
Melvin Fitting
Incompleteness in the land of sets
If you start with the empty set φ you can generate its power set {φ} an the power set of that,{φ,{φ}}. Repeatedly doing this generates the hierarchially finite sets, which have a natural representation as binary integers. But the natural numbers can also be generated by taking φ to be zero and defining each new number as the set of numbers generated so far. The book uses these two ways of linking numbers and sets to provide a useful way of looking at representation of numbers, or of logical statements, in terms of sets. It goes on to give a proof of Tarski's theorem and then to look at computability and axiomatics. This leads to a demonstration of Gödel's first incompleteness theorem.Up to this point the book wouldn't be too hard to follow for someone for someone knowing a bit of mathematical logic. The final three chapters delve deeper into the incompleteness theorems, looking at the relationship between soundness, provability and truth and at whether the proofs are constructive. There is also a look at how Rosser removed the need to assume ωcompleteness and at Gödel's second incompleteness theorem. These chapters need a bit more work, but they do make some deep results accessible to the reader in a concise and pretty much selfcontained book.