What did Godel Prove?

It's hard being a logician sometimes. You tell people that you work in logic and philosophy of mathematics, and they immediately assume that you either have nothing comprehensible or nothing interesting to say to them. Sometimes both. There's definitely a view of the world that there are Mathy People and The Rest of Us, and all this math stuff can be shied away from like mad. But that rant doesn't actually have anything to do with the book that I'm talking about.

Kurt Godel, in the 1930's, proved two of the most important results for logic and the philosophy of mathematics, (the nice thing about a relatively young discipline is that you can pick out these milestones more easily), which were the Incompleteness Theorems for arithmetic. Arithmetic, if it is consistent, cannot prove its own consistency. And furthermore, there isn't any systematic way of laying out axioms for arithmetic so that you could prove everything. These are just rough statements, admittedly, but it's hard to make them precise if you don't know much about logic. And that's probably why these theorems get so abused. (They really do.)

So when I get students in my intro logic class wondering what's up with this Godel guy I mentioned (I show them pictures of famous logicians and mathematicians sometimes), I went back to this book called Godel's Proof, by Ernest Nagel and James Newman, which I'd once read when I was trying to understand the proof for the first time. It strikes me as a really solid introduction to the proof for readers without much technical background. You have to be willing to read some logical notation, naturally, but they give a lot of intuition behind things, and suppress enough of the picky details (and the proof itself has many) to let the big ideas come through. And for anyone who really wants more details, they put a lot of extras in the footnotes.

And there you go. I teach logic for a living, so I want to review the occasional logic book. Math and logic, unlike most other academic disciplines, seem to be things that people will happily profess to hating. (How often do you hear someone say, "I hate math. I failed Math 11." without any sense that they might actually want to be good at math. Ok, now how often do you hear someone say, "I hate reading. I failed English 11." without remorse at their lack of ability to read? See my point?)

Math is interesting and makes science work, and logic is what makes you make sense when you say things. So read about this stuff too.