Principia Mathematica and Axioms

Question

I have been trying to understand how mathematicians achieve the truth through axiomatic systems. Surely, if one searches a bit more, it will get to know how axiomatic systems work, generally. After that, I got to know, for example, why there are geometries, instead of one (the Euclidian) taught at school; Quantum Mechanics vs. Newton's ... and so on.... However, I really would like to know how can axiomatic systems be created. Then, I searched a bit more and got Principia Mathematica (PM), Whitehead & Russell, to read, but comments on the book disappointeded me.

So the questions are:

1) Which book should I read so as to understand how axiomatic systems work? Also, one that enables me creating my own, for example.

2) Is there a more complete book than Principia Mathematica? Recently, I read Gödel's theorem and understood the basic concept: there is no axiomatic system that is FULLY COMPLETE. But it can have a more complete book than PM.

Answer 1

You come up with your own axioms, and see where they lead. I have never created my own set of axioms, but historically, mathematicians have taken existing axiomatic systems and tried changing one (or more) of the axioms, and seeing if they can build a self-consistent system. The most famous example of this is Riemann replacing the parallel postulate from geometry, leading to Riemannian geometry. Other examples lead to broader conceptions of numbers.

You might appreciate: Defending the Axioms: On the Philosophical Foundations of Set Theory by Penelope Maddy.

Answer 2

Well, axioms are conceived of by thinking about what would be useful given a set of concepts, and which seem obviously true, but which also seem to not be falsifiable.

However, we need to be careful when defining axioms, as some have historically been shown to produce contradictions, as was the case with Russell's Paradox. One proposed axiom for set theory was that for any property of a mathematical object, there exists a set of all objects with that property. Bertrand Russell showed you can derive a contradiction from that proposed axiom by considering the set of all sets S with the property that S is not a member of itself (and I'll leave it up to you to reason out why or read into it).

I unfortunately don't have much in the way of books, but I can definitely recommend Axiomatic Set Theory by Patrick Suppes, or possibly even Naive Set Theory by Paul R. Halmos. The former is more in-depth, but the latter definitely shows you that you can derive quite a bit from just a few axioms and some definitions.