cause of a course he took from a charismatic instructor who was confined
to a wheelchair. Gödel was highly conscious of his own health prob-
lems—a consciousness that was later to prove his undoing, so it is possi-
ble that the instructor’s condition had a significant impact on Gödel’s
decision.
Mathematicians in Europe generally have to overcome two hurdles on
the road to a tenured professorship: the doctoral dissertation (as do Amer-
ican mathematicians), and the habilitation (thankfully not required of
American mathematicians), which is an additional noteworthy perform-
ance after the doctorate has been awarded. Gödel had become interested
in mathematical logic, and his doctoral dissertation consisted of a proof
that a system of predicate logic proposed in part by Hilbert was com-
plete—every true result in the system was provable. This result was a
considerable leap beyond Post’s demonstration that propositional logic
was consistent—and Gödel’s proof used mathematical induction to estab-
lish the result. For his habilitation, Gödel decided to go after really big
game—the consistency of arithmetic, number two on Hilbert’s list of
twenty-three problems.
In August 1930, having completed his work, Gödel submitted a contrib-
uted paper for a mathematics conference that featured an address by
Hilbert entitled “Logic and the Understanding of Nature.” Hilbert was
still on the trail of axiomatizing physics and proving arithmetic was con-
sistent, and he ended his speech with supreme confidence: “We must
know. We shall know.” It is somewhat ironic that Gödel’s contributed pa-
per at the same conference contained results, delivered in a twenty-minute
talk, that were to dash forever Hilbert’s dream of “We shall know.” In an
address delivered far from the limelight (or what passes for limelight at a
mathematics conference), Gödel announced his result that one of two
conditions must exist: either arithmetic included propositions that could
not be proved (now known as undecidable propositions), or that Peano’s
axioms were inconsistent. To this day, no one has shown that Peano’s axi-
oms are inconsistent, and despite the lingering uncertainty you can get
almost infinite odds from any mathematician that they are not. This re-
sult is known as Gödel’s incompleteness theorem.
Unlike Einstein’s theory of relativity, which took the world of physics by
storm and was accepted almost immediately, the mathematics commu-
nity initially did not appreciate the significance of Gödel’s work. Nonethe-
less, during the ensuing five years or so, his results gained widespread
recognition and acceptance. He continued to do impressive work in math-
ematical logic, despite encountering problems in terms of his health. Al-
though Gödel was not a Jew, he could easily have been mistaken for one
Even Logic Has Limits 123