User andrew ostergaard - MathOverflow

Answer by Andrew Ostergaard for undergraduate logic textbook

2011-12-26T02:01:12Z

The best introduction to logic that I have seen is Kenneth Kunen's recent book, "The Foundations of Mathematics" (ISBN: 978-1-904987-14-7), published in 2009. The book provides a brief introduction to axiomatic set theory, model theory, and computability theory; and it culminates with a proof of Godel's incompleteness theorems and Tarski's theorem on the non-definability of truth. There are also a couple brief discussions of the philosophy of mathematics; these are given from the perspective of the working mathematician, and they are used to motivate the material. And they are very helpful. In fact, the most salient thing about this book is that it is exceptionally clear, well-written, and easy to learn from. (Kunen also wrote "Set Theory: An Introduction To Independence Proofs" which is also exceptionally clear, well-written, and easy to learn from). The book's only prerequisite is the mathematical maturity that an Introduction to Analysis course would provide, and it is available (new) on amazon.com for less than $25.

Answer by Andrew Ostergaard for Any suggestions for a course in Mathematical Logic?

2011-12-26T01:55:28Z

Kenneth Kunen recently wrote a wonderful introduction to Mathematical Logic, called "The Foundations of Mathematics" (ISBN: 978-1-904987-14-7), published in 2009. The book's only prerequisite is the mathematical maturity that an Introduction to Analysis course would provide, so it sounds like your students would be prepared. The book provides a brief introduction to axiomatic set theory, model theory, and computability theory; and it culminates with a proof of Godel's incompleteness theorems and Tarski's theorem on the non-definability of truth. There are also a couple brief discussions of the philosophy of mathematics; these are given from the perspective of the working mathematician, and they are used to motivate the material. And they are very helpful. In fact, the most salient thing about this book is that it is exceptionally clear, well-written, and easy to learn from. (Kunen also wrote "Set Theory: An Introduction To Independence Proofs" which is also exceptionally clear, well-written, and easy to learn from). Your students will be grateful for the fact that this book is available (new) on amazon.com for less than $25.

Is the chain homotopy category, K(Ab), an Abelian category? By Ab, I mean the category of Abelian groups.

2010-04-29T00:33:21Z

Let A be an Abelian category.

From this category, we can form the chain complex category Ch(A). The objects of Ch(A) are chain complexes of objects of A. The morphisms of Ch(A) are chain maps. Ch(A) is an Abelian category for every Abelian category A.

Now from Ch(A), we can form the chain homotopy category K(A). The objects of K(A) are just objects of Ch(A), but the morphisms of K(A) are chain homotopy classes of chain maps. Thus, K(A) is a quotient of Ch(A).

It turns out that K(A) is an additive category for any Abelian category A. From asking people, I seem to get the impression that K(A) is not always abelian, but I've had a hard time showing this. If all objects of A are projective (e.g. if A is the category of vector spaces over some field k), then K(A) will be Abelian.

I've been trying to show that K(Ab) is not Abelian (where Ab is the category of Abelian groups). More specifically, I've been trying to show this by showing the following (which may or may not be true):

Let X be a chain complex with the group of integers in dimension 0, and zero in every other dimension. Let f be the chain map from X to X, that sends each integer x to 2x. I've been trying to show that in K(Ab), the homotopy class of this chain map does not have a cokernel.

Any answers, suggestions, hints, or comments would be greatly appreciated!