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June 23rd, 2014, 10:56 AM   #1
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H&B's 2nd axiom for successor function contradicts 1st?

Hi all,
I'm reading Charles Petzold's "The Annotated Turing" for fun, and I'm no mathematician, so this might be a silly question. I have a question about page 226, where Hilbert and Bernay's axioms for the successor function are defined. Specifically, the second axiom:
(Ǝx)(y)-S(y,x)
I don't agree with this axiom: what's an example of a number with no successor?
Doesn't this directly contradict the first axiom:
(x)(Ǝy)S(x,y)

?
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June 23rd, 2014, 01:12 PM   #2
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(Ǝx)(y)-S(y,x)
I don't agree with this axiom: what's an example of a number with no successor?
There is an x such that for all y, x is not the successor of y. This is an axiom for the natural numbers: there is some element (usually 0 in modern formulations) which is not a successor.

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Doesn't this directly contradict the first axiom:
(x)(Ǝy)S(x,y)
For all x, there is some y such that y is the successor of x. Not the same as the above because of the order of quantifiers and the order of the arguments. 0 may have no predecessor but it and all other elements have successors.
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June 24th, 2014, 03:46 AM   #3
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Thanks. I didn't know these only applied to the natural numbers, so yes, they both make sense in that regard.
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June 24th, 2014, 05:43 AM   #4
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Thanks. I didn't know these only applied to the natural numbers, so yes, they both make sense in that regard.
You could say they define the natural numbers.
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