High-speed intro to Bayes's rule

(This is a high-speed introduction to [https://arbital.com/p/1lz](https://arbital.com/p/1lz) for people who want to get straight to it and are good at math.  If you'd like a gentler or more thorough introduction, try starting at the [Bayes' Rule Guide](https://arbital.com/p/1zq) page instead.)

# Percentages, frequencies, and waterfalls

Suppose you're screening a set of patients for a disease, which we'll call Diseasitis.%note:Lit. "inflammation of the disease".% Your initial test is a tongue depressor containing a chemical strip, which usually turns black if the patient has Diseasitis.

- Based on prior epidemiology, you expect that around 20% of patients in the screening population have Diseasitis.
- Among patients with Diseasitis, 90% turn the tongue depressor black.
- 30% of the patients without Diseasitis will also turn the tongue depressor black.

What fraction of patients with black tongue depressors have Diseasitis?

%%hidden(Answer): 3/7 or 43%, quickly obtainable as follows:  In the screened population, there's 1 sick patient for 4 healthy patients.  Sick patients are 3 times more likely to turn the tongue depressor black than healthy patients.  $(1 : 4) \cdot (3 : 1) = (3 : 4)$ or 3 sick patients to 4 healthy patients among those that turn the tongue depressor black, corresponding to a probability of $3/7 = 43\%$ that the patient is sick.%%

(Take your own stab at answering this question, then please click "Answer" above to read the answer before continuing.)

Bayes' rule is a theorem which describes the general form of the operation we carried out to find the answer above.  In the form we used above, we:

- Started from the *prior odds* of (1 : 4) for sick versus healthy patients;
- Multiplied by the *likelihood ratio* of (3 : 1) for sick versus healthy patients blackening the tongue depressor;
- Arrived at *posterior odds* of (3 : 4) for a patient with a positive test result being sick versus healthy.

Bayes' rule in this form thus states that **the prior odds times the likelihood ratio equals the posterior odds.**

We could also potentially see the positive test result as **revising** a *prior belief* or *prior probability* of 20% that the patient was sick, to a *posterior belief* or *posterior probability* of 43%.

To make it clearer that we did the correct calculation above, and further pump intuitions for Bayes' rule, we'll walk through some additional visualizations.

## Frequency representation

The *frequency representation* of Bayes' rule would describe the problem as follows:  "Among 100 patients, there will be 20 sick patients and 80 healthy patients."

![prior frequency](https://i.imgur.com/gvcoqQN.png?0)

"18 out of 20 sick patients will turn the tongue depressor black.  24 out of 80 healthy patients will blacken the tongue depressor."

![posterior frequency](https://i.imgur.com/0XDbrYi.png?0)

"Therefore, there are (18+24)=42 patients who turn the tongue depressor black, among whom 18 are actually sick. (18/42)=(3/7)=43%."

(Some experiments show %%note: E.g. "[Probabilistic reasoning in clinical medicine](https://faculty.washington.edu/jmiyamot/p548/eddydm%20prob%20reas%20i%20clin%20medicine.pdf)" by David M. Eddy (1982).%% that this way of explaining the problem is the easiest for e.g. medical students to understand, so you may want to remember this format for future use.  Assuming you can't just send them to Arbital!)

## Waterfall representation

The *waterfall representation* may make clearer why we're also allowed to transform the problem into prior odds and a likelihood ratio, and multiply (1 : 4) by (3 : 1) to get posterior odds of (3 : 4) and a probability of 3/7.

The following problem is isomorphic to the Diseasitis one:

"A waterfall has two streams of water at the top, a red stream and a blue stream.  These streams flow down the waterfall, with some of each stream being diverted off to the side, and the remainder pools at the bottom of the waterfall."

![unlabeled waterfall](https://i.imgur.com/D8EhY65.png?0)

"At the top of the waterfall, there's around 20 gallons/second flowing from the red stream, and 80 gallons/second flowing from the blue stream.  90% of the red water makes it to the bottom of the waterfall, and 30% of the blue water makes it to the bottom of the waterfall.  Of the purplish water that mixes at the bottom, what fraction is from the red stream versus the blue stream?"

![labeled waterfall](https://i.imgur.com/QIrtuVU.png?0)

We can see from staring at the diagram that the *prior odds* and *likelihood ratio* are the only numbers we need to arrive at the answer:

- The problem would have the same answer if there were 40 gallons/sec of red water and 160 gallons/sec of blue water (instead of 20 gallons/sec and 80 gallons/sec).  This would just multiply the total amount of water by a factor of 2, without changing the ratio of red to blue at the bottom.
- The problem would also have the same answer if 45% of the red stream and 15% of the blue stream made it to the bottom (instead of 90% and 30%).  This would just cut down the total amount of water by a factor of 2, without changing the *relative* proportions of red and blue water.

![wide vs narrow waterfall](https://i.imgur.com/6FOndjc.png?0)

So only the *ratio* of red to blue water at the top (prior odds of the proposition), and only the *ratio* between the percentages of red and blue water that make it to the bottom (likelihood ratio of the evidence), together determine the *posterior* ratio at the bottom:  3 parts red to 4 parts blue.

## Test problem

Here's another Bayesian problem to attempt.  If you successfully solved the earlier problem on your first try, you might try doing this one in your head.

10% of widgets are bad and 90% are good.  4% of good widgets emit sparks, and 12% of bad widgets emit sparks.  What percentage of sparking widgets are bad?

%%hidden(Answer):
- There's $1 : 9$ bad vs. good widgets.  (9 times as many good widgets as bad widgets; widgets are 1/9 as likely to be bad as good.)
- Bad vs. good widgets have a $12 : 4$ relative likelihood to spark, which simplifies to $3 : 1.$  (Bad widgets are 3 times as likely to emit sparks as good widgets.)
- $(1 : 9) \cdot (3 : 1) = (3 : 9) \cong (1 : 3).$  (1 bad sparking widget for every 3 good sparking widgets.)
- Odds of $1 : 3$ convert to a probability of $\frac{1}{1+3} = \frac{1}{4} = 25\%.$ (25% of sparking widgets are bad.) %%

(If you're having trouble using odds ratios to represent uncertainty, see [this intro](https://arbital.com/p/561) or [this page](https://arbital.com/p/1rb).)

# General equation and proof

To say exactly what we're doing and prove its validity, we need to introduce some notation from [probability theory](https://arbital.com/p/1rf).

If $X$ is a proposition, $\mathbb P(X)$ will denote $X$'s probability, our quantitative degree of belief in $X.$

$\neg X$ will denote the negation of $X$ or the proposition "$X$ is false".

If $X$ and $Y$ are propositions, then $X \wedge Y$ denotes the proposition that both X and Y are true.  Thus $\mathbb P(X \wedge Y)$ denotes "The probability that $X$ and $Y$ are both true."

We now define [conditional probability](https://arbital.com/p/1rj):

$$\mathbb P(X|Y) := \dfrac{\mathbb P(X \wedge Y)}{\mathbb P(Y)}   \tag*{(definition of conditional probability)}$$

We pronounce $\mathbb P(X|Y)$ as "the conditional probability of X, given Y".  Intuitively, this is supposed to mean "The probability that $X$ is true, *assuming* that proposition $Y$ is true".

Defining conditional probability in this way means that to get "the probability that a patient is sick, given that they turned the tongue depressor black" we should put all the sick *plus* healthy patients with positive test results into a bag, and ask about the probability of drawing a patient who is sick *and* got a positive test result from that bag.  In other words, we perform the calculation $\frac{18}{18+24} = \frac{3}{7}.$

![diseasitis frequency](https://i.imgur.com/0XDbrYi.png?0)

Rearranging [the definition of conditional probability](https://arbital.com/p/1rj), $\mathbb P(X \wedge Y) = \mathbb P(Y) \cdot \mathbb P(X|Y).$  So to find "the fraction of all patients that are sick *and* get a positive result", we multiply "the fraction of patients that are sick" times "the probability that a sick patient blackens the tongue depressor".

We're now ready to prove Bayes's rule in the form, "the prior odds times the likelihood ratio equals the posterior odds".

The "prior odds" is the ratio of sick to healthy patients:

$$\frac{\mathbb P(sick)}{\mathbb P(healthy)}  \tag*{(prior odds)}$$

The "likelihood ratio" is how much more *relatively* likely a sick patient is to get a positive test result (turn the tongue depressor black), compared to a healthy patient:

$$\frac{\mathbb P(positive | sick)}{\mathbb P(positive | healthy)} \tag*{(likelihood ratio)}$$

The "posterior odds" is the odds that a patient is sick versus healthy, *given* that they got a positive test result:

$$\frac{\mathbb P(sick | positive)}{\mathbb P(healthy | positive)} \tag*{(posterior odds)}$$

Bayes's theorem asserts that *prior odds times likelihood ratio equals posterior odds:*

$$\frac{\mathbb P(sick)}{\mathbb P(healthy)} \cdot \frac{\mathbb P(positive | sick)}{\mathbb P(positive | healthy)} = \frac{\mathbb P(sick | positive)}{\mathbb P(healthy | positive)}$$

We will show this by proving the general form of Bayes's Rule.  For any two hypotheses $H_j$ and $H_k$ and any piece of new evidence $e_0$:

$$
\frac{\mathbb P(H_j)}{\mathbb P(H_k)}
\cdot
\frac{\mathbb P(e_0 | H_j)}{\mathbb P(e_0 | H_k)}
=
\frac{\mathbb P(e_0 \wedge H_j)}{\mathbb P(e_0 \wedge H_k)}
=
\frac{\mathbb P(e_0 \wedge H_j)/\mathbb P(e_0)}{\mathbb P(e_0 \wedge H_k)/\mathbb P(e_0)}
=
\frac{\mathbb P(H_j | e_0)}{\mathbb P(H_k | e_0)}
$$

In the Diseasitis example, this corresponds to performing the operations:

$$
\frac{0.20}{0.80}
\cdot
\frac{0.90}{0.30}
=
\frac{0.18}{0.24}
=
\frac{0.18/0.42}{0.24/0.42}
=
\frac{0.43}{0.57}
$$

Using red for sick, blue for healthy, grey for a mix of sick and healthy patients, and + signs for positive test results, the proof above can be visualized as follows:

![bayes venn](https://i.imgur.com/YBc2nYo.png?0)

%todo: less red in first circle (top left).  in general, don't have prior proportions equal posterior proportions graphically!%

## Bayes' theorem

An alternative form, sometimes called "Bayes' theorem" to distinguish it from "Bayes' rule" (although not everyone follows this convention), uses absolute probabilities instead of ratios.  The [law of marginal probability](https://arbital.com/p/marginal_probability) states that for any set of [mutually exclusive and exhaustive](https://arbital.com/p/1rd) possibilities $\{X_1, X_2, ..., X_i\}$ and any proposition $Y$:

$$\mathbb P(Y) = \sum_i \mathbb P(Y \wedge X_i)   \tag*{(law of marginal probability)}$$

Then we can derive an expression for the absolute (non-relative) probability of a proposition $H_k$ after observing evidence $e_0$ as follows:

$$
\mathbb P(H_k | e_0)
= \frac{\mathbb P(H_k \wedge e_0)}{\mathbb P(e_0)}
= \frac{\mathbb P(e_0 \wedge H_k)}{\sum_i P(e_0 \wedge H_i)}
= \frac{\mathbb P(e_0 | X_k) \cdot \mathbb P(X_k)}{\sum_i \mathbb P(e_0 | X_i) \cdot \mathbb P(X_i)}
$$

The equation of the first and last terms above is what you will usually see described as Bayes' theorem.

To see why this decomposition might be useful, note that $\mathbb P(sick | positive)$ is an *inferential* step, a conclusion that we make after observing a new piece of evidence.  $\mathbb P(positive | sick)$ is a piece of *causal* information we are likely to have on hand, for example by testing groups of sick patients to see how many of them turn the tongue depressor black.  $\mathbb P(sick)$ describes our state of belief before making any new observations.  So Bayes' theorem can be seen as taking what we already believe about the world (including our prior belief about how different imaginable states of affairs would generate different observations), plus an actual observation, and outputting a new state of belief about the world.

## Vector and functional generalizations

Since the proof of Bayes' rule holds for *any pair* of hypotheses, it also holds for relative belief in any number of hypotheses.  Furthermore, we can repeatedly multiply by likelihood ratios to chain together any number of pieces of evidence.

Suppose there's a bathtub full of coins:

- Half the coins are "fair" and have a 50% probability of coming up Heads each time they are thrown.
- A third of the coins are biased to produce Heads 25% of the time (Tails 75%).
- The remaining sixth of the coins are biased to produce Heads 75% of the time.

You randomly draw a coin, flip it three times, and get the result **HTH**.  What's the chance this is a fair coin?

We can validly calculate the answer as follows:

$$
\begin{array}{rll}
  & (3 : 2 : 1) & \cong (\frac{1}{2} : \frac{1}{3} : \frac{1}{6}) \\
\times & (2 : 1 : 3) & \cong ( \frac{1}{2} : \frac{1}{4} : \frac{3}{4} ) \\
\times & (2 : 3 : 1) & \cong ( \frac{1}{2} : \frac{3}{4} : \frac{1}{4} ) \\
\times & (2 : 1 : 3) & \\
= & (24 : 6 : 9) & \cong (8 : 2 : 3) \cong (\frac{8}{13} : \frac{2}{13} : \frac{3}{13})
\end{array}
$$

So the posterior probability the coin is fair is 8/13 or ~62%.

This is one reason it's good to know the [odds form](https://arbital.com/p/1x5) of Bayes' rule, not just the [probability form](https://arbital.com/p/554) in which Bayes' theorem is often given.%%note: Imagine trying to do the above calculation by repeatedly applying the form of the theorem that says: $$\mathbb P(H_k | e_0) = \frac{\mathbb P(e_0 | X_k) \cdot \mathbb P(X_k)}{\sum_i \mathbb P(e_o | X_i) \cdot \mathbb P(X_i)}$$ %%

We can generalize further by writing Bayes' rule in a functional form.  If $\mathbb O(H_i)$ is a relative belief vector or relative belief function on the variable $H,$ and $\mathcal L(e_0 | H_i)$ is the likelihood function giving the relative chance of observing evidence $e_0$ given each possible state of affairs $H_i,$ then relative posterior belief $\mathbb O(H_i | e_0)$ is given by:

$$\mathbb O(H_i | e_0) = \mathcal L(e_0 | H_i) \cdot \mathbb O(H_i)$$

If we [normalize](https://arbital.com/p/1rk) the relative odds $\mathbb O$ into absolute probabilities $\mathbb P$ - that is, divide through $\mathbb O$ by its sum or integral so that the new function sums or integrates to $1$ - then we obtain Bayes' rule for probability functions:

$$\mathbb P(H_i | e_0) \propto \mathcal L(e_0 | H_i) \cdot \mathbb P(H_i)  \tag*{(functional form of Bayes' rule)}$$

# Applications of Bayesian reasoning

This general Bayesian framework - prior belief, evidence, posterior belief - is a lens through which we can view a *lot* of formal and informal reasoning plus a large amount of entirely nonverbal cognitive-ish phenomena.%%note:This broad statement is widely agreed.  Exactly *which* phenomena are good to view through a Bayesian lens is sometimes disputed.%%

Examples of people who might want to study Bayesian reasoning include:

- Professionals who use statistics, such as scientists or medical doctors.
- Computer programmers working in the field of machine learning.
- Human beings trying to think.

The third application is probably of the widest general interest.

## Example human applications of Bayesian reasoning

Philip Tetlock found when studying "superforecasters", people who were especially good at predicting future events:

"The superforecasters are a numerate bunch: many know about Bayes' theorem and could deploy it if they felt it was worth the trouble. But they rarely crunch the numbers so explicitly. What matters far more to the superforecasters than Bayes' theorem is Bayes' core insight of gradually getting closer to the truth by constantly updating in proportion to the weight of the evidence." — Philip Tetlock and Dan Gardner, [_Superforecasting_](https://arbital.com/p/https://en.wikipedia.org/wiki/Superforecasting)

This is some evidence that *knowing about* Bayes' rule and understanding its *qualitative* implications is a factor in delivering better-than-average intuitive human reasoning.  This pattern is illustrated in the next couple of examples.

### The OKCupid date.

One realistic example of Bayesian reasoning was deployed by one of the early test volunteers for a much earlier version of a guide to Bayes' rule.  She had scheduled a date with a 96% OKCupid match, who had then cancelled that date without other explanation.  After spending some mental time bouncing back and forth between "that doesn't seem like a good sign" versus "maybe there was a good reason he canceled", she decided to try looking at the problem using that Bayes thing she'd just learned about.  She estimated:

- A 96% OKCupid match like this one, had prior odds of 2 : 5 for being a desirable versus undesirable date.  (Based on her prior experience with 96% OKCupid matches, and the details of his profile.)
- Men she doesn't want to go out with are 3 times as likely as men she might want to go out with to cancel a first date without other explanation.

This implied posterior odds of 2 : 15 that this was an undesirable date, which was unfavorable enough not to pursue him further.%%note: She sent him what might very well have been the first explicitly Bayesian rejection notice in dating history, reasoning that if he wrote back with a Bayesian counterargument, this would promote him to being interesting again.  He didn't write back.%%

The point of looking at the problem this way is not that she knew *exact* probabilities and could calculate that the man had an exactly 88% chance of being undesirable.  Rather, by breaking up the problem in that way, she was able to summarize what she thought she knew in compact form, see what those beliefs already implied, and stop bouncing back and forth between imagined reasons why a good date might cancel versus reasons to protect herself from potential bad dates.  An answer *roughly* in the range of 15/17 made the decision clear.

### Internment of Japanese-Americans during World War II

From Robyn Dawes's [Rational Choice in an Uncertain World](https://amazon.com/Rational-Choice-Uncertain-World-Robyn/dp/0155752154):

> Post-hoc fitting of evidence to hypothesis was involved in a most grievous chapter in United States history: the internment of Japanese-Americans at the beginning of the Second World War.  When California governor Earl Warren testified before a congressional hearing in San Francisco on February 21, 1942, a questioner pointed out that there had been no sabotage or any other type of espionage by the Japanese-Americans up to that time.  Warren responded, "I take the view that this lack [subversive activity](https://arbital.com/p/of) is the most ominous sign in our whole situation. It convinces me more than perhaps any other factor that the sabotage we are to get, the Fifth Column activities are to get, are timed just like Pearl Harbor was timed... I believe we are just being lulled into a false sense of security."

You might want to take your own shot at guessing what Dawes had to say about a Bayesian view of this situation, before reading further.

%%hidden(Answer):Suppose we put ourselves into the shoes of this congressional hearing, and imagine ourselves trying to set up this problem.

- The [prior](https://arbital.com/p/1rm) [odds](https://arbital.com/p/1rb) that there would be a conspiracy of Japanese-American saboteurs.
- The [likelihood](https://arbital.com/p/56t) of the observation "no visible sabotage or any other type of espionage", *given* that a Fifth Column actually existed.
- The likelihood of the observation "no visible sabotage from Japanese-Americans", in the possible world where there is *no* such conspiracy.

As soon as we set up this problem, we realize that, whatever the probability of "no sabotage" being observed if there is a conspiracy, the likelihood of observing "no sabotage" if there *isn't* a conspiracy must be even higher.  This means that the likelihood ratio:

$$\frac{\mathbb P(\neg \text{sabotage} | \text {conspiracy})}{\mathbb P(\neg \text {sabotage} | \neg \text {conspiracy})}$$

...must be *less than 1,* and accordingly:

$$
\frac{\mathbb P(\text {conspiracy} | \neg \text{sabotage})}{\mathbb P(\neg \text {conspiracy} | \neg \text{sabotage})}
<
\frac{\mathbb P(\text {conspiracy})}{\mathbb P(\neg \text {conspiracy})}
\cdot
\frac{\mathbb P(\neg \text{sabotage} | \text {conspiracy})}{\mathbb P(\neg \text {sabotage} | \neg \text {conspiracy})}
$$

Observing the total absence of any sabotage can only decrease our estimate that there's a Japanese-American Fifth Column, not increase it.  (It definitely shouldn't be "the most ominous" sign that convinces us "more than any other factor" that the Fifth Column exists.)

Again, what matters is not the *exact* likelihood of observing no sabotage given that a Fifth Column actually exists.  As soon as we set up the Bayesian problem, we can see there's something *qualitatively* wrong with Earl Warren's reasoning.%%

# Further reading

This has been a very brief and high-speed presentation of Bayes and Bayesianism.  It should go without saying that a vast literature, nay, a universe of literature, exists on Bayesian statistical methods and Bayesian epistemology and Bayesian algorithms in machine learning.  Staying inside Arbital, you might be interested in moving on to read:

## More on the technical side of Bayes' rule

- A sadly short list of [example Bayesian word problems](https://arbital.com/p/22w).  Want to add more?  (Hint hint.)
- [__Bayes' rule: Proportional form.__](https://arbital.com/p/1zm) The fastest way to present a step in Bayesian reasoning in a way that will sound sort of understandable to somebody who's never heard of Bayes.
- [__Bayes' rule: Log-odds form.__](https://arbital.com/p/1zh) A simple transformation of Bayes' rule reveals tools for measuring degree of belief, and strength of evidence.
- [__The "Naive Bayes" algorithm.__](https://arbital.com/p/1zg)  (Scroll down to the middle.)  The original simple Bayesian spam filter.
- [__Non-naive multiple updates.__](https://arbital.com/p/1zg)  (Scroll down past Naive Bayes.)  How to avoid double-counting the evidence, or worse, when considering multiple items of *correlated* evidence.
- [__Laplace's Rule of Succession.__](https://arbital.com/p/21c)  The classic example of an [inductive prior](https://arbital.com/p/21b).

## More on intuitive implications of Bayes' rule

- [__A Bayesian view of scientific virtues.__](https://arbital.com/p/220) Why is it that science relies on bold, precise, and falsifiable predictions? Because of Bayes' rule, of course.
- [__Update by inches.__](https://arbital.com/p/update_by_inches) It's virtuous to change your mind in response to overwhelming evidence. It's even more virtuous to shift your beliefs a little bit at a time, in response to *all* evidence (no matter how small).
- [__Belief revision as probability elimination.__](https://arbital.com/p/1y6) Update your beliefs by throwing away large chunks of probability mass.
- [__Shift towards the hypothesis of least surprise.__](https://arbital.com/p/552) When you see new evidence, ask: which hypothesis is *least surprised?*
- [__Extraordinary claims require extraordinary evidence.__](https://arbital.com/p/21v) The people who adamantly claim they were abducted by aliens do provide *some* evidence for aliens. They just don't provide quantitatively *enough* evidence.
- [__Ideal reasoning via Bayes' rule.__](https://arbital.com/p/) Bayes' rule is to reasoning as the [Carnot cycle](https://arbital.com/p/https://en.wikipedia.org/wiki/Carnot_cycle) is to engines: Nobody can be a perfect Bayesian, but Bayesian reasoning is still the theoretical ideal.
- [__Likelihoods, p-values, and the replication crisis.__](https://arbital.com/p/4xx)  Arguably, a large part of the replication crisis can ultimately be traced back to the way journals treat p-values, and a large number of those problems can be summed up as "P-values are not Bayesian."

Joe Huang
Brie Hoffman
Eric Rogstad
Gijs van Dam
Rihn H
Mike Totman
Connor Flexman
Eliezer Yudkowsky


Axiom of Choice

"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 \[equivalence between the Axiom of Choice and the statement 'every infinite set A has the same cardinality as AxA'\](https://arbital.com/p/the) 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*

[https://arbital.com/p/toc:](https://arbital.com/p/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 [vector spaces](https://arbital.com/p/-3w0) of arbitrary [dimension](https://arbital.com/p/-dimension_mathematics), and [ultraproducts](https://arbital.com/p/-ultraproduct)). 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](https://arbital.com/p/-axiom_mathematics), it cannot be proven and must instead be assumed.
In particular, it is an axiom of [set theory](https://arbital.com/p/-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](https://arbital.com/p/-constructive_mathematics) 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://arbital.com/p/-https://wiki.lesswrong.com/wiki/Map_and_Territory_](https://arbital.com/p/-https://wiki.lesswrong.com/wiki/Map_and_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](https://arbital.com/p/-axiom_mathematics) of choice states that, given a collection of *[non-empty](https://arbital.com/p/-5zc)* [sets](https://arbital.com/p/-3jz), there is a [function](https://arbital.com/p/-3jy) which selects a single element from each of the sets.

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

In [logical notation](https://arbital.com/p/-logical_notation),
$$
\forall_X
\left(
\left[\in X} Y \not=  \emptyset \right](https://arbital.com/p/\forall_{Y)
\Rightarrow
\left[
\left](https://arbital.com/p/\exists)
\right)
$$

#Axiom Unnecessary for Finite Collections of Sets#
For a [finite set](https://arbital.com/p/-5zy)  $X$ containing only [finite](https://arbital.com/p/-5zy) non-empty sets, the axiom is actually provable (from the [Zermelo-Fraenkel axioms](https://arbital.com/p/-zermelo_fraenkel_axioms) of set theory ZF), and hence does not need to be given as an [axiom](https://arbital.com/p/-axiom_mathematics). 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](https://arbital.com/p/-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 [countable](https://arbital.com/p/-2w0) 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).

[Georg Cantor](https://arbital.com/p/-https://en.wikipedia.org/wiki/Georg_Cantor) introduced the concept of [transfinite numbers](https://arbital.com/p/-transfinite_number)
and different [cardinalities of infinity](https://arbital.com/p/-4w5) in a 1874
[paper](https://arbital.com/p/https://en.wikipedia.org/wiki/Georg_Cantor%27s_first_set_theory_article) (which contains his infamous
[Diagonalization Argument](https://arbital.com/p/-https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument))
 and along with this sparked the introduction of [set theory](https://arbital.com/p/-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.

[Ernst Zermelo](https://arbital.com/p/-https://en.wikipedia.org/wiki/Ernst_Zermelo) attempted to
develop an [axiomatic](https://arbital.com/p/-axiom_system) 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 [Zermelo Axioms of Set Theory](https://arbital.com/p/-https://en.wikipedia.org/wiki/Zermelo_set_theory).
Mathematicians, [Abraham Fraenkel](https://arbital.com/p/-https://en.wikipedia.org/wiki/Abraham_Fraenkel) and
[Thoralf Skolem](https://arbital.com/p/-https://en.wikipedia.org/wiki/Thoralf_Skolem)  improved this system (independently of each other)
into its modern version, the [Zermelo Fraenkel Axioms of Set Theory](https://arbital.com/p/-https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
).

In 1914, [Felix Hausdorff](https://arbital.com/p/https://en.wikipedia.org/wiki/Felix_Hausdorff) proved
[Hausdorff's paradox](https://arbital.com/p/https://en.wikipedia.org/wiki/Hausdorff_paradox). The ideas
behind this proof were used in 1924 by [Banach](https://arbital.com/p/-https://en.wikipedia.org/wiki/Stefan_Banach
Stefan) and [Alfred Tarski](https://arbital.com/p/-https://en.wikipedia.org/wiki/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, [Kurt Gödel](https://arbital.com/p/-https://en.wikipedia.org/wiki/Kurt_G%C3%B6del) proved that
the Axiom of Choice is consistent with the rest of the ZF axioms.

Finally, in 1963, [Paul Cohen](https://arbital.com/p/-https://en.wikipedia.org/wiki/Paul_Cohen) developed a revolutionary
mathematical technique called [forcing](https://arbital.com/p/-forcing_mathematics), 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
[Generalized Continuum Hypothesis](https://arbital.com/p/-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](https://arbital.com/p/-constructive_mathematics) 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
[Per Martin-Löf](https://arbital.com/p/-https://en.wikipedia.org/wiki/Per_Martin-L%C3%B6f)
at [this webpage](https://arbital.com/p/-http://comjnl.oxfordjournals.org/content/49/3/345.full).

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

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

#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](https://arbital.com/p/-product_of_sets) 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 [https://arbital.com/p/-3w0](https://arbital.com/p/-3w0) can be intuitively thought of as a collection of [vectors](https://arbital.com/p/-vector) 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 [https://arbital.com/p/-vector_space_dimension](https://arbital.com/p/-vector_space_dimension). For a finite-dimensional vector space, it is possible to find a [basis](https://arbital.com/p/-vector_space_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 functions](https://arbital.com/p/-continuous_function) $f: C \rightarrow C$ from a [https://arbital.com/p/-closed_disk](https://arbital.com/p/-closed_disk) $C$ onto itself has a [https://arbital.com/p/-fixed_point](https://arbital.com/p/-fixed_point) $x_0$. (In full generality, $C$ may be any [https://arbital.com/p/-5xr](https://arbital.com/p/-5xr) [compact](https://arbital.com/p/-compact_mathematics) [https://arbital.com/p/-3jz](https://arbital.com/p/-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 [real numbers](https://arbital.com/p/-4bc), not necessarily [integers](https://arbital.com/p/-48l)), 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 [natural numbers](https://arbital.com/p/-45h) $\mathbb{N}$ and let each of the mathematical structures $A_n$ be numbered).

An [https://arbital.com/p/-ultrafilter](https://arbital.com/p/-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 [https://arbital.com/p/-cofinite](https://arbital.com/p/-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 [ultraproducts](https://arbital.com/p/-ultraproduct). 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 [https://arbital.com/p/-first_order_logic](https://arbital.com/p/-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 [https://arbital.com/p/-nonstandard_real_numbers](https://arbital.com/p/-nonstandard_real_numbers).

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

#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 [Stefan Banach](https://arbital.com/p/-https://en.wikipedia.org/wiki/Stefan_Banach) and
[Alfred Tarski](https://arbital.com/p/-https://en.wikipedia.org/wiki/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 [Here](https://arbital.com/p/-https://www.youtube.com/watch?v=s86-Z-CbaHA).

The video [
Infinity shapeshifter vs. Banach-Tarski paradox](https://arbital.com/p/-https://www.youtube.com/watch?v=ZUHWNfqzPJ8) 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 [Zermelo-Fraenkel axioms of Set Theory](https://arbital.com/p/-https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory)
(ZF)
introduce a fundamental concept, called a [set](https://arbital.com/p/-3jz), and a [relation](https://arbital.com/p/-3nt) between
sets, called
the [element of](https://arbital.com/p/-5xy) 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 [axioms](https://arbital.com/p/-axiom_mathematics) 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](https://arbital.com/p/-function_mathematics) can be interpreted as a specific kind of set. In particular,
it is a set of [ordered pairs](https://arbital.com/p/-ordered_pair) (more generally, [ordered n-tuples](https://arbital.com/p/-tuple), 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
[Zermelo-Fraenkel one from Wikipedia](https://arbital.com/p/-https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory).

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](https://arbital.com/p/-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 [Platonic](https://arbital.com/p/-https://en.wikipedia.org/wiki/Platonism)
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 [models](https://arbital.com/p/-model_theory). However, these are not physical or even computational models,
but models of a more abstract kind.

For example, a model of [group theory](https://arbital.com/p/-3g8) is a specific [group](https://arbital.com/p/-3gd),
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 [cyclic group on 3 elements](https://arbital.com/p/-47y)
and the [symmetric group](https://arbital.com/p/-497) $S_3$. 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,
[this
StackExchange answer](https://arbital.com/p/-http://math.stackexchange.com/questions/531516/meta-theory-when-studying-set-theory) 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 [
Kunen's text](https://arbital.com/p/-http://vanilla47.com/PDFs/Cryptography/Mathematics/Set%20Theory%20PDFs/SET%20THEORY.pdf). The consistency of choice is the easier direction, and involves constructing
something called [Gödel's constructible universe of sets](https://arbital.com/p/-constructible_universe) (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://arbital.com/p/-https://en.wikipedia.org/wiki/Paul_Cohen_](https://arbital.com/p/-https://en.wikipedia.org/wiki/Paul_Cohen_)
called [forcing](https://arbital.com/p/-forcing_mathematics).

#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](https://arbital.com/p/-disjoint_set)
(any two sets in $C$ do not [intersect](https://arbital.com/p/-5sb)),
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 [product](https://arbital.com/p/-5zs) of non-empty sets is non-empty.
 - Tarski's Theorem for Binary [Products](https://arbital.com/p/-product_mathematics)
 (from the quote at the start of this article) that $A$ is [bijective](https://arbital.com/p/-bijection) to $A \times A$.
 - Every [surjective function](https://arbital.com/p/-4bg) has a [right inverse](https://arbital.com/p/-inverse_of_function).
 - Given two sets, either
 - Every [vector space](https://arbital.com/p/-3w0) has a [basis](https://arbital.com/p/-vector_space_basis)
 - Every [connected graph](https://arbital.com/p/-connected_graph) has a [spanning tree](https://arbital.com/p/-spanning_tree).
 - For any pair of sets have comparable[cardinality](https://arbital.com/p/-4w5): 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 [Wikipedia page](https://arbital.com/p/-https://en.wikipedia.org/wiki/Axiom_of_choice#Equivalents).

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 [partially ordered set](https://arbital.com/p/-3rb) (poset) for which every [chain](https://arbital.com/p/-chain_order_theory) has an [upper bound](https://arbital.com/p/-upper_bound_mathematics)
has a [maximal element](https://arbital.com/p/-maximal_mathematics).

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 [linearly independent](https://arbital.com/p/-linear_independence) 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](https://arbital.com/p/-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 [large enough infinite](https://arbital.com/p/-transfinite) 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 [natural numbers](https://arbital.com/p/-45h) $\mathbb{N}$, 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 [linearly ordered set](https://arbital.com/p/540) is called well-ordered if any of its non-empty subsets has a least element.

For example, the [natural numbers](https://arbital.com/p/-45h) $\mathbb{N}$ 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 [real numbers](https://arbital.com/p/-4bc) (and in fact the positive [rational numbers](https://arbital.com/p/-4zq))
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 [bijective](https://arbital.com/p/-bijection) 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](https://arbital.com/p/-union_mathematics) of [countably many](https://arbital.com/p/-countble_infinity) countable sets is countable.
 - The [axiom of dependent choice](https://arbital.com/p/-axiom_of_dependent_choice) (given a non-empty set $X$ and
([entire](https://arbital.com/p/-entire_relation)) [binary relation](https://arbital.com/p/-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](https://arbital.com/p/-axiom_of_countable_choice) (every countable set of sets
has a choice function).
 - Every [field](https://arbital.com/p/-field_mathematics) has an [algebraic closure](https://arbital.com/p/-algebraic_closure)
 - Existence of non-principal [ultrafilters](https://arbital.com/p/-ultrafilter).
 - Gödel's [completeness theorem](https://arbital.com/p/-completeness_theorem) for first-order logic.
 - [Boolean Prime Ideal Theorem](https://arbital.com/p/-booelan_prime_ideal_theorem) (useful for proving existence
of non-principal ultrafilters and Gödel's completenes theorem).
 - The [Law of excluded middle](https://arbital.com/p/-excluded_middle) for logic.

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

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

The formal statement concerns [cardinality](https://arbital.com/p/-4w5) 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 Hypthosis](https://arbital.com/p/-generalized_continuum_hypthesis) (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 [power set](https://arbital.com/p/-6gl) $P(X)$. 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](https://arbital.com/p/-constructive_mathematics)
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

 - [this section of the Wikipedia page on AC](https://arbital.com/p/-https://en.wikipedia.org/wiki/Axiom_of_choice#Criticism_and_acceptance)
 - [this StackOverflow question](https://arbital.com/p/-http://mathoverflow.net/questions/22927/why-worry-about-the-axiom-of-choice),
 - [this paper by Per Martin-Löf on the history and problems with the Axiom of Choice](https://arbital.com/p/-http://comjnl.oxfordjournals.org/content/49/3/345.full)
 - [this post by Greg Muller on why the Axiom of Choice is wrong](https://arbital.com/p/-https://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/)
 - [this post on constructive mathematics on the Internet Encyclopaedia of Philosophy](https://arbital.com/p/-http://www.iep.utm.edu/con-math/)
 - [this post
on Good Math Bad Math by Mark C. Chu-Carroll](https://arbital.com/p/-http://scienceblogs.com/goodmath/2007/05/27/the-axiom-of-choice/),
 - [this
post by Terrance Tao on the usefulness of the axiom in "concrete" mathematics](https://arbital.com/p/-https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/),
 - [this post regarding constructive mathematics on the University of Canterbury website](https://arbital.com/p/-http://www.math.canterbury.ac.nz/php/groups/cm/faq/)
 - ["interview with a constructivist"](https://arbital.com/p/-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)
 - [this page on Standford Enclyclopaedia on constructive mathematics](https://arbital.com/p/-http://plato.stanford.edu/entries/mathematics-constructive/)
 - [this page on Stanford Encyclopaedia on intuitionistic mathematics](https://arbital.com/p/-http://plato.stanford.edu/entries/intuitionism/)

#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](https://arbital.com/p/-axiom_of_determinancy). This axiom states that for any two-person [game](https://arbital.com/p/-game_mathematics)
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:

 - [Countably many prisoners have to guess hats with no information transfer, yet all but a finite number may go free.](https://arbital.com/p/-https://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/)
 - [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.](https://arbital.com/p/-http://mathoverflow.net/questions/20882/most-unintuitive-application-of-the-axiom-of-choice)
 - [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.](https://arbital.com/p/-http://math.stackexchange.com/questions/613506/real-guessing-puzzle)

Daniel Satanove
Dylan Hendrickson
Yoni Lavi
Eric Bruylant
Mark Chimes
Jaime Sevilla Molina

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.


Low-speed explanation

Use this tag to indicate that a page offers a relatively slower, more gentle, or more wordy explanation.

Note that the speed of an explanation is not the same as its technical level. An explanation can assume a high technical level and still take its time going into details and so deserve the low-speed tag.


High-speed explanation

Use this tag to indicate that a page offers a relatively faster and more terse explanation.

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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 [https://arbital.com/p/-5p2](https://arbital.com/p/-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 [https://arbital.com/p/Fibonacci_sequence](https://arbital.com/p/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 [halting theorem](https://arbital.com/p/46h), 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 ##

[https://arbital.com/p/multiple-choice](https://arbital.com/p/multiple-choice)

[https://arbital.com/p/multiple-choice](https://arbital.com/p/multiple-choice)

[https://arbital.com/p/multiple-choice](https://arbital.com/p/multiple-choice)

[https://arbital.com/p/multiple-choice](https://arbital.com/p/multiple-choice)

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

%start-path([https://arbital.com/p/6c9](https://arbital.com/p/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 \[equivalence between the Axiom of Choice and the statement 'every infinite set A has the same cardinality as AxA'\](https://arbital.com/p/the) 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 [vector spaces](https://arbital.com/p/-3w0) of arbitrary [dimension](https://arbital.com/p/-dimension_mathematics), and [ultraproducts](https://arbital.com/p/-ultraproduct)). 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](https://arbital.com/p/-axiom_mathematics), it cannot be proven and must instead be assumed.
In particular, it is an axiom of [set theory](https://arbital.com/p/-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](https://arbital.com/p/-constructive_mathematics) 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://arbital.com/p/-https://wiki.lesswrong.com/wiki/Map_and_Territory_](https://arbital.com/p/-https://wiki.lesswrong.com/wiki/Map_and_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](https://arbital.com/p/-axiom_mathematics) of choice states that, given a collection of *[non-empty](https://arbital.com/p/-5zc)* [sets](https://arbital.com/p/-3jz), there is a [function](https://arbital.com/p/-3jy) which selects a single element from each of the sets.

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

In [logical notation](https://arbital.com/p/-logical_notation),
$$
\forall_X
\left(
\left[\in X} Y \not=  \emptyset \right](https://arbital.com/p/\forall_{Y)
\Rightarrow
\left[
\left](https://arbital.com/p/\exists)
\right)
$$

#Axiom Unnecessary for Finite Collections of Sets#
For a [finite set](https://arbital.com/p/-5zy)  $X$ containing only [finite](https://arbital.com/p/-5zy) non-empty sets, the axiom is actually provable (from the [Zermelo-Fraenkel axioms](https://arbital.com/p/-zermelo_fraenkel_axioms) of set theory ZF), and hence does not need to be given as an [axiom](https://arbital.com/p/-axiom_mathematics). 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](https://arbital.com/p/-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 [countable](https://arbital.com/p/-2w0) 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).

[Georg Cantor](https://arbital.com/p/-https://en.wikipedia.org/wiki/Georg_Cantor) introduced the concept of [transfinite numbers](https://arbital.com/p/-transfinite_number)
and different [cardinalities of infinity](https://arbital.com/p/-4w5) in a 1874
[paper](https://arbital.com/p/https://en.wikipedia.org/wiki/Georg_Cantor%27s_first_set_theory_article) (which contains his infamous
[Diagonalization Argument](https://arbital.com/p/-https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument))
 and along with this sparked the introduction of [set theory](https://arbital.com/p/-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.

[Ernst Zermelo](https://arbital.com/p/-https://en.wikipedia.org/wiki/Ernst_Zermelo) attempted to
develop an [axiomatic](https://arbital.com/p/-axiom_system) 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 [Zermelo Axioms of Set Theory](https://arbital.com/p/-https://en.wikipedia.org/wiki/Zermelo_set_theory).
Mathematicians, [Abraham Fraenkel](https://arbital.com/p/-https://en.wikipedia.org/wiki/Abraham_Fraenkel) and
[Thoralf Skolem](https://arbital.com/p/-https://en.wikipedia.org/wiki/Thoralf_Skolem)  improved this system (independently of each other)
into its modern version, the [Zermelo Fraenkel Axioms of Set Theory](https://arbital.com/p/-https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
).

In 1914, [Felix Hausdorff](https://arbital.com/p/https://en.wikipedia.org/wiki/Felix_Hausdorff) proved
[Hausdorff's paradox](https://arbital.com/p/https://en.wikipedia.org/wiki/Hausdorff_paradox). The ideas
behind this proof were used in 1924 by [Banach](https://arbital.com/p/-https://en.wikipedia.org/wiki/Stefan_Banach
Stefan) and [Alfred Tarski](https://arbital.com/p/-https://en.wikipedia.org/wiki/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, [Kurt Gödel](https://arbital.com/p/-https://en.wikipedia.org/wiki/Kurt_G%C3%B6del) proved that
the Axiom of Choice is consistent with the rest of the ZF axioms.

Finally, in 1963, [Paul Cohen](https://arbital.com/p/-https://en.wikipedia.org/wiki/Paul_Cohen) developed a revolutionary
mathematical technique called [forcing](https://arbital.com/p/-forcing_mathematics), 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
[Generalized Continuum Hypothesis](https://arbital.com/p/-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](https://arbital.com/p/-constructive_mathematics) 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
[Per Martin-Löf](https://arbital.com/p/-https://en.wikipedia.org/wiki/Per_Martin-L%C3%B6f)
at [this webpage](https://arbital.com/p/-http://comjnl.oxfordjournals.org/content/49/3/345.full).

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

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


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](https://arbital.com/p/theory_mathematics) $T$ is a [well-formed](https://arbital.com/p/well_formed) [sentence](https://arbital.com/p/sentence_mathematics) in the [language](https://arbital.com/p/language_mathematics) 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](https://arbital.com/p/) 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](https://arbital.com/p/) 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](https://arbital.com/p/) 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](https://arbital.com/p/).

#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](https://arbital.com/p/)%%note: Incidentally, a theory whose set of axioms is computable is called [https://arbital.com/p/-axiomatizable](https://arbital.com/p/-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 [https://arbital.com/p/3ft](https://arbital.com/p/3ft) you specify the [induction axiom schemata](https://arbital.com/p/5fz), stating that every sentence of the form $[https://arbital.com/p/P](https://arbital.com/p/P)\rightarrow \forall n. P(n)$ is an axiom of $PA$.

The reason why first order logic can handle infinite sets of axioms is due to its [compactness,](https://arbital.com/p/) which guarantees that every consequence of the theory is a consequence of a finite subset of the theory.