Is Arithmetic Consistent?

Shortly after Andrew Wiles anounced his proof of Fermat's Last Theorem the popular writer Marylin vos Savant published a little book expressing scepticism as to whether we could be certain of the proof's correctness. This led to a discussion on the Internet about doubting the consistency of arithmetic itself.

For the record, the most widely accepted formal basis for arithmetic is called Peano's Axioms (PA). Giuseppe Peano based his system on a specific natural number, 1, and a successor function such that to each natural number x there corresponds a successor x' (also denoted as x+1). He then formulated the properties of the set of natural numbers in five axioms:

  1. 1 is a natural number.
  2. If x is a natural number then x' is a natural number.
  3. If x is a natural number than x' is not 1.
  4. If x' = y' then x = y.
  5. If S is a set of natural numbers including 1, and if for every x in S the successor x' is also in S, then every natural number is in S.
The following web pages summarize the lively discussion that ensued. In addition to the acronym "PA", the reader will also encounter ZFC and ZF, signifying Zermelo-Fraenkel set theory, with and without the axiom of choice.

Part 1: Chick With Ludicrous IQ
Part 2: Walt Whitman Weighs In
Part 3: A Very Very Very Very Very Very Pathetic and Ignorant Book
Part 4: God's Pentium
Part 5: May I Make a Suggestion?
Part 6: Round Up The Usual Proofs
Part 7: Manifestly True
Part 8: Sublime Confidence
Part 9: Concluding Remarks
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