Thursday, March 6, 2014

A Proof of Self-Proof; 9 of 15

          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.”

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