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which he seems to be inviting his viewers to enter. |
Godel |
In the examples we have seen of Strange Loops by Bach and Escher, there is a conflict |
between the finite and the infinite, and hence a strong sense of paradox. Intuition senses |
that there is something mathematical involved here. And indeed in our own century a |
mathematical counterpart was discovered, with the most enormous repercussions. And, |
just as the Bach and Escher loops appeal to very simple and ancient intuitions-a musical |
scale, a staircase-so this discovery, by K. Godel, of a Strange Loop in |
Introduction: A Musico-Logical Offering |
15 |
FIGURE 9. Kurl Godel. |
InlroducUon: A Musico-Logical Offering |
16 |
mathematical systems has its origins in simple and ancient intuitions. In its absolutely |
barest form, Godel's discovery involves the translation of an ancient paradox in |
philosophy into mathematical terms. That paradox is the so-called Epimenides paradox, |
or liar paradox. Epimenides was a Cretan who made one immortal statement: "All |
Cretans are liars." A sharper version of the statement is simply "I am lying"; or, "This |
statement is false". It is that last version which I will usually mean when I speak of the |
Epimenides paradox. It is a statement which rudely violates the usually assumed |
dichotomy of statements into true and false, because if you tentatively think it is true, |
then it immediately backfires on you and makes you think it is false. But once you've |
decided it is false, a similar backfiring returns you to the idea that it must be true. Try it! |
The Epimenides paradox is a one-step Strange Loop, like Escher's Print Gallery. But |
how does it have to do with mathematics? That is what Godel discovered. His idea was to |
use mathematical reasoning in exploring mathematical reasoning itself. This notion of |
making mathematics "introspective" proved to be enormously powerful, and perhaps its |
richest implication was the one Godel found: Godel's Incompleteness Theorem. What the |
Theorem states and how it is proved are two different things. We shall discuss both in |
quite some detail in this book. The Theorem can De likened to a pearl, and the method of |
proof to an oyster. The pearl is prized for its luster and simplicity; the oyster is a complex |
living beast whose innards give rise to this mysteriously simple gem. |
Godel's Theorem appears as Proposition VI in his 1931 paper "On Formally |
Undecidable Propositions in Principia Mathematica and Related Systems I." It states: |
To every w-consistent recursive class K of formulae there correspond recursive |
class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Fig (K) (where v |
is the free variable of r). |
Actually, it was in German, and perhaps you feel that it might as well be in German |
anyway. So here is a paraphrase in more normal English: |
All consistent axiomatic formulations of number theory |
include undecidable propositions. |
This is the pearl. |
In this pearl it is hard to see a Strange Loop. That is because the Strange Loop is buried |
in the oyster-the proof. The proof of Godel's Incompleteness Theorem hinges upon the |
writing of a self-referential mathematical statement, in the same way as the Epimenides |
paradox is a self-referential statement of language. But whereas it is very simple to talk |
about language in language, it is not at all easy to see how a statement about numbers can |
talk about itself. In fact, it took genius merely to connect the idea of self-referential |
statements with number theory. Once Godel had the intuition that such a statement could |
be created, he was over the major hurdle. The actual creation of the statement was the |
working out of this one beautiful spark of intuition. |
Introduction: A Musico-Logical Offering |
17 |
We shall examine the Godel construction quite carefully in Chapters to come, but so that |
you are not left completely in the dark, I will sketch here, in a few strokes, the core of the |
idea, hoping that what you see will trigger ideas in your mind. First of all, the difficulty |
should be made absolutely clear. Mathematical statements-let us concentrate on number- |
theoretical ones-are about properties of whole numbers. Whole numbers are not |
statements, nor are their properties. A statement of number theory is not about a. |
statement of number theory; it just is a statement of number theory. This is the problem; |
but Godel realized that there was more here than meets the eye. |
Godel had the insight that a statement of number theory could be about a statement of |
number theory (possibly even itself), if only numbers could somehow stand for |
statements. The idea of a code, in other words, is at the heart of his construction. In the |
Godel Code, usually called "Godel-numbering", numbers are made to stand for symbols |
and sequences of symbols. That way, each statement of number theory, being a sequence |
of specialized symbols, acquires a Godel number, something like a telephone number or a |