Jaynes: Probability Theory & Gödel's incompleteness theorem

I've recently been dabbling in statistics and probability, and it was only a matter of time before my attention became drawn to the book Probability Theory: The Logic of Science by E. T. Jaynes. In it, Jaynes proposes to start from the smallest possible set of common-sense axioms and proceed to derive more or less the entire theory of probability, demonstrating how desirable properties such as consistency and paradox-free reasoning can thus be achieved for the whole system.

I decided to buy this book, and at 50 pages in (of about 700 pages total) I can already say it will be worth the full price. Here's a quote to show what I mean (chapter 2.6.2, pp. 45-46):

To understand [Gödel's incompleteness theorem], the essential point is the principle of elementary logic that a contradiction A and not-A implies all propositions, true and false. -- Then let A = {A1, A2,..., An} be the system of axioms underlying a mathematical theory and T any proposition, or theorem, deducible from them:

A => T.

Now, whatever T may assert, the fact that T can be deduced from the axioms cannot prove that there is no contradiction in them, since, if there were a contradiction, T could certainly be deduced from them!

This is the essence of the Gödel theorem, as it pertains to our problems. As noted by Fisher (1956), it shows us the intuitive reason why Gödel’s result is true. --

Recommended.