Still on The Annotated Turing: and now (I am up to page 260) I have to admit I am finding the maths part quite hard going, though I guess there is no particular shame in that – after all this was the very bleeding edge of the discipline 70 years ago.
Well, yesterday I pointed out my confusion at Charles Petzold’s statement about these axioms which he said (p 226) defined a successor function:
In this case the second axiom really does indicate that there is exists a number x that has no sucessor (as Petzold originally stated the second axiom to mean). But how could that be true for any natural number?
Indeed on page 223 Petzold states the Peano Axioms “in plain English” – with axiom number 2 as “every number has a successor that is also a number”.
Unfortunately I cannot find a complete copy of the original work online to check what Hilbert and Bernays actually wrote on page 209 of volume one of Grundlagen der Mathematik – perhaps it is in the Birkbeck library and I can check it there.
(I admit I am excited by all this, because I seem to have found an error in a book by someone much cleverer than me: it’s not a case of intellectual one-one-upmanship, I am just pleasantly surprised that I (a) followed the text closely enough and (b) could spot it.)