maldv/crabcanon · Datasets at Hugging Face

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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

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