The Foundations of Mathematics

by Kenneth Kunen

Mathematical logic grew out of philosophical questions regarding the foundations of mathematics, but logic has now outgrown its philosophical roots, and has become an integral part of mathematics in general. This book is designed for students who plan to specialize in logic, as well as for those who are interested in the applications of logic to other areas of mathematics. Used as a text, it could form the basis of a beginning graduate-level course. There are three main Set Theory, Model Theory, and Recursion Theory. The Set Theory chapter describes the set-theoretic foundations of all of mathematics, based on the ZFC axioms. It also covers technical results about the Axiom of Choice, well-orderings, and the theory of uncountable cardinals. The Model Theory chapter discusses predicate logic and formal proofs, and covers the Completeness, Compactness, and Löwenheim-Skolem Theorems, elementary submodels, model completeness, and applications to algebra. This chapter also continues the foundational issues begun in the set theory chapter. Mathematics can now be viewed as formal proofs from ZFC. Also, model theory leads to models of set theory. This includes a discussion of absoluteness, and an analysis of models such as H(κ) and R(γ). The Recursion Theory chapter develops some basic facts about computable functions, and uses them to prove a number of results of foundational importance; in particular, Church's theorem on the undecidability of logical consequence, the incompleteness theorems of Gödel, and Tarski's theorem on the non-definability of truth.

Genres: Mathematics, Logic

262 pages, Paperback

First published January 1, 2009

Reviews

[Hobart Mariner · Rating 5 out of 5]

Really above and beyond the ruck of even the better intro logic/set theory texts, I think because Kunen is trying to do something somewhat different: develop enough model theory, proof theory, set theory, and recursion theory to prove the Incompleteness theorems (as well as Kleene's recursion theorems and Tarski's theorem) within a very limited subset of ZFC, TST. This gets him out of the need to do any Gödel coding, at the expense of developing recursion theory on HF, the class of hereditary finite sets. At first I was a bit annoyed at this but after a while I became HF-pilled and found the development more natural and "correct" than what you might see in a rival text (Enderton, e.g.). He also develops/defines computability as Delta1 on HF - somewhat different from the usual presentations as "obtainable by algorithm/TM/lambda calc/register machine." Again this first annoyed me but after reflection I realized it's a more universal approach, and not just Kunen showing off his set theoretic finesse.

The book serves as a good introduction to axiomatic set theory (including a few peripheral topics like cofinality), model theory and proof theory (including completeness, compactness, LST, etc.), and recursion. It also includes brief and funny digressions on philosophy of math, where Kunen encourages finitist readers to throw the book in the trash.

My only criticism is that the embeddedness of exercises in the text means there are comparatively few problems, and a couple of them should have been spelled out in greater detail within the text itself, because Kunen heavily depends on them in later exposition. Specifically, the undefinability of even numbers in (N, <) and the Ackermann embedding of HF in N are both sketched as exercises rather than worked out in detail, and he cites both of them later on.

[Hyung Mook Kang · Rating 5 out of 5]

This book is very unique to the point that Kunen uses CST for a basic primitive recursive notion rather than a function of natural numbers. He uses the concept "delta 1" instead of the usual recursive definition we know. This book builds a foundation of mathematics from set theory instead of logic, so anyone who's set-theory-buff is good to read this book.