This chapter’s title is modeled after Gödel’s Principia Mathematica. Hofstadter expands on two ideas that form the basis of Gödel’s incompleteness theorem and illustrates them using Typographical Number Theory (TNT). First, Hofstadter shows that TNT can produce self-referential statements. Second, Hofstadter proposes that self-reference and self-scrutiny can be achieved using a single string, helping to focus the computation. Gödel’s numbering system is used to explore the limitations of TNT and how it connects with Gödel’s incompleteness theorem. By encoding paradoxical statements numerically, the system becomes self-referential, alluding to its own theorems. While formal systems are not inherently self-referential, numerical systems can be used to produce limited self-referential statements. These self-referential statements illuminate the limitations of the system itself.

Birthday Cantatatata..

The Tortoise meets Achilles in the woods on Achilles’s birthday. Achilles hums a birthday cantata written by Bach for King Augustus. He tells the Tortoise the song has double meaning for him, because he shares a birthday with Augustus. The Tortoise is skeptical that it is Achilles’s birthday after applying mathematical computation. At the end of the dialogue, the Tortoise tells Achilles that it is also his uncle’s birthday, which is the same as it being his own birthday.