These have been depressing times, despite good vaccine news, no? Grey winter days do not lift the lockdown spirits. So an unproductive period for me. I don’t think I’m alone in this either.
Regrouping, I realize I’ve been trying to juggle too many balls at the same time recently. So — with apologies to Catarina Dutilh Novaes — I’m going to hang fire on blogging chapter-by-chapter about her interesting The Dialogical Roots of Induction (this is such a wide-ranging book, and it would take me too much time to do the homework to do it detailed justice). I might put together some brisker comments later. I’m also going to back off from the idea of doing some podcasts. I need to focus, and since both are downloaded a lot, I’m going to concentrate over the next few months on completing (i) the new version study guide and (ii) the notes on category theory. Which probably won’t make for many interesting blog posts here!
OK; so I have now uploaded the latest version of the partial Logic: A Study Guide, with a new twelve page chapter on elementary set theory. There is an overview of the topics, and I’ve slightly revised my preference-ordering of recommended texts. It’s been fun (and embarrassingly instructive) to revisit some of those basic set theory book; so I hope that some students will find the results useful!
Do you mind adding MacFarlane’s recent book to the guide? It seems a great resource for the more philosophically-minded.
First thanks for the pointer to MacFarlane’s recent book, which I didn’t know about. I’ll take a look at it.
The Guide is mostly a guide to books on formal logic; but maybe there is room for a brief section which would mention books like Mark Sainsbury’s Logical Forms and perhaps MacFarlane (if I like it!) which say more about relevant conceptual issues.
There is a chapter on relevance logic and discussions of intuitionistic logic which may be relevant to topics included in the guide. It may be possible to point toward some specific chapters of the book in relevant places instead of adding another section for philosophical discussions.
Peter, I’m wondering if you have any comments on Gregory Chaitin’s work within the logic community, on what he calls algorithmic complexity theory. He claims that Godel’s incompleteness theorem, the undecidability of the halting problem and related results are easy to prove using ideas from algorithmic complexity theory.
To my knowledge, no logic book has ever discussed these ideas. Although there are a number of books on algorithmic complexity theory which do discuss these ideas, the approach is clearly out of the mainstream of mathematical logic. Any idea why?
Although I’ve never liked Chaitin’s bravado style, the ideas do seem to have merit.
I think I have spotted a typo on page 62, the paragraph starting with (e): The way you define the set, I don’t think it makes sense to write “x is not an element of f(x)”. Perhaps you meant to write “x =/= f(x)” or “x is not an element of f(X)”, whatever the set X is. Sorry, this is coming from someone not at all at home with mathematics, so please ignore this comment, if it is just plain wrong.
f(x) is a set, a subset of the powerset, so it makes sense to ask whether x is a member.
However, I can see that the argument wasn’t phrased as clearly as it could be, so I have updated it. Hopefully, all is clear now!
Sorry, I took f(x) to just mean a particular value, namely the value of x under the function f (I am German, so this is the way I would phrase it in German). I generally read small x as a particular “thing” and capital X as the set that usally then includes x. Thanks for helping those less fluent in math.