Another revised chapter for the Study Guide. And again, there is little substantial change from the previous version, except to significantly cut down the length of the “overviews”, which were getting a bit out of hand! Anyway, here is the latest version of the chapter on computable functions, formal arithmetic, and Gödelian incompleteness.

These revised chapters I am posting are being downloaded a significant number of times, but comments (either here or by email) are few and far between. I‘m rather hoping that that’s because people aren’t finding my overviews on topic areas gruesomely misleading or my recommendations for reading too outlandish! But, as I’ve said before, if you do think I’m leading your students horribly astray (if you are a logic teacher) or think the Guide could be more helpful (if you are a student), now really is the time to say!

David MakinsonNot suitable for the Guide, but perhaps interesting for its author and some blog readers, is the article by James Walsh on incompleteness via ordinal analysis that went up on arXiv a few days ago. Full ref: Tue, 21 Sep 2021,[10] arXiv:2109.09678 [pdf, ps, other], An incompleteness theorem via ordinal analysis, James Walsh.

Even without the technical details, the background idea seems natural: If the consistency of PA can be proven by induction up to epsilon zero but not up to a lower ordinal, then presumably it should correspond in some way with properties of the ordinals up to that limit.

Rowsety MoidThanks!

Peter FIn similar arXiv news, Anton Freud (currently a Postdoc, PhD under Michael Rathjen) recently put up lecture notes for a first course in ordinal analysis that look very well done to me on a first pass: https://arxiv.org/abs/2109.06258. These two would certainly go well together for anyone interested in diving into proof theory.