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:

But my confusion has been added to by his restatement of the axioms on page 250. Except they are not a restatement:

Notice that the second axiom reverses the order of *x *and *y* inside *S*.

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.)

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It should be there is a number that is not the successor of any number, i.e. there is a least natural number (taken to be 0 or 1, depending on your taste. With 0 then you have an additive identity floating about). The first version of the axioms looks right to me.

You are right of course, but what that means is that Charles Petzold misdescribed it on page 226 of the book.