Gödel's Theorems and Zermelo's Axioms: A Firm Foundation of Mathematics

by Lorenz Halbeisen, Regula Krapf

This book provides a concise and self-contained introduction to the foundations of mathematics. The first part covers the fundamental notions of mathematical logic, including logical axioms, formal proofs and the basics of model theory. Building on this, in the second and third part of the book the authors present detailed proofs of Gödel’s classical completeness and incompleteness theorems. In particular, the book includes a full proof of Gödel’s second incompleteness theorem which states that it is impossible to prove the consistency of arithmetic within its axioms. The final part is dedicated to an introduction into modern axiomatic set theory based on the Zermelo’s axioms, containing a presentation of Gödel’s constructible universe of sets. A recurring theme in the whole book consists of standard and non-standard models of several theories, such as Peano arithmetic, Presburger arithmetic and the real numbers.The book addresses undergraduate mathematics students and is suitable for a one or two semester introductory course into logic and set theory. Each chapter concludes with a list of exercises.

Genres: Logic

248 pages, Kindle Edition

Published October 16, 2020

Reviews

[Zhijing · Rating 5 out of 5]

This is a book on which the ETH Course Grundstruktur (basic structures in math) is based.

I view it from the point of languages, as all the contents are about a special type of language, the language of math. There are many analogous parts to natural languages just like English, German, etc, but also some profound things that are unique to mathematics as a language.

How different domains of knowledge connect:
- First-order logic can formalize all of the Set Theory, and hence "virtually all of Mathematics"

Key problem to keep in mind:
- The problem about the limitations of the first-order logic is "finiteness"

What each term entail:
- First-order logic does to have quantification, or statements of infinite lengths, or statements about formulas