(he was once attacked by a gang of thugs who thought he was Jewish), and
when one of his inf luential teachers was murdered by a Nazi student in
1936, Gödel suffered a nervous breakdown. When World War II began,
Gödel left Germany and traveled to America by way of Russia and Japan,
ending up at Princeton.
Health problems, both physical and mental, continued to plague Gödel.
His circle of friends and acquaintances at Princeton was very select—there
were periods during which the only person to whom he spoke was Einstein.
Toward the end of his life, paranoia gained the upper hand, and his health
problems led him to believe that people were trying to poison him. He died
in 1978 from attempting to avoid being poisoned by refusing to eat.
Proofs of Gödel’s Incompleteness Theorem
There are many different ways to go about demonstrating Gödel’s theo-
rem. I have elected to go with demonstrating here that it is plausible and
have given a reference to formal proof in the notes to this chapter that
gives the f lavor of Gödel’s original proof.^4
Gödel took the liar’s paradox, and modified the sentence This statement
is false (which, as we have seen, lies outside the realm of statements that
can be judged to be true or false) to This statement is unprovable. That was
Gödel’s starting point, but by a technique known as Gödel numbering,
which is described in the proof, he linked unprovability of statements to
unprovability of statements about integers in the Peano axioms frame-
work. If the statement This statement is unprovable is unprovable, then it is
true, and the link Gödel established with arithmetic showed that there
exist unprovable statements in number theory. If the statement This state-
ment is unprovable is provable then it is false, and Gödel’s proof linked this
conclusion to the inconsistency of the Peano Axioms.^5
What exactly is meant by the word unprovable? It means just what it says,
that there is no proof that will determine the truth or falsity of the state-
ment. Needless to say, the existence of unprovable statements raises some
questions. There are two schools of thought on the subject. Recall that
the uncertainty principle is interpreted by most physicists to mean that
conjugate variables do not have specifically defined values, not that hu-
mans are just not good enough to measure the specifically defined varia-
bles. One group of mathematicians interpret unprovability in the same
fashion—it isn’t that we aren’t bright enough to prove that a statement is
true or false, it’s that if logic is used as the ultimate arbiter, it is inade-
quate to the task. Others view an unprovable statement as one that is in-
herently true or false, but the system of logic used just doesn’t reach far
enough to discern it.
124 How Math Explains the World