Two thousand years ago, Euclid set a standard for rigor in geometrical proofs. The rest of mathematics has never succeeded in reaching this standard.

The required properties of a satisfactory formal system are that it be

**complete**-- it must be possible either to prove or to disprove any proposition that can be expressed in the system.**consistent**-- it must*not*be possible to both prove and disprove a proposition in the system.

A number of mathematicians have attempted to put mathematics on a firmer, more logical
footing. A major effort was mounted at the end of the 19th century by Alfred North
Whitehead and Bertrand Russell; their *Principia Mathematica* attempted to use
axiomatic set theory to form a foundation for all of mathematics. This attempt foundered
when it was discovered that set theory is not consistent.

Here is the now-famous problem that demolished the *Principia Mathematica*.
Consider the set of all sets that do not have themselves as a member. Is this set a member
of itself?

Kurt Gödel explored the very notions of completeness and consistency. He invented a
numbering scheme *(Gödel numbers)* that allowed him to express proofs as numbers
(much as we might consider a computer program to be a very large binary number). He was
able to prove the following result:

If it is possible to prove, within a formal system, that the system is consistent, then the formal system is

not, in fact, consistent.

Or, equivalently,

If a formal system is consistent, then it is impossible to prove (within the system) that it is consistent.

This result sets very definite limits on the kinds of things that we *can* know.
In particular, it shows that any attempt to prove mathematics consistent is foredoomed to
failure.

Copyright © 1996 by David Matuszek

Last modified Apr 17, 1996