Looking at my lecture notes Gödel Without (Too Many) Tears, they now seem rather patchy. They could certainly be clearer in a number of places, and could be better arranged too. Since they are downloaded two thousand times a year or so — and people have said that they find them useful — it seems worth making the effort to revisit the notes and improve them where I can.
So here is the first installment of a revised version, the brief Preface, and Chapter 1 on Incompleteness, the Very Idea. There are now eighteen short chapters (some very short), and I’ll be posting them at the rate of a chapter a day, starting now (with weekends off, and a composite update on the next three Mondays). You can either read along slowly, or join the party later.
I have tried to keep the notes, as before, at under a third the length of IGT2. So you get some headline news without too much (distracting?) extra detail. Comments and corrections are most welcome, indeed encouraged — except from Gödel cranks! In particular, don’t hesitate to let me know about typos. I will try to respond to comments and queries day by day (depending how few/many there are).
You can comment by replying to the blog post for the relevant chapter. Three points about using the comments system:
- If the system does not recognize your email address as a real one and that of someone who has had comments approved before, your comment will be held in a moderation queue (for obvious enough reasons!).
- If your comment is approved, your email address won’t be published, however. (Comments which use offensive names won’t be published even if the comment is a sensible one. Yes, people are that silly!)
- If your comment is more than a sentence or two long, it is probably worth editing it off-line and checking it before copying, pasting, and posting (notionally you have a few minutes to edit, but this doesn’t work for everyone).
Alternatively, you can mail peter underscore smith at logicmatters dot net.
[Added: I’ve already corrected Chap 1 to remove some typos — thanks to David Furcy!]
9 thoughts on “Gödel Without Tears, slowly, 1”
A very very minor correction in page 1: According to “Logical Dilemmas. The life and work of Kurt Gödel” by John W. Dawson, Jr., page 149, G. left Austria for the USA in January 1940.
Yes! Thanks! What was I thinking?
You insist on the fact that a “formal system” (starting with a “language”) is given with an interpretation. Ok with that, but then Definition 5 seems awkward to me, and Gödel’s completeness theorem with it : if a given interpretation is fixed, then what does “any model” mean ? Of course I know what you mean, and you are careful to write “any model (re-)interpreting…” but I think it is confusing for the anaware reader — your intended audience, after all. It is probably not going to happen, but I think it would be clearer to really distinguish syntax and semantics ; this way it would be easier to see what belongs to which… That being said, always a pleasure to read. Thanks.
p. 1: “The title of Gödel’s great 1931 paper is ‘On formally undecidable propositions
of Principia Mathematica and related systems I’”. He wrote in german, though.
Good point! “is” –> “translates as”.
Should the term be ‘closed timelike curves’ rather than ‘timeline’? (p. 1)
Thank you for posting the revised chapters like this. Here in Melbourne we are in lockdown again, so having this these to read is keeping me sane.
And making your other books available to people in the way you have is a really really lovely and generous thing to do.
Thanks for that! And yes, oops, it should be “closed timelike curves”.
p iv: Should also explain “wff” (as it does “iff”)
p 1: “Always a perfectionist, after the mid 1940s he more or less stopped publishing.” ends the story a bit abruptly.
p 1, last par of 1.1: programme or program?
p 3, par 1: “to make the idea of effective decidability in good shape.” Not sure that’s grammatical. “be in good shape” might be grammatical, though awkward.
p 5: “then there will be a formal deduction” — perhaps not clear how that fits with the last par of §1.2
p 6: “However, trivially, T is not a complete theory.” — better to say “negation complete” there too?
I still think, btw, as in my 1st comment on “Gödel news!”, that “semantically complete” (used in IGT) is useful terminology and usefully contrasts with “negation complete”. You can then have (as IGT does) semantically complete logic and negation-complete theory, reinforcing the contrast. I think that works better than just having “complete logic” and “complete theory”, or “complete logic” and “negation-complete theory”.
Thanks, as usual, for this! (And on the final point, on further reflection you I think you are probably right!)