This Chapter's title is an adaptation of the title of Gödel’s 1931 article, in which his Incompleteness-Theorem was first published. The two major parts of Gödel’s proof are gone through carefully. It is shown how the assumption of consistency of TNT forces one to conclude that TNT (or any similar system) is incomplete. Relations to Euclidean and non-Euclidean geometry are discussed.Implications for the philosophy of mathematics are gone into with some care.
- Which of the two examples on p.440 is a TNT-proof-pair?
- Translate the following statements of meta-TNT into TNT:
- `0=0` is not a theorem of TNT
- `~0=0` is a theorem of TNT
- `~0=0` is not a theorem of TNT
- JÕSCHŨ is a theorem of TNT
- META-JÕSCHŨ is a theorem of TNT
- META-META-JÕSCHŨ is a theorem of TNT
- META-META-META-JÕSCHŨ is a theorem of TNT
- Which of the examples on p. 444 checks?