Here, in a very slightly revised form — and I’m not going to have time to revise them properly for a while — are the handouts for three “Back to Basics” lectures on incompleteness that I gave at the Cameleon workshop a few weeks ago. There’s nothing excitingly original here, with the possible exception of mistakes! But maybe if you are looking for one story about the shape of the wood which doesn’t get too distracted by all the trees, then the handouts could be useful.
Many thanks for the comments: they look very useful: as soon as I have a chance to check things I’ll make any changes they prompt!
There are some things in section 1.5 that I’m finding a bit confusing.
Perhaps I’m just not clear on what it means to be a true sentence, as opposed to being a true L_A sentence. (I take it that a true L_A sentence is one that is true in the standard model.)
When I look at theorems 1.5.4 and 1.5.5, it’s not immediately clear which differences in wording are significant and which are not:
Theorem 1.5.4 Any II_1 generic Godel sentence for an w-consistent
normal theory T is true.
Theorem 1.5.5 There are normal (and w-consistent) theories with undecidable generic Godel sentences which are false L_A sentences.
Are they both talking about theories that are both normal and w-consistent? If so, why is the w-consistent part parenthesised in 1.5.5?
How significant is the II_1 mentioned in the first theorem but not in the second?
What about the “undecidable”, mentioned only in the 2nd theorem?
Could 1.5.4 and 1.5.5 be rewritten to “parallel” each other more closely?