Gödel's First Theorem, from Gödel 1931 to Kleene 1943

As 'homework', before writing more of the second edition of my Gödel book, I'm reading through the literature to see how others have handled the First Incompleteness Theorem, both in the early papers from Gödel on, and then in the later textbook tradition. How do is the Theorem stated? How is it proved?

I've started writing notes on the expository history, and here's the first (19 page) instalment. The notes don't at all aim to be comprehensive, though I'd like to know about significant omissions as I go along. They have been written, as much as anything, as a rather detailed aide-memoire for myself (and a source of bits and pieces I might use). I have done some joining up of the dots to make them tolerably readable, but I certainly haven't put in the time to spell out everything out in the way a beginning student might want. Still, you shouldn't need much background to follow the twists and turns.

The notes come in three parts. Part 1 looks at early papers by the Founding Fathers. Part 2, to follow, will look at three pivotal works, Mostowki's Sentences Undecidable in Formalized Arithmetic and Kleene's Introduction to Metamathematics (both from 1952), and then Tarski, Mostowski and Robinson's Undecidable Theories (1953). Part III will continue the story on through some sixty years of textbooks.

Make what use of these notes that you will. Though the usual warning applies in spades, as I'm no historian: caveat lector! Remember how very easy it to LaTeX your work. Just because what you write then looks very pretty doesn't mean that it is any more authoritative …