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On Authority

Some Flaws in the Analogy
 > Gödel's Theorem
An Example

## Gödel's Theorem

Actually, I lied (in What Is Law?)—it was the following idea that first popped into my head.

Actions that are right but not legal (i.e., the absurdities of law) are like mathematical statements that are true but not provable.

Hofstadter's figure 18 provided the idea of having a complex boundary and a nicely-structured subset; the bit about rectangles I made up later.

The fact that there really are mathematical statements that are true but not provable is known as Gödel's theorem. Apart from pointing out that this is a very disturbing fact, there's no point in my saying anything about it—you should just read Gödel, Escher, Bach.

Well, I will mention one thing, because it amuses me: what if Gödel's theorem itself were true but not provable? (In reality, the theorem is provable. The proof involves statements that are self-referential, hence my amusement.)

What I'd like to do now is compare and constrast the three different domains.

In the geometrical domain, the set of points is divided into symmetrical black and white areas, with the exception that the point in the exact center of the figure must be either gray or asymmetrical. A series of rectangles is used as an approximation to the black area, and, in the limit, matches it exactly.

In the legal domain, the set of actions is divided into right and wrong actions, but, as discussed in Some Flaws in the Analogy, there is no concept of symmetry, and there may also be gray areas. Legal actions are used as an approximation to right actions, but the approximation is finite and inexact.

In the mathematical domain, the set of statements is divided into true and false statements; the two are exactly symmetrical, with no gray areas. Provable statements are used as an approximation to true statements, but even though the approximation is an infinite series, the two do not match exactly, not even in the limit.

Here's a table that summarizes all the above.

 domain geometrical legal mathematical point action statement black right true white wrong false symmetrical? almost n/a X gray? a little X - rectangles legal provable infinite? X - X exact? X - -