turning coffee into theorems,” as Erdos drank prodigious quantities.^11
Goodstein’s theorem^12 is reminiscent of this problem; it defines a se-
quence (called a Goodstein sequence) recursively (the next term is de-
fined by doing something to the previous term in the sequence, just as in
the above unsolved problem), although the sequence it defines is not as
simple to state as the one in the Collatz conjecture. It can be shown that
every Goodstein sequence terminates at 0—although one cannot show
this using the Peano axioms alone; one must use an additional axiom, the
axiom of infinity, from Zermelo-Fraenkel set theory. As such, it is an in-
teresting proposition that would be undecidable using only the Peano ax-
ioms—as opposed to the uninteresting undecidable propositions used by
Gödel in his original proof. It is also worth pointing out that the provabil-
ity of Goodstein’s theorem in a stronger version of set theory lends cre-
dence to the point of view that these theorems are inherently true or false,
and that an adequate system of logic can determine it.
So, unless another graduate student with the talent of Gödel shows up
to prove that the overwhelming consensus of mathematicians is wrong,
and that the Peano axioms are actually inconsistent, mathematicians will
continue to rely on mathematical induction. It is one of the most useful
tools available—and the existence of undecidable propositions is a small
price to pay for such a valuable tool. If some graduate student comes
along to accomplish such an unlikely task, he or she can be assured of
two things: a Fields Medal, and the undying hatred of the mathematical
community, whom he or she will have deprived of one of the most valua-
ble weapons in its arsenal.
NOTES
1. S e e h t t p : / / w w w - g r o u p s. d c s. s t - a n d. a c. u k / ~ h i s t o r y / B i o g r a p h i e s / H i l b e r t. h t m l. T h i s
may have been the last time in human history when there were polymaths who
could make truly significant contributions in several fields. In addition to Hilbert,
Henri Poincaré (of Poincaré conjecture fame) also did important work in both
mathematics and physics.- See http://en .wikipedia.org/wiki/ Hilbert’s_problems. This contains a list of all
twenty-three problems along with their current state. Most of the ones not dis-
cussed in this book are fairly technical, but number three is easily understood—
given two polyhedra of equal volumes, can you cut the first into a finite number
of pieces and reassemble it into the second? That this could not be done was
shown by Max Dehn. - A. K. Dewdney, Beyond Reason (Hoboken, N.J.: John Wiley & Sons, 2004). The
proof of the consistency of propositional logic is given on pp. 150 –152. - Ibid. The proof of the impossibility theorem is given on pp.153–158. See
130 How Math Explains the World