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?