Editing 982: Set Theory

{{comic
| number    = 982
| date      = November 25, 2011
| title     = Set Theory
| image     = set_theory.png
| titletext = Proof of Zermelo's well-ordering theorem given the Axiom of Choice: 1: Take S to be any set. 2: When I reach step three, if S hasn't managed to find a well-ordering relation for itself, I'll feed it into this wood chipper. 3: Hey, look, S is well-ordered.
}}

==Explanation==
This comic is a pun on the phrase "{{w|Proof by Intimidation}}" which normally is a jocular term used mainly in mathematics. It refers to a style of presenting a purported mathematical proof by giving an argument loaded with jargon and appeals to obscure results, so that the audience is simply obliged to accept it, lest they have to admit to their ignorance and lack of understanding.

However, in this comic, "Proof by Intimidation" is taken to mean that by intimidating the elements within a set, they will conform to the proof (or, as the title text says, they will become "well-ordered"). This is accomplished by believing that the elements can be {{w|anthropomorphize}}d such that they feel fear. The idea of executing as an example was discussed by Sun Tzu in the ancient book {{w|The Art Of War}}.

This interpretation of the term "Proof by Intimidation" bears great resemblance to {{w|Argumentum ad baculum|argument from the stick}}, which is a fallacious form of reasoning of the form
1.  If not P, I will do you harm.
2.  Therefore, P.
This form of fallacy has the distinction, if properly applied, of never being called out as fallacious.  Ponytail, however, is threatening the proposition itself, rather than her audience, bringing a level of absurdity to the situation.

The {{w|axiom of choice}} (which has been referenced previously in [[804: Pumpkin Carving]]) says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin. It was later referenced in the title text of [[1724: Proofs]], another comic about a math class with a similar theme on how teachers teach their student mathematical proofs.

In the title text, the well-ordering theorem states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. This is also known as {{w|Zermelo's theorem}} and is equivalent to the Axiom of Choice. The woodchipper is a reference to the 1996 film {{w|Fargo (film)|Fargo}}, where a character uses one to dispose of a body.

It might seem there is another layer to the joke: if you can feed the set to the wood-chipper, that defines an ordering on the set (the order in which the elements are fed to the wood chipper).  However, that doesn't actually work, because the resulting ordering is not necessarily well-ordered. For example, consider the set of positive real numbers. You can imagine feeding half a number line to a wood chipper from the end near zero. This defines the standard less-than ordering, but it is not a well-ordering because it does not define a least element. For any positive number x, x/2 went into the wood chipper first. The set may be motivated to find a well-ordering, but it won't be the standard one.

==Transcript==
:[Ponytail stands at a blackboard, facing away from it. She has a pointer in her hand, and written on the blackboard is some set theory math, although one of the set elements is being pointed into a guillotine.]
:Ponytail: The axiom of choice allows you to select one element from each set in a collection
:Ponytail: and have it ''executed'' as an example to the others.

:[Caption below the panel:]
:My math teacher was a big believer in Proof by Intimidation.

{{comic discussion}}

[[Category:Comics featuring Ponytail]]
[[Category:Math]]
[[Category:Logic]]
[[Category:Puns]]