*How heavy is your theory?*
There
is an information-theory version of Gödel’s Theorems; informally, it says that
“you can’t prove a twenty-pound theorem with a ten-pound theory”. Starting with
a definition of the information content of a theory, it proves that no theory
can prove any statement of greater content.

Löb’s
theorem also has an information-theory version. Informally, it says that “you
can prove a zero-pound theorem even with a zero-pound theory”. The vanity of
faith simplifies the proof of its necessity.

*Paradoxes of Orthodoxy*
Self-doubt
and self-shame are paradoxical in form, but fairly orthodox in their truth
values. Self-doubt, which can’t prove itself, is equivalent to the proposition
that you can’t prove everything; self-shame, which refutes itself, is
equivalent to the proposition that you can refute anything.

On
the other hand, self-pride and self-trust are orthodox in form but paradoxical
in truth values. They affirm themselves in opposite ways, and you’d think
self-pride was making the more modest claim. It only says that it’s true in

*some*mathematical model; self-trust says that it’s true in*every*mathematical model. But of these two Pollyanna-ish self-affirmations, the one making the more sweeping claim is the true one, due to its emptiness.
What’s
worse, self-trust is ‘obvious’, but only after you’ve learned Gödel’s two
Incompleteness Theorems! Löb’s Theorem is like the math professor who, during a
lecture, said, “This proposition is obvious,” then asked himself, “

*Is*it obvious?” He left the hall and retired to his office. A half-hour later he returned to say, “Yes, it*is*obvious.”
## No comments:

## Post a Comment