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Gödel Numbers (1969)
J.W. Swanson

Contributed by Vijay Fafat

The story revolves around an ancient stone artifact found near Cairo which has engraved markings of slanted lines. In an incredible non-sequitor, one of the characters in the story guesses that the numbers must represent a gigantic "Godel number", factorizable in the form (2^k2)*(3^k3)*(5^k5)*…(pn^kn) with non-zero exponents kj. The string of k’s is then supposed to be holding a secret message. The author mentions that the number system used is base-12 but then ends up reverting to base 10 while counting the number of lines on the stones. The inscribed "Godel number" is conjectured to be 17^19 + 39^(27^45) + 12^[41^(37^43)] (not sure if this IS a Godel number. Quite doubtful). The decoded message shows the geographical location of an alien/ancient civilizations library of knowledge, also encoded in Godel numberings on thousands of stone artifacts. Indeed, part of the knowledge base includes information on curing lymphosarcoma….Godel’s incompleteness theorems play no part in the story but the author has taken a brief stab at explaining it. There is some Russian espionage thrown in as well but that story thread is left dangling and incomplete.

The story assumes that in 1968, we had the capability of factoring such large numbers easily, which we don’t even now. The story really should have been set in the future and some made-up theorem used to get around this difficulty of factoring. Further, since the message is designed to survive "millions of years", it would have been a nice touch to have a much older artifact, show that the date of its creation was obtained based on encoded information about relative star positions and then have the physical location of the library obtained by correcting for continental drift. Having the artifact dated at 3000 BC is anachronistic if the originator were an earth civilization. In any case, the author has done a nice job of explaining the deciphering of the map.

Overall, a very good idea for math sci fi. Would have worked well in a Dan-Brown-type novel.

For those who may not know, Kurt Gödel's famous theorem (which essentially states that any axiomatic system including arithmetic is either inconsistent or contains propositions which can neither be proved nor disproved from the axioms) involves a technique for turning statements in a formal system (which themselves could be either true or false) into numbers. This clever trick allowed him to do meta-mathematics with mathematics. A brilliant idea, and one well worth exploiting in fiction.

First appeared in the March 1969 issue of Galaxy.

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Works Similar to Gödel Numbers
According to my `secret formula', the following works of mathematical fiction are similar to this one:
  1. PopCo by Scarlett Thomas
  2. Artifact by Gregory Benford
  3. Sphere by Michael Crichton
  4. Count to a Trillion by John C. Wright
  5. The Logic Pool by Stephen Baxter
  6. Cryptonomicon by Neal Stephenson
  7. The Lure by Bill Napier
  8. Improbable by Adam Fawer
  9. Gödel's Sunflowers by Stephen Baxter
  10. Ground Zero Man (The Peace Machine) by Bob Shaw
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GenreScience Fiction, Adventure/Espionage,
TopicComputers/Cryptography, Logic/Set Theory,
MediumShort Stories,

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