Mathematical logic is an area where it’s very easy to get bogged with the details and lose the big picture. For someone intensely interested in logic, this is hardly acceptable.
For this reason, we have spent one month (April and May) working on a summary of Set Theory, Logic and their Limitations, a textbook on First-Order Logic by Moshé Machover. This paper summarizes the book’s last 4 chapters:
- Chapter 7 – Propositional Logic (up to soundness and completeness of PropCal)
- Chapter 8 – First-Order Logic (e.g., completeness of FopCal, Compactness Theorem, Löwenheim-Skolem Theorem)
- Chapter 9 – Recursion Theory (e.g., recursiveness, enumerability, MRDP Theorem)
- Chapter 10 – Limitative Results (e.g., Skolem’s Theorem, Tarski’s Theorem, Church’s Theorem, Gödel’s Incompleteness Theorems)
Substantial efforts were invested in synthesizing materials into an elegant and coherent structure, and in brushing up the terminology. To make certain sections easier and avoid unnecessary introduction of abstract concepts, we have sometimes resorted to rewriting the section. For example, the Generalized Tarski’s Undecidability Theorem has been rephrased differently using a new definition of truth definition and numeralwise representation of the diagonal function — just to name a few.
For the record, a summary is never a substitute for an actual textbook, but for those who have went through Set Theory, Logic and their Limitations carefully, going through this summary will speed up the process of reviewing, while at the same time providing additional insights.
Summary of Machover’s Set Theory, Logic and their Limitations – Chapters 7-10: A 43-page summary of Machover’s Set Theory, Logic and their Limitations, with linkable table of contents (preview the table of contents here!)