Page 117 of Atiyah-MacDonald

I think there is a problem with page 117 of "Introduction to Commutative Algebra" by Atiyah and MacDonald. If P(M,t)=0, the definition of d(M) and the proof of Corollary 11.2 make no sense.

Edit of Dec. 1, 2010. This is easy to fix.

Today I reread Section I.4 of Cohen's "Set Theory and the Continuum Hypothesis" ...

Today I reread Section I.4 of Cohen's "Set Theory and the Continuum Hypothesis". I was devastated. I wonder how people would react to this text if they didn't know who wrote it.

Leap of Faith

Standard mathematics doesn't require any leap of faith, whereas works such as Gödel's and Cohen's require to believe in the consistency of some formalization of mathematics.

Peak

I think that mathematics reached its peak when Cantor stated the Continuum Hypothesis problem, and started regressing when Hilbert launched his foundation program.

Consistency

There is a question I've never seen discussed. It's the question of evaluating how likely a given formalization of mathematics is of being consistent.

Results such as Gödel's and Cohen's are always stated under appropriate consistency assumptions.

So the question of knowing whether such consistency assumptions are realistic is inescapable.

This question can be posed in a slightly more precise form:

Let T be a mathematical theory. (I'm using Bourbaki's terminology.) Let L(T) be the length of the shortest proofs of a relation of the form "R and (not R)" if such proofs exist. Otherwise, say that L(T) is infinite.

For all practical purposes, the fact that L(T) is very large or is infinite makes no difference.

But I think that, for any reasonable mathematical theory, L(T) is much more likely to be very large than to be infinite.

It would be interesting to have a lower bound for L(T).

The mathematical question whose answer I most ardently wish I knew

Is the Continuum Hypothesis true, false or undecidable in Bourbaki's set theory?

About "The Continuum Hypothesis is an Open Problem"

The text "The Continuum Hypothesis is an Open Problem" is much too long for its contents. It suffices to tell the reader:

"Compare the following books:

(1) P. Cohen, Set theory and the Continuum Hypothesis.

(2) S.C. Kleene, Introduction to Metamathematics.

(3) N. Bourbaki, Theory of Sets. (French original: Théorie des ensembles.)"