Any books on mathematical logic that can be serve as supplementary reading for Roger Godement's Algebra Chapter 0?

Question

I am from a non-mathematics background but the course that I am taking in probability class is built on measure theory. I found many theorems stated in the book hard to comprehend and I think that's partly due to my insufficient maths background. I am self-learning real analysis currently and think it may be equally important to systematically learn abstract algebra if I really want to know the logic behind the probability class I am currently taking.

I picked Algebra by Roger Godement but find even the first chapter very hard to comprehend. For example, I find myself completely lost proceeding to the following section of the text.

Let $\mathrm{R}$ be a relation, $\mathrm{A}$ a mathematical object, and $x$ a letter (i.e., a "totally indeterminate" mathematical object). In the assembly of letters and fundamental signs which constitutes the relation $\mathrm{R},$ replace the letter $x$ wherever it occurs by the assembly A. One of the criteria for forming relations is that the assembly so obtained is again a relation, which is denoted $(*)$ by the notation $$ (\mathrm{A} \mid x) \mathrm{R} $$ and is called the relation obtained by substituting A for $x$ in $\mathrm{R},$ or by giving $x$ the value $\mathrm{A}$ in $\mathrm{R}$. The mathematical object $\mathrm{A}$ is said to satisfy the relation $\mathrm{R}$ if the relation $(\mathrm{A} \mid x) \mathrm{R}$ is true. It goes without saying that if the letter $x$ does not appear at all in the assembly $\mathrm{R},$ then the relation $(\mathrm{A} \mid x) \mathrm{R}$ is just $\mathrm{R},$ and in this case to say that $\mathrm{A}$ satisfies R means that $R$ is true.

However, I do appreciate the textbook that is self-contained and appreciate the author devoted to mathematical reasoning so rigorously at the beginning of the chapter. I tried to find some textbook about mathematical logic but they are either too abstract or not thorough enough that seems to start from the most fundamental (i.e. from axiom and the most basic rule).

I have read relevant posts on the subject I am asking but can't decide the material right for me. I am wondering if there are any materials or textbooks that introduce mathematical logic rigorously and serve as a supplementary text for me to understand the first chapter of the book? If there really isn't any textbook that is not too abstract but rigorous enough, I am wondering if there are any other textbooks on abstract algebra that start from mathematical logic and build the whole system from the scratch?

Answer 1

I liked “A Transition to Advanced Mathematics” by Douglas Smith. It covers a lot of transitional knowledge like predicate calculus, sets, relations, functions, cardinality, and abstract algebra. I found it to be very easy to understand in transition from calculus to upper level maths.

Answer 2

Godement's Algebra is an excellent book, except for the parts on logic at the beginning.

Godement presents an extremely abbreviated exposition of the logical foundations given for mathematics by Bourbaki's Set Theory, which are different from those usually used by logicians. Godement's exposition is so telegraphic that much of the time it takes considerable effort to understand what is being said. However, to be fair, I find that this is also often the case where authors purport to provide an introduction to logic in the first few pages of a book on something else.

A number of years ago, I actually read the parts on logic in Bourbaki's Set Theory in detail and found that with sufficient so-called "mathematical maturity," it is possible to understand them as a system for formalizing mathematics, whatever the merits of this system relative to others. If you want to understand the details of this system, read Bourbaki, not Godement.

My feeling is that it is not necessary to study logic before studying abstract algebra or analysis. Instead, what is important is to have acquired sufficient mathematical maturity by studying other topics in mathematics, particularly rigorous calculus.

If you do want to read the kind of introduction to logic that is sometimes presented as a prelude to more advanced mathematics, I would recommend the book An Introduction to Mathematical Reasoning by Eccles.

If you want to study logic as a subject in itself (not as a prelude to algebra or analysis), then the level will necessarily be higher than in Eccles because logic will be treated as a mathematical subject in its own right, and the same mathematical maturity will be necessary as for studying algebra. One good book for this is Mathematical Logic by Cori and Lascar.