fibonacci_sequence], i.e. returns the $n$th Fibonacci number on input $n$. If we try to write a Fibonacci-checker program that tells us whether programs compute the Fibonacci sequence, we're going to have a hard time, because by Rice's Theorem it's impossible!

There might be some programs for which we can say "yes, this program definitely computes the Fibonacci sequence," but we can't write a general procedure that always tells whether some program computes the Fibonacci sequence; there are some very convoluted programs that compute it, but not obviously, and if you modify the Fibonacci-checker to handle some of them, there will always be even more convoluted programs that it doesn't work on.

We make a distinction between *programs* and *(partial) functions computed by programs*. There are usually many programs that compute the same function. Rice's Theorem only cares about the partial functions; it's possible to test statements about programs, like "has fewer than 100 lines." But this kind of statement won't always tell you something about the function the program computes.

# Proof

To prove Rice's Theorem, we'll use the [46h halting theorem], which says that you can't write a program that can always tell whether some program will halt on some input. Here's how the proof will go: suppose you come up with a program that checks whether programs compute partial functions with some property, e.g. you successfully write a Fibonacci-checker. We'll then use the Fibonacci-checker to design a Halt-checker, which can tell when a program halts. But this is impossible by the Halting theorem, so this couldn't be the case: your Fibonacci-checker must not work!

We need to make a couple of assumptions. First, that the empty function, meaning the function "computed" by a program that always enters an infinite loop regardless of its input, does not satisfy the property. If it does, we can replace the property by its opposite (say, replace "computes the Fibonacci sequence" with "doesn't compute the Fibonacci sequence"). If we can tell whether a program computes a function that *doesn't* have the property, then we can tell when it does, simply be reversing the output.

Second, we assume there is *some* program that does compute a function with the property, i.e. a program that the checker says "yes" to. If there isn't, we could have the checker always say "no." Say the program $s$ computes a function with the property.

 Now suppose you have a checker $C$ that, when given a program as input, returns whether the program computes a function with some property of interest. We want to test whether some program $M$ halts, when given input $x$.

We start by building a new program, called $Proxy_s$, which does the following on some input $z$:

- Run $M$ on input $x$.
- Run $s$ on input $z$, and return the result.

If $M$ halts on $x$, $Proxy_s$ moves on to the second step, and returns the same value as $s$. Since they return the same value (or no value, because of an infinite loop) on every possible input, $s$ and $Proxy_s$ compute the same function. So $Proxy_s$ computes a function with the property.

If $M$ doesn't halt on $x$, $Proxy_s$ never finishes the first step, and runs forever. So $Proxy_s$ always ends up in an infinite loop, which means it computes the empty function. So $Proxy_s$ computes a function that doesn't have the property.

What happens if we plug $Proxy_s$ as input into $C$? If $C$ says "yes," $Proxy_s$ computes a function with the property, which means $M$ halts on $x$. If $C$ says "no," $Proxy_s$ doesn't compute a function with the property, so $M$ doesn't halt on $x$.

So we can tell whether $M$ halts on $x$! All we have to do is write this proxy function, and ask $C$ whether its function has the property. But the Halting theorem says we can't do this! That means $C$ can't exist, and we can't write a checker to see whether programs compute functions with the property.

As an example, suppose we have a Fibonacci-checker $fib\_checker$. The following Python program computes the Fibonacci numbers, and so $fib\_checker$ says "yes" to it:

    def fib(n):
        a, b = 0, 1
        for i in range(n):
            a, b = b, a+b
        return a

Then, for some program $M$ and input $x$, $Proxy_{fib}$ is the program:

    def Proxy_fib(z):
        M(x)
        return fib(z)

If $fib\_checker$ works, this program says whether $M$ halts on $x$:

    def halts(M,x)
        return fib_checker(Proxy_fib)

Expanding this, we have the program:

    def halts(M,x)
        return fib_checker('''
                            M(x)
                            a, b = 0, 1
                            for i in range(n):
                                a, b = b, a+b
                            return a
                          ''')

If $M$ halts on $x$, the program inside the triple quotes computes the Fibonacci sequence, so $fib\_checker$ returns $true$, and so does $halts$. If $M$ doesn't halt on $x$, the program in the triple quotes also doesn't halt, and returns the empty function, which isn't the Fibonacci sequence. So $fib\_checker$ returns $false$, and so does $halts$.


Axiom of Choice: Guide

## Learning the Axiom of Choice ##

[multiple-choice(q_level_of_math_logic): What do you know about set theory, mathematical logic and axiom systems?
a: Almost nothing
wants: [axiom_of_choice_definition_intutive] [axiom_of_choice_finite_sets] [banach_tarski_paradox] [axiom_of_choice_function_existence] [axiom_of_choice_how_can_something_be_neither_true_nor_false]
-wants: [6c8]
b: Well, I've used them for other maths, but haven't studied them directly.
wants: [axiom_of_choice_definition_intutive] [axiom_of_choice_finite_sets] [axiom_of_choice_function_existence] [axiom_of_choice_how_can_something_be_neither_true_nor_false]
-wants: [6c8] [banach_tarski_paradox]
c: I have a good grasp of it, but would like some explanation anyway.
wants: [6c8] [axiom_of_choice_finite_sets]  [axiom_of_choice_function_existence] [axiom_of_choice_how_can_something_be_neither_true_nor_false]
-wants: [6cb] [banach_tarski_paradox]
d: I have a good grasp of it and don't need to hear more.
wants: [6c8]
-wants: [6cb] [axiom_of_choice_finite_sets]  [axiom_of_choice_function_existence] [banach_tarski_paradox][axiom_of_choice_how_can_something_be_neither_true_nor_false]
]

[multiple-choice(q_axiom_of_choice_history): This axiom has a rich and interesting history.  How much do you want to learn about?
a: Give me all of the juicy history side-facts!
wants: [6c7] [6c9]
b: Eh, give me a short intro.
wants: [6c7]
-wants: [6c9]
c: Just stick to the mathematics, please.
-wants: [6c7] [6c9]
]

[multiple-choice(q_axiom_of_choice_history): This axiom has a rich and interesting history.  How much do you want to learn about?
a: Give me all of the juicy history side-facts!
wants: [6c7] [6c9]
b: Eh, give me a short intro.
wants: [6c7]
-wants: [6c9]
c: Just stick to the mathematics, please.
-wants: [6c7] [6c9]
]

[multiple-choice(q_axiom_of_choice_mathematics): How heavy should the maths be that we show you?
a: Lay it on me! Give me the formal logical notation.
wants: [6c8]
-wants: [6cb]
b: Just give me an intuitive description with words. No mathematical notation.
-wants: [6c8]
wants: [6cb]
]

%%%box:
You will get the following pages:
%%wants-requisite([6c7]):
Basic intro %%
%%wants-requisite([6c9]):
History and controversy %%
%%wants-requisite([6c8]):
Definition (Formal) %%
%%wants-requisite([6c7]):
Definition (Intuitive) %%
%%wants-requisite([6cb]):

%start-path([6c9])%
%%
%%%

Plan for this guide:

Axiom of Choice: Guide

Conditional paragraphs for concepts being described on later pages.

Questions on what the main page should look like

1 Introduction
2 Getting the Heavy Maths out the Way: Definitions
3 Axiom Unnecessary for Finite Collections of Sets
4 Controversy: Mathematicians Divided! Counter-Intuitive Results, and The History of the Axiom of Choice
5 So, What is this Choice Thing Good for Anyways?
6 Physicists Hate Them! Find out How Banach and Tarski Make Infinity Dollars with this One Simple Trick!
7 How Something Can Exist Without Actually Existing: The Zermelo Fraenkel Axioms and the Existence of a Choice Function
8 How Something can be Neither True nor False:
9 A Rose by Any Other Name: Alternative Characterizations of AC
10 Zorn's Lemma? I hardly Know her!
11 Getting Your Ducks in a Row, or, Rather, Getting Your Real Numbers in a Row: The Well-Ordering Principle
12 AC On a Budget: Weaker Versions of the Axiom
13 And In Related News: The Continuum Hypothesis
14 Axiom of Choice Considered Harmful: Constructive Mathematics and the Potential Pitfalls of AC
15 Choosing Not to Choose: Set-Theoretic Axioms Which Contradict Choice
16 I Want to Play a Game: Counterintuitive Strategies Using AC

-Guide Questions-

I'd also like the corresponding pages to show or hide some information based on what is chosen here.

Choose one of the pregenerated paths, or customize your own.
a. Comprehensive path: Learn more than you wanted to know!
	Add all pages
b. Substantial path: All the most important stuff
	Add 1,2,3,4,5,6,7,8,10,14
c. Compact path: Only the very important stuff
	Add 2, 3, 5, 7, 10
d. First-time path: Learn the important basics without getting too bogged down.
	Add 1. 2 (intuitive), 2 (definition), 3, 4, 5, 6, 7, 8, 10
e. Overview path: Just get a taste of the axiom
	Add 1, 2 (intuitive), 2 (definition), 3, 4, 5, 7
f. Custom Path

If the user chooses to customize a path:
1. What do you know about set theory, mathematical logic and axioms?
a. Almost nothing
	Add Definitions (Intuitive)
	Add 3. Finite Sets
	Add 6. Banach-Tarski
	Add 7. ZF Axioms and Existence of a Choice Function
	Add 8. How can something be neither True nor False
b. Well, I've used them for other maths, but haven't studied them directly.
	Add 2. Definitions
	Add 3. Finite Sets
	Add 7. ZF Axioms and Existence of a Choice Function
	Add 8. How can something be neither True nor False
c. I have a good grasp of it, but would like some explanation anyway.
	Add 2. Definitions
	Add 3. Finite Sets
	Add 7. ZF Axioms and Existence of a Choice Function
	Add 8. How can something be neither True nor False
d. I have a good grasp of it and don't need to hear more.
	Add 2. Definitions

2. This axiom has a rich and interesting history. How much do you want to learn about?
a. Give me all of the juicy history side-facts!
	Adds 1. Introduction
	Adds 4. Controversy (History)
b. Eh, give me a short intro.
	Adds 1. Intro
c. Just stick to the mathematics, please.

-Questions 3-5 are revealed if, and only if, the answer to Question 1 is b,c, or d.-

32. How much detail would you like to read about the mathematics related to the axiom?
a. I'd like to know a lot of the detail.
	Adds 5. What is this Choice Thing Good For
	Adds 9. Alternative Characterizations
	Adds 10. Zorn's Lemma
	Adds 12. Weaker Versions of the Axiom
	Adds 13. Continuum Hypthesis
b. I only want the most important extra details.
	Adds 5. What is this Choice thing good for
	Adds 10. Zorn's Lemma
c. I only want the absolute essentials.

4. How much would you like to know about constructive mathematics and its relation to the axiom of choice?
a. I don't care to read about it right now.
b. I don't know what that is, please tell me about it.
	Adds 14. Constructive Mathematics and Pitfalls of AC
	Potentially adds a link to an intro on constructivsm.
c. I know what it is, but I would like to hear more about how it relates to Axiom of Choice.
	Adds 14. Constructive Mathematics and Pitfalls of AC
5. How much do you care about the paradoxes the axiom implies?
a. I don't care about them, I just want to know about the axiom itself.
b. Give me a very basic overview.
	Adds 6. Banach-Tarski Paradox
d. Just tell me some interesting ones.
	Adds 6. Banach-Tarksi Paradox
	Adds 16. Counterintuitive strategies using AC
c. Tell me the whole story!
	Adds 6. Banach-Tarski Paradox
	Adds 11. Well-Ordering Principle
	Adds 15. Axioms which contradict choice
	Adds 16. Counterintuitive strategies using AC

O


Axiom of Choice: Introduction

"The Axiom of Choice is necessary to select a set from an infinite number of pairs of socks, but not an infinite number of pairs of shoes." — *Bertrand Russell,  Introduction to mathematical philosophy*

"Tarski told me the following story. He tried to publish his theorem \[the equivalence between the Axiom of Choice and the statement 'every infinite set A has the same cardinality as AxA'\] in the Comptes Rendus Acad. Sci. Paris but Fréchet and Lebesgue refused to present it. Fréchet wrote that an implication between two well known propositions is not a new result. Lebesgue wrote that an implication between two false propositions is of no interest. And Tarski said that after this misadventure he never again tried to publish in the Comptes Rendus."
- *Jan Mycielski, A System of Axioms of Set Theory for the Rationalists*

#Obligatory Introduction#
The Axiom of Choice, the most controversial axiom of the 20th Century.

The axiom states that a certain kind of function, called a `choice' function, always exists. It is called a choice function, because, given a collection of non-empty sets, the function 'chooses' a single element from each of the sets. It is a powerful and useful axiom, asserting the existence of useful mathematical structures (such as bases for [-3w0 vector spaces] of arbitrary [-dimension_mathematics dimension], and [-ultraproduct ultraproducts]). It is a generally accepted axiom, and is in wide use by mathematicians. In fact, according to Elliott Mendelson in Introduction to Mathematical Logic (1964) "The status of the Axiom of Choice has become less controversial in recent years. To most mathematicians it seems quite plausible and it has so many important applications in practically all branches of mathematics that not to accept it would seem to be a wilful hobbling of the practicing mathematician. "

Neverless, being an [-axiom_mathematics axiom], it cannot be proven and must instead be assumed.
In particular, it is an axiom of [-set_theory set theory] and it is not provable from the other axioms (the Zermelo-Fraenkel axioms of Set Theory). In fact many mathematicians, in particular [-constructive_mathematics constructive] mathematicians, reject the axiom, stating that it does not capture a 'real' or 'physical' property, but is instead just a mathematical oddity, an artefact of the mathematics used to approximate reality, rather than reality itself. In the words of the LessWrong community: the constructive mathematicians would claim it is a statement about [-https://wiki.lesswrong.com/wiki/Map_and_Territory_(sequence) the map, and not the territory].

Historically, the axiom has experienced much controversy. Before it was shown to be independent of the other axioms, it was believed either to follow from them (i.e., be 'True') or lead to a contradiction (i.e., be 'False'). Its independence from the other axioms was, in fact, a very surprising result at the time.


Axiom of Choice: Definition (Formal)

#Getting the Heavy Maths out the Way: Definitions#
Intuitively, the [-axiom_mathematics axiom] of choice states that, given a collection of *[-5zc non-empty]* [-3jz sets], there is a [-3jy function] which selects a single element from each of the sets.

More formally, given a set $X$ whose [-5xy elements] are only non-empty sets, there is a function
$$
f: X \rightarrow \bigcup_{Y \in X} Y
$$
from $X$ to the [-5s8 union] of all the elements of $X$ such that, for each $Y \in X$, the [-3lh image] of $Y$ under $f$ is an element of $Y$, i.e., $f(Y) \in Y$.

In [-logical_notation logical notation],
$$
\forall_X
\left(
\left[\forall_{Y \in X} Y \not=  \emptyset \right]
\Rightarrow
\left[\exists
\left( f: X \rightarrow \bigcup_{Y \in X} Y \right)
\left(\forall_{Y \in X}
\exists_{y \in Y} f(Y) = y \right) \right]
\right)
$$

#Axiom Unnecessary for Finite Collections of Sets#
For a [-5zy finite set]  $X$ containing only [-5zy finite] non-empty sets, the axiom is actually provable (from the [-zermelo_fraenkel_axioms Zermelo-Fraenkel axioms] of set theory ZF), and hence does not need to be given as an [-axiom_mathematics axiom]. In fact, even for a finite collection of possibly infinite non-empty sets, the axiom of choice is provable (from ZF), using the [-axiom_of_induction axiom of induction]. In this case, the function can be explicitly described. For example, if the set $X$ contains only three, potentially infinite, non-empty sets $Y_1, Y_2, Y_3$, then the fact that they are non-empty means they each contain at least one element, say $y_1 \in Y_1, y_2 \in Y_2, y_3 \in Y_3$. Then define $f$ by $f(Y_1) = y_1$, $f(Y_2) = y_2$ and $f(Y_3) = y_3$. This construction is permitted by the axioms ZF.

The problem comes in if $X$ contains an infinite number of non-empty sets. Let's assume $X$ contains a [-2w0 countable] number of sets $Y_1, Y_2, Y_3, \ldots$. Then, again intuitively speaking, we can explicitly describe how $f$ might act on finitely many of the $Y$s (say the first $n$ for any natural number $n$), but we cannot describe it on all of them at once.

To understand this properly, one must understand what it means to be able to 'describe' or 'construct' a function $f$. This is described in more detail in the sections which follow. But first, a bit of background on why the axiom of choice is interesting to mathematicians.


Axiom of Choice: History and Controversy

#Controversy: Mathematicians Divided! Counter-Intuitive Results, and The History of the Axiom of Choice#
Mathematicians have been using an intuitive concept of a set for probably as long as mathematics has been practiced.
At first, mathematicians assumed that the axiom of choice was simply true (as indeed it is for finite collections of sets).

[-https://en.wikipedia.org/wiki/Georg_Cantor Georg Cantor] introduced the concept of [-transfinite_number transfinite numbers]
and different [-4w5 cardinalities of infinity] in a 1874
[https://en.wikipedia.org/wiki/Georg_Cantor%27s_first_set_theory_article paper] (which contains his infamous
[-https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument Diagonalization Argument])
 and along with this sparked the introduction of [-set_theory set theory].
 In 1883, Cantor introduced a principle called the 'Well-Ordering Princple'
(discussed further in a section below) which he called a 'law of thought' (i.e., intuitively true).
He attempted to prove this principle from his other principles, but found that he was unable to do so.

[-https://en.wikipedia.org/wiki/Ernst_Zermelo Ernst Zermelo] attempted to
develop an [-axiom_system axiomatic] treatment of set theory. He
 managed to prove the Well-Ordering Principle in 1904 by introducing a new principle: The Principle of Choice.
This sparked much discussion amongst mathematicians. In 1908 published a paper containing responses to this debate,
as well as a new formulation of the Axiom of Choice. In this year, he also published his first version of
the set theoretic axioms, known as the [-https://en.wikipedia.org/wiki/Zermelo_set_theory Zermelo Axioms of Set Theory].
Mathematicians, [-https://en.wikipedia.org/wiki/Abraham_Fraenkel Abraham Fraenkel] and
[-https://en.wikipedia.org/wiki/Thoralf_Skolem Thoralf Skolem]  improved this system (independently of each other)
into its modern version, the [-https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
 Zermelo Fraenkel Axioms of Set Theory].

In 1914, [https://en.wikipedia.org/wiki/Felix_Hausdorff Felix Hausdorff] proved
[https://en.wikipedia.org/wiki/Hausdorff_paradox Hausdorff's paradox]. The ideas
behind this proof were used in 1924 by [-https://en.wikipedia.org/wiki/Stefan_Banach
Stefan Banach] and [-https://en.wikipedia.org/wiki/Alfred_Tarski Alfred Tarski]
to prove the more famous Banach-Tarski paradox (discussed in more detail below).
This latter theorem is often quoted as evidence of the falsehood of the axiom
of choice.

Between 1935 and 1938, [-https://en.wikipedia.org/wiki/Kurt_G%C3%B6del Kurt Gödel] proved that
the Axiom of Choice is consistent with the rest of the ZF axioms.

Finally, in 1963, [-https://en.wikipedia.org/wiki/Paul_Cohen Paul Cohen] developed a revolutionary
mathematical technique called [-forcing_mathematics forcing], with which he proved that the
axiom of choice could not be proven from the ZF axioms (in particular, that the negation of AC
is consistent with ZF). For this, and his proof of the consistency of the negation of the
[-continuum_hypothesis Generalized Continuum Hypothesis] from ZF, he was awarded a fields medal
in 1966.

This axiom came to be accepted in the general mathematical community, but was rejected by the
[-constructive_mathematics constructive] mathematicians as being fundamentally non-constructive.
However, it should be noted that in many forms of constructive mathematics,
there are *provable* versions of the axiom of choice.
The difference is that in general in constructive mathematics, exhibiting a set of non-empty sets
(technically, in constructive set-theory, these should be 'inhabited' sets) also amounts to
exhibiting a proof that they are all non-empty, which amounts to exhibiting an element for all
of them, which amounts to exhibiting a function choosing an element in each. So in constructive
mathematics, to even state that you have a set of inhabited sets requires stating that you have a choice
function to these sets proving they are all inhabited.

Some explanation of the history of the axiom of choice (as well as some of its issues)
can be found in the
paper "100 years of Zermelo's axiom of choice: what was the problem with it?"
by the constructive mathematician
[-https://en.wikipedia.org/wiki/Per_Martin-L%C3%B6f Per Martin-Löf]
at [-http://comjnl.oxfordjournals.org/content/49/3/345.full this webpage].

(Martin-Löf studied under [-https://en.wikipedia.org/wiki/Andrey_Kolmogorov Andrey Kolmogorov] of
 [-5v Kolmogorov complexity] and has made contributions to [-3qq information theory],
[-statistics mathematical_statistics], and [-mathematical_logic mathematical_logic], including developing a form of
intuitionistic [-3sz]).

A nice timeline is also summarised on [-http://plato.stanford.edu/entries/axiom-choice/index.html#note-6
The Stanford Encyclopaedia of Philosophy].


Axiom of Choice Definition (Intuitive)


Concept

Add this tag to pages which represent a concept. A concept page doesn't have an explanation, it merely describes what the concept is. Take a look at the text that comes before the table of contents on any Wikipedia page to see what a good example of a concept page might contain. Concepts are also expected to be used as requisites.


Axiom

An **axiom** of a [theory_mathematics theory] $T$ is a [well_formed well-formed] [sentence_mathematics sentence] in the [language_mathematics language] of the theory that we assume to be true without a formal justification.

#Models
Models of a certain theory are going to be those mathematical objects in which the axioms hold, so they can be used to pin down the mathematical structures we want to talk about.

Normally, when we want to reason about a particular aspect of the world we have to try to figure out a sufficiently descriptive set of axioms which are satisfied by the thing we want to reason about. Then we can use [ deduction rules] to deduce consequences of those axioms, which will also be satisfied by the thing in question.

For example, we may want to model how viral videos spread across the internet. Then we can make some assumptions about this situation. For example, we may consider that the internet is a [ graph] in which each node is a person, and its edges are friendships. We may further assume that the edges have a weight between 0 and 1 representing the probability that in a time step a person will tell its friend about the video. Then we can use this model to figure out how kitten videos end up on your twitter feed.

This is a particularly complex model with many assumptions behind. Formalizing all those assumptions and turning them into axioms would be a pain in the ass, but they are still there, albeit hidden.

For example, there might be an axiom in the language of [ first order logic] stating that $\forall w. weight(w)\rightarrow 0<w \wedge w < 1$; that is, every weight in the graph is between $0$ and $1$.

In the ideal case, we want to write down enough axioms so that the only model satisfying them is the one we want to study. However, when dealing with first order logic there will be many occasions in which this is simply not possible, no matter how many axioms we add to our theory.

One result showing this is the [ Skolem-Löwenheim theorem].

#Axiom schematas
For proper deduction to work as intended, the set of axioms of a theory do not have to be strictly finite, but just [ computable]%%note: Incidentally, a theory whose set of axioms is computable is called [-axiomatizable]%%.

In particular, we can specify an infinite amount of axioms in one go by specifying an **axiom schemata**, or particular form of a sentence which will be an axioms.

For example, in [3ft] you specify the [5fz induction axiom schemata], stating that every sentence of the form $[P(0) \wedge \forall n. P(n)\rightarrow P(n+1)]\rightarrow \forall n. P(n)$ is an axiom of $PA$.

The reason why first order logic can handle infinite sets of axioms is due to its [ compactness,] which guarantees that every consequence of the theory is a consequence of a finite subset of the theory.


Odds: Technical explanation

Odds express relative belief: we write "the odds for X versus Y are $17 : 2$" when we think that proposition X is 17/2 = 8.5 times as likely as proposition Y.%%note: The colon denotes that we are forming a set of odds. It does not denote division, as it might in French or German.%%

Odds don't say anything about how likely X or Y is in absolute terms. X might be "it will hail tomorrow" and Y might be "there will be a hurricane tomorrow." In that case, it might be the case that the odds for X versus Y are $17 : 2$, despite the fact that both X and Y are very unlikely. [1lz Bayes' rule] is an example of an important operation that makes use of relative belief.

Odds can be expressed between many different propositions at once. For example, let Z be the proposition "It will rain tomorrow," the odds for X vs Y vs Z might be $(17 : 2 : 100).$ When odds are expressed between only two propositions, they can be expressed using a single [-ratio]. For example, above, the odds ratio between X and Y is 17/2, the odds ratio between X and Z is 17/100, and the odds ratio between Y and Z is 2/100 = 1/50. This asserts that X is 8.5x more likely than Y, and that Z is 50x more likely than Y. When someone says "the odds ratio of sick to healthy is 2/3", they mean that the odds of sickness vs health are $2 : 3.$

[toc:]

# Formal definition

Given $n$ propositions $X_1, X_2, \ldots X_n,$ a set of odds between the propositions is a list $(x_1, x_2, \ldots, x_n)$ of non-negative [4bc real] numbers. Each $x_i$ in the set of odds is called a "term." Two sets of odds $(x_1, x_2, \ldots, x_n)$ and $(y_1, y_2, \ldots, y_n)$ are called "equivalent" if there is an $\alpha > 0$ such that $ \alpha x_i = y_i$ for all $i$ from 1 to $n.$

When we write a set of odds using colons, like $(x_1 : x_2 : \ldots : x_n),$ it is understood that the '=' sign denotes this equivalence. Thus, $(3 : 6) = (9 : 18).$

A set of odds with only two terms can also be written as a fraction $\frac{x}{y},$ where it is understood that $\frac{x}{y}$ denotes the odds $(x : y).$ These fractions are often called "odds ratios."

# Example

Suppose that in some forest, 40% of the trees are rotten and 60% of the trees are healthy.  There are then 2 rotten trees for every 3 healthy trees, so we say that the relative *odds* of rotten trees to healthy trees is 2 : 3. If we selected a tree at random from this forest, the *probability* of getting a rotten tree would be 2/5, but the *odds* would be 2 : 3 for rotten vs. healthy trees.

![2 sick trees, 3 healthy trees](https://i.imgur.com/GVZnz2c.png?0)

# Conversion between odds and probabilities

Consider three propositions, $X,$ $Y,$ and $Z,$ with odds of $(3 : 2 : 6).$ These odds assert  that $X$ is half as probable as $Z.$

When the set of propositions are [1rd mutually exclusive and exhaustive], we can convert a set of odds into a set of [1rf probabilities] by [1rk normalizing] the terms so that they sum to 1.  This can be done by summing all the components of the ratio, then dividing each component by the sum:

$$(x_1 : x_2 : \dots : x_n) = \left(\frac{x_1}{\sum_{i=1}^n x_i} : \frac{x_2}{\sum_{i=1}^n x_i} : \dots : \frac{x_n}{\sum_{i=1}^n x_i}\right)$$

For example, to obtain probabilities from the odds ratio 1/3, w write:

$$(1 : 3) = \left(\frac{1}{1+3}:\frac{3}{1+3}\right) = ( 0.25 : 0.75 )$$

which corresponds to the probabilities of 25% and 75%.

To go the other direction, recall that $\mathbb P(X) + \mathbb P(\neg X) = 1,$ where $\neg X$ is the negation of $X.$  So the odds for $X$ vs $\neg X$ are $\mathbb P(X) : \mathbb P(\neg X)$ $=$ $\mathbb P(X) : 1 - \mathbb P(X).$ If Alexander Hamilton has a 20% probability of winning the election, his odds for winning vs losing are $(0.2 : 1 - 0.2)$ $=$ $(0.2 : 0.8)$ $=$ $(1 : 4).$

# Bayes' rule

Odds are exceptionally convenient when reasoning using [1lz Bayes' rule], since the [1rm prior] odds can be term-by-term multiplied by a set of [relative_likelihoods relative likelihoods] to yield the [1rp posterior] odds.  (The posterior odds in turn can be normalized to yield posterior probabilities, but if performing repeated updates, it's [1zg more convenient] to multiply by all the likelihood ratios under consideration before normalizing at the end.)

$$\dfrac{\mathbb{P}(H_i\mid e_0)}{\mathbb{P}(H_j\mid e_0)} = \dfrac{\mathbb{P}(e_0\mid H_i)}{\mathbb{P}(e_0\mid H_j)} \cdot \dfrac{\mathbb{P}(H_i)}{\mathbb{P}(H_j)}$$

As a more striking illustration, suppose we receive emails on three subjects:  Business (60%), personal (30%), and spam (10%).  Suppose that business, personal, and spam emails are 60%, 10%, and 90% likely respectively to contain the word "money"; and that they are respectively 20%, 80%, and 10% likely to contain the word "probability".  Assume for the sake of discussion that a business email containing the word "money" [naive_bayes is thereby no more or less likely] to contain the word "probability", and similarly with personal and spam emails.  Then if we see an email containing both the words "money" and "probability":

$$(6 : 3 : 1) \times (6 : 1 : 9) \times (2 : 8 : 1) = (72 : 24 : 9) = (24 : 8: 3)$$

...so the posterior odds are 24 : 8 : 3 favoring the email being a business email, or roughly 69% probability after [1rk normalizing].

# Log odds

The odds $\mathbb{P}(X) : \mathbb{P}(\neg X)$ can be viewed as a dimensionless scalar quantity $\frac{\mathbb{P}(X)}{\mathbb{P}(\neg X)}$ in the range $[0, +\infty]$.  If the odds of Alexander Hamilton becoming President are 0.75 to 0.25 in favor, we can also say that Andrew Jackson is 3 times as likely to become President as not.  Or if the odds were 0.4 to 0.6, we could say that Alexander Hamilton was 2/3rds as likely to become President as not.

The **log odds** are the logarithm of this dimensionless positive quantity, $\log\left(\frac{\mathbb{P}(X)}{\mathbb{P}(\neg X)}\right),$ e.g., $\log_2(1:4) = \log_2(0.25) = -2.$  Log odds fall in the [range_notation range] $[-\infty, +\infty]$ and are finite for probabilities inside the range $(0, 1).$

When using a log odds form of [1lz Bayes' rule], the posterior log odds are equal to the prior log odds plus the log likelihood.  This means that the change in log odds can be identified with [ the strength of the evidence].  If the probability goes from 1/3 to 4/5, our odds have gone from 1:2 to 4:1 and the log odds have shifted from -1 bits to +2 bits.  So we must have seen evidence with a strength of +3 bits (a likelihood ratio of 8:1).

The convenience of this representation is what Han Solo refers to in *Star Wars* when he shouts:  "Never tell me the odds!", implying that he would much prefer to be told the logarithm of the odds ratio.

## Direct representation of infinite certainty

In the log odds representation, the probabilities $0$ and $1$ are represented as $-\infty$ and $+\infty$ respectively.

This exposes the specialness of the classical probabilities $0$ and $1,$ and the ways in which these "infinite certainties" sometimes behave qualitatively differently from all finite credences.  If we don't start by being absolutely certain of a proposition, it will require infinitely strong evidence to shift our belief all the way out to infinity.  If we do start out absolutely certain of a proposition, no amount of ordinary evidence no matter how great can ever shift us away from infinity.

This reasoning is part of the justification of [Cromwell's rule](https://en.wikipedia.org/wiki/Cromwell%27s_rule) which states that probabilities of exactly $0$ or $1$ should be avoided except for logical truths and falsities (and maybe [not even then](http://lesswrong.com/lw/mo/infinite_certainty/)).  It also demonstrates how log odds are a good fit for measuring *strength of belief and evidence,* even if classical probabilities are a better representation of *degrees of caring* and betting odds.

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Countability

The [-3jz set] of *counting numbers*, or of *positive integers*, is the set $\mathbb{Z}^+ = \{1, 2, 3, 4, \ldots\}$.

A set $S$ is called *countable* or *enumerable* if there exists a [4bg surjection] from the counting numbers onto $S$.

### Example: The rational numbers ###

The set of *rational numbers*, $\mathbb Q$, is the set of integer fractions $\frac{p}{q}$ in reduced form; the greatest common divisor of $p$ and $q$ is one, with $q > 0$.

**Theorem** The rational numbers are countable.

The proof is, essentially, that $\mathbb Z^+ \times \mathbb Z^+$ is isomorphic to $\mathbb Z$; we count in a roughly spiral pattern centered at zero.

**Proof**  Define the *height* of $\frac{a}{b}$ to be $|a| + |b|$.  We may count the rational numbers in order of height, and ordering by $a$, and then $b$, when the heights are the same.  The beginning of this counting is $0 / 1$, $-1 / 1$, $1 / 1$, $-2 / 1$, $-1 / 2$, $1 / 2$, $2 / 1$, $\ldots$   Since there are at most $(2d+1)^2$ rational numbers of height less than or equal to $d$, a rational number with height $d$ is mapped on to by one of the counting numbers up to $(2d+1)^2$; every rational number is mapped onto by this counting.  Thus, the rational numbers are countable.  $\square$

*Note*: It is not hard to extend this proof to show that $(\mathbb Z^+)^n$ is countable for any finite $n$.

**Theorem** If there exists a surjection $f$ from a countable set $A$ to a set $B$, then $B$ is countable.
**Proof** By definition of countable, there exists an enumeration $E$ of $A$. Now, $E\circ f$ is an enumeration of $B$, so $B$ is countable.

##Exercises

>Show that the set $\Sigma^*$ of [ finite words] of an enumerable [ alphabet] is countable.

%%hidden(Show solution):
First, we note that since $\mathbb N^n$ is countable, the set of words of length $n$ for each $n\in \mathbb N$ is countable.

Let $E_n: \mathbb N  \to \mathbb N^n$ stand for an enumeration of $\mathbb N ^n$, and $(J_1,J_2)(n)$ for an enumeration of $\mathbb N^2$.

Consider the function $E: \mathbb N \to \Sigma^* , n\hookrightarrow E_{J_1(n)}(J_2(n))$ which maps every number to a word in $\Sigma^*$. Then a little thought shows that $E$ is an enumeration of $\Sigma^*$.

$\square$
%%

>Show that the set $P_\omega(A)$ of finite subsets of an enumerable set $A$ is countable.

%%hidden(Show solution):
Let $E$ be an enumeration of $A$.

Consider the function $E': \mathbb N^* \to P_\omega(A)$ which relates a word $n_0 n_1 n_2 ... n_r$ made from natural numbers to the set $\{a\in A:\exists m\le k E(n_m)=a\}\subseteq A$. Clearly $E'$ is an enumeration of $P_\omega(A)$.
%%

>Show that the set of [ cofinite subsets] of an enumerable set is countable.

%%hidden(Show solution):
Simply consider the function which relates each cofinite set with its complementary.
%%


Real numbers are uncountable

We present a variant of Cantor's diagonalization argument to prove the  [4bc real numbers] are [2w0 uncountable].  This [constructive_proof constructively proves] that there exist [uncountable_set uncountable sets] %%note:  Since the real numbers are an example of one.%%.

We use the decimal representation of the real numbers.  An overline ( $\bar{\phantom{9}}$ ) is used to mean that the digit(s) under it are repeated forever.  Note that $a.bcd\cdots z\overline{9} = a.bcd\cdots (z+1)\overline{0}$ (if $z < 9$; otherwise, we need to continue carrying the one); $\sum_{i=k}^\infty 10^{-k} \cdot 9 = 1 \cdot 10^{-k + 1} + \sum_{i=k}^\infty 10^{-k} \cdot 0$.  Furthermore, these are the only equivalences between decimal representations; there are no other real numbers with multiple representations, and these real numbers have only these two decimal representations.

**Theorem** The real numbers are uncountable.

**Proof** Suppose, for [46z contradiction], that the real numbers are [-6f8 countable]; suppose that $f: \mathbb Z^+ \twoheadrightarrow \mathbb R$ is a surjection.  Let $r_n$ denote the $n^\text{th}$ decimal digit of $r$, so that the fractional part of $r$ is $r_1r_2r_3r_4r_5\ldots$  Then define a real number $r'$ with $0 \le r' < 1$ so that $r'_n$ is 5 if $(f(n))_n \ne 5$, and 6 if $(f(n))_n = 5$.  Then there can be no $n$ such that $r' = f(n)$ since $r'_n \ne (f(n))_n$.  Thus $f$ is not surjective, contradicting our assumption, and $\mathbb R$ is uncountable. $\square$

Note that choosing 5 and 6 as our allowable digits for $r'$ side-steps the issue that $0.\overline{9} = 1.\overline{0}$.  %%


Empty set

[summary: The empty set, $\emptyset$, is the set with no elements. For every object $x$, $x$ is not in $\emptyset$.]

The empty set, $\emptyset$, is the set with no elements.
Why, *a priori*, should this set exist at all?
Well, if we think of sets as "containers of elements", the idea of an empty container is intuitive: just imagine a box with nothing in it.

# Definitions

## Definition in [ZF]

In the set theory [ZF], there are exactly two axioms which assert the existence of a set *ex nihilo*; all of the rest of set theory builds sets from those given sets, or else postulates the existence of more sets.

There is the [-axiom_of_infinity], which asserts the existence of an infinite set, and there is the [-empty_set_axiom], which asserts that an empty set exists.

In fact, we can deduce the existence of an empty set even without using the empty set axiom, as long as we are allowed to use the [-axiom_of_comprehension] to select a certain specially-chosen subset of an infinite set.
More formally: let $X$ be an infinite set (as guaranteed by the axiom of infinity).
Then select the subset of $X$ consisting of all those members $x$ of $X$ which have the property that $x$ contains $X$ and also does not contain $X$.

There are no such members, so we must have just constructed an empty set.

## Definition by a universal property

The empty set [5zr has a definition] in terms of a [-600]: the empty set is the unique set $X$ such that for every set $A$, there is exactly one map from $X$ to $A$.
More succinctly, it is the [-initial_object] in the [-category_of_sets] (or in the [-614]).

# Uniqueness of the empty set

The [-axiom_of_extensionality] states that two sets are the same if and only if they have exactly the same elements.
If we had two empty sets $A$ and $B$, then certainly anything in $A$ is also in $B$ ([vacuous_truth vacuously]), and anything in $B$ is also in $A$, so they have the same elements.
Therefore $A = B$, and we have shown the uniqueness of the empty set.

# A common misconception: $\emptyset$ vs $\{ \emptyset \}$

It is very common for people to start out by getting confused between $\emptyset$ and $\{\emptyset\}$.
The first contains no elements; the second is a set containing exactly one element (namely $\emptyset$).
The sets don't [499 biject], because they are finite and have different numbers of elements.

# [vacuous_truth Vacuous truth]: a whistlestop tour

The idea of vacuous truth can be stated as follows:

> For *any* property $P$, everything in $\emptyset$ has the property $P$.

It's a bit unintuitive at first sight: it's true that everything in $\emptyset$ is the Pope, for instance.
Why should this be the case?

In order for there to be a counterexample to the statement that "everything in $\emptyset$ is the Pope", we would need to find an element of $\emptyset$ which was not the Pope. %%note: Strictly, we'd only need to show that such an element must exist, without necessarily finding it.%%
But there aren't *any* elements of $\emptyset$, let alone elements which fail to be the Pope.
So there can't be any counterexample to the statement that "everything in $\emptyset$ is the Pope", so the statement is true.


Free group universal property

[summary: The [-5kg] may be defined by a [-600], allowing [-4c7] to talk about free groups. The universal property is a helpful way to think about free groups even without any category theory: it opens the way to considering [5j9 group presentations], which are very interesting objects in their own right (for example, because they're easy to compute with).]

The [-600] of the [-5kg] basically tells us that "the definition of the free group doesn't depend (up to isomorphism) on the exact details of the set $X$ we picked; only on its [-4w5]", which is morally a very useful thing to know.
You may skip down to the next subheading if you might be scared of category theory, but the property itself doesn't need category theory and is helpful.

The universal property is the technical [4c7 category-theoretic] fact that [free_group_functor_left_adjoint_to_forgetful the free-group functor is left adjoint to the forgetful functor], and it is not so immediately useful as the other more concrete properties on this page, but it is exceedingly important in category theory as a very natural example of a [adjoint_functor pair of adjoint functors] and as an example for the [-general_adjoint_functor_theorem].

# Statement and explanation
The universal property which characterises the free group is:

> The free group $FX$ on the set $X$ is the group, unique up to isomorphism, such that for any group $G$ and any [3jy function of sets] $f: X \to G$ %%note:Here we're slightly abusing notation: we've written $G$ for the [-3gz] of the group $G$ here.%%, there is a unique [-47t] $\overline{f}: FX \to G$ such that $\overline{f}(\rho_{a_1} \rho_{a_2} \dots \rho_{a_n}) = f(a_1) \cdot f(a_2) \cdot \dots \cdot f(a_n)$.

This looks very opaque at first sight, but what it says is that $FX$ is the unique group such that:

> Given any target group $G$, we can extend any map $f: X \to G$ to a unique homomorphism $FX \to G$, in the sense that whenever we're given the image of each generator (that is, member of $X$) by $f$, the laws of a group homomorphism force exactly where every other element of $FX$ must go.
That is, we can specify homomorphisms from $FX$ by specifying where the generators go, and moreover, *every* possible such specification does indeed correspond to a homomorphism.

# Why is this a non-trivial property?

Consider the cyclic group $C_3$ with three elements; say $\{ e, a, b\}$ with $e$ the identity and $a + a = b$, $a+b = e = b+a$, and $b+b = a$.
Then this group is generated by the element $a$, because $a=a$, $a+a = b$, and $a+a+a = e$.
Let us pick $G = (\mathbb{Z}, +)$.
We'll try and define a map $f: C_3 \to \mathbb{Z}$ by $a \mapsto 1$.

If $C_3$ had the universal property of the free group on $\{ e, a, b\}$, then we would be able to find a homomorphism $\overline{f}: C_3 \to \mathbb{Z}$, such that $\overline{f}(a) = 1$ (that is, mimicking the action of the set-function $f$).
But in fact, no such homomorphism can exist, because if $\overline{f}$ were such a homomorphism, then $\overline{f}(e) = \overline{f}(a+a+a) = 1+1+1 = 3$ so $\overline{f}(e) = 3$, which contradicts that [49z the image of the identity under a group homomorphism is the identity].

In essence, $C_3$ "has extra relations" (namely that $a+a+a = e$) which the free group doesn't have, and which can thwart the attempt to define $\overline{f}$; this is reflected in the fact that $C_3$ fails to have the universal property.

A proof of the universal property may be found [free_group_satisfies_universal_property elsewhere].


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