Open david-a-wheeler opened 3 years ago
I think the current wording in Section 1.1.6 of the book (which is not exactly what you quote above: a chunk might have been lost in a copy-paste) can stay as is. It is indeed informal, but that whole part of the book is an informal exposition anyway, so I think this does not justify modifying posthumously Norm's words. Maybe one can add a footnote:
(free from contradiction).\footnote{
G\¨odel's second incompleteness theorem actually states that every recursively
axiomatizable formal system containing some basic arithmetic (for instance,
Robinson or Peano arithmetic) cannot prove its own consistency unless it is
inconsistent.}
(many of these terms require a precise definition, but this is not the purpose of this book).
The book says:
What Goedel proved is more subtle. As noted by Norman Megill, "A proof of consistency of PA is possible within a system stronger than PA such as ZFC, but Godel showed it's not possible within PA itself unless PA is inconsistent. If you are interested, see http://timothychow.net/consistent.pdf ".
We were trying to make things simple, but in this case I think we could be more precise without making it impossible to understand.