From The Annotated Turing: now reached page 226 and it is still good.

I think Charles Petzold has made a mistake – has he? Please read on and let me know.

Petzold says these three axioms come from Hilbert and Bernays and they mean:

- Every member has a successor
- There exists a number that does not have a successor
- That is
*r *is a successor to *x* and *y* and *x* is the successor to *s*, then *y* is also the successor to *s*.

But surely the second axiom actually means there exists an *x* which is not a successor of *y*?

Comments please.

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The second statement should read “there exists a number which does not have a predecessor”, this is what we call zero.